CZECH TECHNICAL UNIVERSITY IN PRAGUE

Faculty of Nuclear Sciences and Physical Engineering

DIPLOMA THESIS

SEMICLASSICAL ANALYSIS OF TIME DEPENDENT QUANTUM SYSTEMS

Ondřej Lev

Supervisor: Prof. Ing. Pavel Šťovíček, DrSc.

Consultant: Prof. Pierre Duclos, Université de Toulon et du Var, France

Trutnov

5.5. 2002

I declare that I wrote this diploma thesis independently using the listed references. Ondřej Lev

1. Preface

During the last three years when I have been dealing with mathematical physics, I encountered six functional analytic methods:

1) KAM algorithm [D1]; applications of

2) WKB in Feynman integral and Geometric asymptotics [Sch], [GS]

3) Dyson expansion and

4) Egorov theorem [BR], [Lev] on quantum ball;

5) Egorov theorem for time dependent case [Lev],

6) Matrix elements investigated with WKB. I decided to include in this thesis three of them. Matrix elements investigated with WKB because it is my last work, Egorov theorem for time dependent case because new improvements appeared and finally KAM, although not semiclassical method it was inspiration for starting semiclassical analysis of matrix elements. As a consequence of this decision this thesis consists of three more or less independent parts. The last part contains the results of paper [D2] and application of this improved KAM technique on two models. The quantity V_{knm} defined by (79), in fact matrix element of perturbation V in eigen-basis of unperturbed Floquet Hamiltonian, is crucial for Theorem 4.1 to be applicable. Section 3 gives a new statement of Egorov theorem for time dependent systems, based on ideas of [BR], [Lev] and Joachim Asch, who showed direct approach to time dependent generalization. Also assumptions on the form and growth of time dependent Hamiltonian are weaken. The heart of thesis, section 2, contains application of WKB (although nonstandard one, see 2.1. Introduction) in investigation of asymptotic behavior of

W_{n, l} :=< W Ψ_{n}, Ψ_{l} >,

for large n, l, where Ψ_{n} is the normalised Dirichlet L^{2}(ℝ_{+}) solution of

(−Δ + x^{α})Ψ_{n}(x) = E_{n} Ψ_{n}(x) , α > 1.

Asymptotic behavior is important for finiteness of quantities like sup_{n} ∑_{l} W_{n, l} or sup_{n} ∑_{l} \frac{W_{n, l}}{E_{n}−E_{l}}, which are important in KAM, Dyson or adiabatic analysis of time dependent systems. The result of this method is precise as far as computations of leading terms of normalization constant and the diagonal of matrix W_{n, l} are concerned, with the numerical values being derived, but less precise for offdiagonal elements, which are only estimated from above (see Theorem 2.9). Unfortunately, there is no sufficient condition yet implying that W is a Shur-Holmgren operator (sup_{n} ∑_{l} W_{n, l} < ∞). This is the beginning of the work which started 3 month ago within consultations with prof. Pierre Duclos and Dr. Michel Vittot during my stage in CPT, Marseille and continued in Prague within meetings with prof. Pavel Šťovíček, my supervisor. Possible continuation could be an improvement of estimate (67), computing further terms in (13), (16), a generalization to asymptotically homogeneous potentials, taking Neumann condition instead of Dirichlet and using it to treat the true symmetric well on the whole line.

Acknowledgments I would like to thank to Dr. Joachim Asch (CPT, Marseille) for consultations and ideas in Egorov theorem problematic, prof. Pierre Duclos and Dr. Michel Vittot for consultations and ideas in WKB problematic and their care during my stage in CPT, Marseille. Finally, I would like to thank very much to prof. Pavel Šťovíček for his perfect leadership during all three years.

2. Behavior of matrix elements investigated with WKB

2.1. Introduction. WKB analysis is well-known method in quantum mechanics, which allows us to construct wave functions in semiclassical regime. This method exists also in mathematics, under name Liouville-Green method. For details for further discussion one can consult [Ol] Chapters 6 and 11. Let us consider equation

(−Δ + x^{α} − E)Ψ(x) = 0

in E → ∞ regime. Using scaling trick (17) this equation becomes equivalent to

(−ℏ^{2}Δ + x^{α} − 1)Φ(x) = 0 ,

where ℏ → 0. Standard WKB yields the solution

Φ(x) = \frac{1}{\sqrt[4]{1 − x^{α}}} sin (\frac{1}{ℏ} ∫_{0}^{x} \sqrt{1 − y^{α}} dy) for x < 1

\frac{1}{\sqrt[4]{x^{α} − 1}} exp (−\frac{1}{ℏ} ∫_{1}^{x} \sqrt{y^{α} − 1} dy) for x > 1 ,

which is diverging in turning point 1. The problem is that standard WKB is in fact unitary transformation between the solutions of

(−ℏ^{2}Δ + V)Φ = 0

and

(−ℏ^{2}Δ + C)ζ = 0 .

Constant C may be chosen +1, where V > 0, or −1, where V < 0, i.e. in classically forbidden or permitted region. The solution is there exponentially decaying or oscillating respectively. The divergence of standard WKB solution is caused by the fact that the unitary mapping between solutions of equations

(−ℏ^{2}Δ + V)Φ = 0

and

(−ℏ^{2}Δ + v)ζ = 0 ,

where v is some function, conserves the sign of V. Because our problem V (x) = x^{α} − 1 changes the sign, it looks more naturally trying to map the equation

(−ℏ^{2}Δ + x^{α} − 1)Ψ(x) = 0

to

(−ℏ^{2}Δ + x − 1)ζ(x) = 0 .

If we proceed this mapping, Airy functions Ai and Bi appear. Because we are interested in L^{2}(ℝ_{+}) solutions, we exclude Bi. Now our new solution has no singularity! Other advantage is that error is estimated uniformly on ℝ_{+}. We see that this approach is convenient for our problematic. Now if we ask for Dirichlet boundary condition at origin, we will obtain Bohr-Sommerfeld quantization condition and Energy E become labeled by a natural number n.

So at the moment we have two Dirichlet L^{2}(ℝ_{+}) solutions and we would like to compute matrix elements

\frac{< W Ψ_{n}, Ψ_{l} >}{\sqrt{< Ψ_{n}, Ψ_{n} >< Ψ_{l}, Ψ_{l} >}},

for some perturbation W. This expression looks very complicated, but numerically calculated examples α = 1, 2 showed us the way, the most important thing is the behavior in the neighborhood of the origin. The section is divided into several parts: computation of normalization constant, diagonal and offdiagonal matrix elements. The approach is given in details, what is interesting, is that in fact only mathematical tool, which is used is integration by parts. The results are formulated in the beginning of each subsection and at the end of this section is given summary of all results, stated as Theorem 2.9.

Convention: We will use (for simplicity) the letter K for all positive constants which depend only on α, although they may vary. This section is dealing only with α > 1 case.

2.2. Results due to F.W.J. Olver. Due to Theorem 3.1 , Chapter 11 of [Ol] the only L^{2}(ℝ^{+}) solution of equation (E > 0)

(−Δ + x^{α})Ψ(x) = EΨ(x)  (1)

can be written in the form

Ψ(x) = (\frac{s(x)}{x^{α} − E} )^{\frac{1}{4}} (Ai(s(x)) + ε(x)) ,  (2)

where

s(x) :=  (\frac{3}{2} ∫_{E^{\frac{1}{α}}}^{x} \sqrt{y^{α} − E} dy)^{\frac{2}{3}} , for x ≥ E^{\frac{1}{α}}

− (\frac{3}{2} ∫_{x}^{E^{\frac{1}{α}}} \sqrt{E − y^{α}} dy)^{\frac{2}{3}} , for x < E^{\frac{1}{α}} .

ε is an error, which is controlled by the inequality

\frac{|ε(x)|}{M(s(x))} , \frac{|ε′(x)|}{N(s(x))}\sqrt{\frac{s(x)}{x^{α} − E}} ≤ \frac{1}{G(s(x))} (exp(\frac{8}{π^{2}} E^{− \frac{α+2}{2α}} V(x, ∞)) − 1) ,  (3)

where V is given by the formula

V(x, y) := ∫_{x}^{y} \frac{1}{\sqrt{|s̃(z)|}} |\frac{4V V′′ − 5V^{′2}}{16V ^{\frac{5}{2}}} + \frac{5\sqrt{|V |}}{16|s̃|^{3}} | (z) dz = ∫_{x}^{y} |\frac{2s̃′(z)s̃′′′(z) − 3s̃′′^{2}(z)}{4\sqrt{|s̃(z)|}s̃′^{3}(z)}| dz.

We used short notation

V (x) := x^{α} − 1 , s̃(x) :=  (\frac{3}{2} ∫_{1}^{x} \sqrt{y^{α} − 1} dy)^{\frac{2}{3}} , for x ≥ 1  (4)
− (\frac{3}{2} ∫_{x}^{1} \sqrt{1 − y^{α}} dy)^{\frac{2}{3}} , for x < 1.

To investigate possible boundedness of V it is useful to notice that s̃s̃^{′2} = V. If we take into account the behavior of V and s̃

s̃(x) ∼^{∞} x^{\frac{α+2}{3}} , s̃(x) ∼^{1} x − 1,

we conclude (using the fact that s̃ ∈ C^{∞}) that V is a bounded function on ℝ^{+}. Let c be negative root of the equation

Ai(c) = Bi(c)

of the smallest absolute value (c = −0.36605). The functions G, M, N are defined by prescriptions

M := \sqrt{2 Ai Bi}  for x ≥ c , M := \sqrt{Ai^{2} + Bi^{2}} for x ≤ c
G := \sqrt{\frac{Bi}{Ai}}  for x ≥ c , G := 1 for x ≤ c

N := \sqrt{\frac{Ai′^{2} Bi^{2} + Ai^{2} Bi′^{2}}{Ai Bi}}  for x ≥ c , N := \sqrt{Ai′^{2} + Bi′^{2}} for x ≤ c

By graphical analysis with Mathematica P. Duclos and M. Vittot obtained the following bounds of Airy functions and their derivatives, which are valid on ℝ.

| Ai | ≤ 2\frac{1}{\sqrt{π}A ω} , | Bi | ≤ 2\frac{A}{\sqrt{π}ω}
| Ai′ | ≤ \frac{ω}{\sqrt{π}A} , | Bi′ | ≤ \frac{A ω}{\sqrt{π}}  (5)

Auxiliary functions A and ω are defined by

A(s) := e^{^{\frac{2}{3}} s_{+} ^{\frac{3}{2}}} , ω(s) := |s|^{\frac{1}{4}} + \frac{1}{2}

s_{+} := max(s, 0). Putting the definitions of G, M, N, the estimates above (5) and the estimates of error (3) together we obtain

e^{\frac{2}{3} s_{+}(x)^{\frac{2}{3}}} (|s(x)|^{\frac{1}{4}} + \frac{1}{2}) |ε(x)| ≤ \frac{2}{\sqrt{π}} (exp(\frac{8}{π^{2}} E^{− \frac{α+2}{2α}} V(x, ∞)) − 1)

e^{\frac{2}{3} s_{+}(x)^{\frac{2}{3}}} (|s(x)|^{\frac{1}{4}} + \frac{1}{2})^{−1} |ε′(x)| ≤ \frac{2}{\sqrt{π}} \sqrt{\frac{x^{α} − E}{s(x)}} (exp(\frac{8}{π^{2}} E^{− \frac{α+2}{2α}} V(x, ∞)) − 1) .

Finally using the boundedness of V on ℝ^{+}, we conclude the following estimates of error

|ε(x)| ≤ KE^{− \frac{α+2}{2α}} |s(x)|^{− \frac{1}{4}} e^{− \frac{2}{3} s_{+}(x)^{\frac{3}{2}}}  (6)
|ε′(x)| ≤ KE^{− \frac{α+2}{2α}} \sqrt{\frac{x^{α} − E}{s(x)}} (\frac{1}{2} + |s(x)|^{\frac{1}{4}} ) e^{− \frac{2}{3} s_{+}(x)^{\frac{3}{2}}}  (7)

2.3. Bohr-Sommerfeld. If we want (2) to be Dirichlet solution on ℝ^{+}, i.e.

Ψ(0) = 0  (8)

we have to satisfy the condition

Ai(s(0)) + ε(0) = 0

Using the well-known asymptotics of Ai

Ai(−z) = z^{− \frac{1}{4}} \frac{1}{\sqrt{π}} cos(\frac{2}{3}z^{\frac{2}{3}} − \frac{π}{4} ) (1 + O(z^{− \frac{3}{2}} ))  (9)

Ai(z) = \frac{z^{− \frac{1}{4}}}{2\sqrt{π}} e^{− \frac{2}{3} z^{\frac{3}{2}}} (1 + O(z^{− \frac{3}{2}} ))  (10)

and (6), we obtain

\frac{1}{\sqrt{π}|s(0)|^{\frac{1}{4}}} cos (\frac{2}{3}|s(0)|^{\frac{3}{2}} − \frac{π}{4} ) (1 + O(E^{− \frac{α+2}{2α}})) = −ε(0)

⟺ sin (\frac{2}{3}|s(0)|^{\frac{3}{2}} + \frac{π}{4} ) = −\sqrt{π}|s(0)|^{\frac{1}{4}} ε(0) (1 + O(E^{− \frac{α+2}{2α}}))

⟺ \frac{2}{3}|s(0)|^{\frac{3}{2}} + \frac{π}{4} = πn + O(E^{− \frac{α+2}{2α}})

which yields Bohr-Sommerfeld quantization condition

∫_{0}^{E^{\frac{1}{α}}} \sqrt{E − x^{α}} dx = π(n − \frac{1}{4}) + O(E^{− \frac{α+2}{2α}}),  (11)

or equivalently

∫_{0}^{E^{\frac{1}{α}}} \sqrt{E − x^{α}} dx = π(n − \frac{1}{4}) + O(\frac{1}{n}).

Remark 2.1. This formula, well-known as the Bohr-Sommerfeld quantization condition, may be seen as the defining relation between quantum mechanical Energy E and a natural number n. What is interesting is that this formula is here rigorously proven for the class of homogeneous potentials x^{α}, using Liouville-Green method. This result can be similarly proven to much more general class of potentials.

To satisfy (8), we define the Energy behavior by

E_{n} = (\frac{π}{∫_{0}^{1} \sqrt{1 − x^{α}}dx}(n − \frac{1}{4}))^{\frac{2α}{α+2}} + O ((\frac{1}{n})^{\frac{2α}{α+2}} ) .  (12)

2.4. Normalization constant and diagonal elements. In this section we will compute the leading terms of normalization constant of the solution (2) and of diagonal matrix elements of some regular perturbation with respect to functions (2). Firstly we will formulate the result of this subsection, which will be proved at the end of it.

2.4.1. \underline{Result}. Theorem 2.2. Let α > 1 and a function Ψ be defined by (2). This means that Ψ is the only L^{2}(ℝ+) solution of equation

(−Δ + x^{α})Ψ(x) = E Ψ(x).

Then for normalization constant holds

C_{E}^{−2} :=< Ψ, Ψ >= ∫_{0}^{∞} |Ψ|^{2} = \frac{E^{\frac{2−α}{2α}}}{2π} ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O(E^{\frac{16−10α}{18α}}).  (13)

Suppose moreover that a perturbation W satisfies the following decay properties (< x >= \sqrt{1 + x^{2}}).

W, W′ ∈ L^{1}(ℝ^{+}), < x >^{r} W ∈ L^{∞}(ℝ^{+}) , for some r > 1  (14)

Then it holds true that

\frac{< W Ψ, Ψ >}{< Ψ, Ψ >} = E^{− \frac{1}{α}} (∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} )^{−1} (∫_{ℝ_{+}} W + O (E^{− min\{ \frac{α+2}{18α} , \frac{r−1}{α} \}}))  (15)

If we want Ψ to satisfy Dirichlet boundary condition (8), then Energy has to be restricted by the Bohr-Sommerfeld quantization condition (12) and then the quantity (15) becomes a diagonal matrix element

\frac{< W Ψ_{n}, Ψ_{n} >}{< Ψ_{n}, Ψ_{n} >} = (\frac{π(n − \frac{1}{4})}{∫_{0}^{1} \sqrt{1 − x^{α}}dx})^{− \frac{2}{α+2}} (∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} )^{−1} (∫_{ℝ_{+}} W + O (n^{−t})) , (16)

where t := min\{\frac{1}{9}, \frac{2(r−1)}{α+2}\} > 0 Corollary 2.3. A straightforward consequence of (13) is that the quantum density of probability

p_{Q}(x) := \frac{1}{\sqrt{∫_{ℝ+} |Ψ|^{2}}} |Ψ(x)|^{2}

tends to the classical one

p_{C}(x) := \frac{1}{∫_{0}^{1}(1 − y^{α})^{− \frac{1}{2}} dy} \frac{χ_{[0,1]}(x)}{\sqrt{1 − x^{α}}}

in L^{1}(ℝ_{+}) limit, or equivalently

∫_{ℝ_{+}} |p_{Q} − p_{C}| ⟶^{E→∞} 0

Remark 2.4. It is interesting that both integrals in (16) are important classical quantities. (This is the reason, why we didn’t write their explicit numerical values, although they are known.) The first one

∫_{0}^{1} \sqrt{1 − x^{α}}dx

is the volume of area {(q, p)|E(q, p) ≤ 1} in phase space. The second one

∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}}

is proportional to the period of the classical motion with Energy E = 1.

Because there is only one index n in this subsection, which labels the Energy E_{n} according to the Bohr-Sommerfeld condition (12), we will use scaling to replace E_{n}, which is assumed to be large, by the small parameter ℏ. The only reason of this scaling is the manipulation with ℏ becomes more comfortable. It is of course possible to pass through everything without scaling, like in the part dedicated to offdiagonal elements where it is necessary, because of the presence of both indeces n, l.

2.4.2. \underline{Scaling}. Consider equation

(−Δ + x^{α} − E)Ψ(x) = 0.

For any λ this is equivalent to the equation

(−λ^{−2−α}Δ + x^{α} − λ^{−α}E)Ψ(λx) = 0

We define the scaling by the relations

λ^{−α}E := 1 , λ^{−2−α} := ℏ^{2} , Φ(x) := λ^{\frac{1}{2}} Ψ(λx)

which means that our problem (1) is transformed to

(−ℏ^{2}Δ + x^{α} − 1)Φ(x) = 0 , E = ℏ^{− \frac{2α}{2+α}} , Φ(x) = ℏ^{− \frac{β}{2}} Ψ(ℏ^{−β}x) , β := \frac{2}{α + 2} . (17)

We see that investigation of the behavior with respect to the large parameter E is equivalent to the one with respect to small parameter ℏ. By rescaling (2), (6) we conclude that exact L^{2}(ℝ+) solution of equation

(−ℏ^{2}Δ + V)Φ = 0  (18)

can be written in the following form (we remind the definition of V and s̃ (4) ):

Φ(x) = ℏ^{\frac{α−4}{3(α+2)}} (\frac{s̃(x)}{V (x)})^{\frac{1}{4}} (Ai(ℏ^{− \frac{2}{3}} s̃(x)) + ε(ℏ, x)),  (19)

where the error term ε is controlled by the inequality

|ε(ℏ, x)| ≤ Kℏ^{\frac{7}{6}} |s̃(x)|^{− \frac{1}{4}} exp(−\frac{2}{3}s̃_{+}(x)^{\frac{3}{2}} )  (20)

(Again we defined s̃_{+} := max(s̃, 0)). For simplicity, we define new function

Φ̃(x) := (\frac{s̃(x)}{V (x)})^{\frac{1}{4}} (Ai(ℏ^{− \frac{2}{3}} s̃(x)) + ε(ℏ, x)) ⟹ Φ = ℏ^{\frac{α−4}{3(α+2)}} Φ̃  (21)

The quantities we are interested in, are transformed by

< Ψ, Ψ >=< Φ, Φ >= ℏ^{\frac{2(α−4)}{3(α+2)}} < Φ̃, Φ̃ >= E^{\frac{4−α}{3α}} < Φ̃, Φ̃ >  (22)
\frac{< W Ψ, Ψ >}{< Ψ, Ψ >} = \frac{< W (\frac{.}{ℏ^{β}}) Φ, Φ >}{< Φ, Φ >} = \frac{< W (\frac{.}{ℏ^{β}}) Φ̃, Φ̃ >}{< Φ̃, Φ̃ >}  (23)

In this section we will not need the estimate of ε′.

2.4.3. \underline{Normalization constant}. Theorem 2.5. For the function Φ̃ given by (21) it is true that

∫_{0}^{∞} |Φ̃|^{2} = \frac{ℏ^{\frac{1}{3}}}{2π} ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O(ℏ^{\frac{4}{9}} ).

Proof. Let us define

ϕ(x) := (\frac{s̃(x)}{V (x)})^{\frac{1}{4}} Ai(ℏ^{− \frac{2}{3}} s̃(x))  (24)

We would like to compute leading term of integral

I(ℏ) := ∫_{0}^{∞} |ϕ(x)|^{2}dx.  (25)

This expression looks very complicated, for example for α = 2 is the function s̃ equal to

s̃(x) = − (\frac{3}{4}(arcsin x + x\sqrt{1 − x^{2}}))^{\frac{2}{3}} , for x ≤ 1.

What is important, is that the function s̃ maps the point 1 to the origin. But the point 1 is nothing but the turning point (of the classical problem) of our rescaled potential V = x^{α} −1. Now because we are interested in the ℏ → 0 regime, we conclude that there are three imported area of very different behavior of Ai(ℏ^{− \frac{2}{3}} s̃) (we suppose that 0 < γ < \frac{2}{3}).

1) for x ∈ [ 0, 1−ℏ^{γ}] is the argument of Airy function ℏ^{− \frac{2}{3}} s̃(x) mapped to the “neighborhood” of −∞, [−ℏ^{− \frac{2}{3}} K, −ℏ^{− \frac{2}{3} +γ}K]. We use asymptotic (9), Ai is fast oscillating.

2) x ∈ [1 − ℏ^{γ}, 1 + ℏ^{γ}]. This small window of “regular” regime of Ai, where the function is only bounded, is in fact mapped to the whole line. But the influence to integrals like I(ℏ) is controlled by ℏ^{γ} and can be suppressed.

3) x ∈ [1 − ℏ^{γ}, ∞) the argument of Airy function, ℏ^{− \frac{2}{3}} s̃(x) is mapped to the neighborhood of +∞, [ℏ^{− \frac{2}{3} +γ}K, +∞). We use asymptotic (10), Ai is exponentially decaying.

Because of the exponential decay of Ai in the area 3) and because of the fact that we can control the influence of the “window” (area 2) by choosing γ, we will see that the main contribution of integral (25) will come from the oscillating area 1). This strategy will be used throughout all computations, but the use of exponential decay will be necessary only in the part dealing with off-diagonal matrix elements. If not necessary we will omit it to simplify the text. The “regular” behavior of function ϕ in the neighborhood of the turning point is the reason why we chose this type of model solution instead of usual WKB. It is well-known that usual WKB (see [Ol] Chapter 6) is diverging in the turning point. Choosing our type of model solution we can omit this fact and following the explained strategy we obtain in fact the usual WKB as asymptotic of Ai (see (9), (10)).

To compute the leading term of normalization constant it is sufficient to split integral I(ℏ) into following three parts (as we said we will omit exponential decay in the area 3) ).

I(ℏ) = ∫_{1}^{∞} + ∫_{1−ℏ^{γ}}^{1} + ∫_{0}^{1−ℏ^{γ}} ,

where γ > 0 will be defined later. Let us introduce an auxiliary function

g_{α}(x) := \frac{s̃(x)}{V (x)}.  (26)

It is easy to see that g_{α} is bounded on ℝ^{+} for α > 1. So we may estimate (using s̃s̃^{′2} = V )

∫_{1}^{∞} |ϕ(x)|^{2}dx = ∫_{1}^{∞}(\frac{s̃(x)}{V (x)})^{\frac{1}{2}} Ai^{2}(ℏ^{− \frac{2}{3}} s̃(x))dx
= ∫_{1}^{∞}(\frac{s̃(x)}{V (x)}) Ai^{2} (ℏ^{− \frac{2}{3}} s̃(x)) s̃′(x)dx
≤ sup_{[1,+∞)} g_{α} ∫_{1}^{∞} Ai^{2} (ℏ^{− \frac{2}{3}} s̃(x)) s̃′(x)dx
= ℏ^{\frac{2}{3}} sup_{[1,+∞)} g_{α} ∫_{0}^{∞} Ai^{2}(z)dz.  (27)

Because of the boundedness of g_{α} and Airy function Ai we may easily conclude

∫_{1−ℏ^{γ}}^{1} |ϕ(x)|^{2}dx = O(ℏ^{γ}).  (28)

Remaining (and dominant) is this term

∫_{0}^{1−ℏ^{γ}} (\frac{s̃(x)}{V (x)})^{\frac{1}{2}} Ai^{2}(ℏ^{− \frac{2}{3}} s̃(x))dx.

We will follow the explained strategy to the area 1). An easy computation gives

s̃(1 − ℏ^{γ}) = −(\frac{3}{2} ∫_{1−ℏ^{γ}}^{1} \sqrt{1 − y^{α}}dy)^{\frac{2}{3}} = −(\frac{3}{2}\sqrt{α})^{\frac{2}{3}} ℏ^{γ} + O(ℏ^{\frac{4γ}{3}} ).

So, as we said, the argument lies in fact in a “neighborhood” of −∞. As was announced, we will use asymptotics (9) of Ai. Its application yields

∫_{0}^{1−ℏ^{γ}} (\frac{s̃(x)}{V (x)})^{\frac{1}{2}} Ai^{2}(ℏ^{− \frac{2}{3}} s̃(x))dx

= \frac{ℏ^{\frac{1}{3}}}{π} ∫_{0}^{1−ℏ^{γ}} \frac{cos^{2}(\frac{1}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{4})}{\sqrt{1 − x^{α}}}(1 + O(ℏ^{1− \frac{3}{2} γ}))dx

= \frac{ℏ^{\frac{1}{3}}}{π} ∫_{0}^{1−ℏ^{γ}} \frac{1}{\sqrt{1 − x^{α}}} cos^{2}(\frac{1}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{4} ) dx +

+ O(ℏ^{1− \frac{3}{2} γ}) ℏ^{\frac{1}{3}} ∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}}  (29)

The second term is of order ℏ^{\frac{1}{3}} O(ℏ^{1− \frac{3}{2} γ}). Because the argument of cos^{2} is large, it causes fast oscillating. We will use identity cos^{2}(a) = \frac{1}{2}(1 + cos(2a)) to compute “average” and the rest, which is negligible thanks to the fast oscillation.

∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}} cos^{2}(\frac{1}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{4} )

= \frac{1}{2} ∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}} + \frac{1}{2} ∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}} cos(\frac{2}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{2} )

From the first integral will come the leading term

∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}} = ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} − ∫_{1−ℏ^{γ}}^{1} \frac{dx}{\sqrt{1 − x^{α}}} = ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O(ℏ^{\frac{γ}{2}} )  (30)

To suppress the second one, we will integrate by parts

∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}} sin(\frac{2}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy) = ∫_{0}^{1−ℏ^{γ}} sin(\frac{2}{ℏ}f(x))\frac{f′(x)}{(f′(x))^{2}} dx

= −\frac{ℏ}{2} [\frac{cos(\frac{2f}{ℏ})}{(f^{′2})}]_{0}^{1−ℏ^{γ}} + \frac{ℏ}{2} ∫_{0}^{1−ℏ^{γ}} cos(\frac{2}{ℏ}f(x))\frac{f′′(x)}{(f′(x))^{3}} dx

We defined an auxiliary function

f(x) := ∫_{x}^{1} \sqrt{1 − y^{α}}dy.  (31)

The fact that

\frac{f′′(x)}{(f′(x))^{3}} = −\frac{αx^{α−1}}{2(1 − x^{α})^{2}} ,

allows us to see

∫_{0}^{1−ℏ^{γ}} \frac{dx}{\sqrt{1 − x^{α}}} cos(\frac{2}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{4} ) = O(ℏ^{1−γ}) + O(ℏ^{1−2γ})  (32)

Collecting all these informations (27) - (32) yields

I(ℏ) = \frac{ℏ^{\frac{1}{3}}}{2π} ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O(ℏ^{\frac{2}{3}} ) + O(ℏ^{γ}) + ℏ^{\frac{1}{3}} (O(ℏ^{\frac{γ}{2}} ) + O(ℏ^{1−2γ}))

Now choosing γ := \frac{4}{9}, we finally obtain the leading term of normalization constant

I(ℏ) = ∫_{0}^{∞}(\frac{s̃(x)}{V (x)})^{\frac{1}{2}} Ai^{2}(ℏ^{− \frac{2}{3}} s̃(x))dx = \frac{ℏ^{\frac{1}{3}}}{2π} ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O(ℏ^{\frac{4}{9}} )  (33)

To finish the proof we have to show that the contribution of error to normalization constant for the exact solution of (18) is negligible.

∫_{0}^{∞} |Φ̃|^{2} = ∫_{0}^{∞} |ϕ|^{2} + ∫_{0}^{∞}(\frac{s̃}{V})^{\frac{1}{2}} |ε|^{2} + 2 ∫_{0}^{∞}(\frac{s̃}{V})^{\frac{1}{2}} Re(\overline{Ai(ℏ^{− \frac{2}{3}} s̃)} ε)

Thanks to the boundedness of Airy function and g_{α} (see (26)), we have easily (with help of (20))

∫_{0}^{∞}(\frac{s̃}{V})^{\frac{1}{2}} |ε|^{2} = O(ℏ^{\frac{7}{3}} )  (34)
∫_{0}^{∞}(\frac{s̃}{V})^{\frac{1}{2}} Re(\overline{Ai(ℏ^{− \frac{2}{3}} s̃)} ε) = O(ℏ^{\frac{7}{6}} ).  (35)

So finally, using the result (33) we have this expression of normalization constant for arbitrary α > 1

∫_{0}^{∞} |Φ̃|^{2} = \frac{ℏ^{\frac{1}{3}}}{2π} ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O(ℏ^{\frac{4}{9}} ).  (36)

This is nothing but the statement of Theorem 2.5 .

2.4.4. \underline{Diagonal matrix elements}. In this section we will follow again the strategy explained in the beginning of the Proof of Theorem 2.5 to compute leading term of diagonal matrix elements. Because the order of the leading term will be ℏ^{\frac{1}{3} + \frac{2}{2+α}}, what is more then in the case of normalization constant, we will have to assume some decay properties of the perturbation W, to estimate better the rest. First of all let us formulate the result of this subsection. Theorem 2.6. Suppose that the perturbation W satisfies decay properties (14). Then for “rescaled diagonal matrix elements” holds true that

< W (\frac{.}{ℏ^{β}}) Φ̃, Φ̃ >= \frac{ℏ^{\frac{1}{3} +β}}{2π} (∫_{0}^{∞} W + O (ℏ^{min\{β(r−1), \frac{α}{2+α} \}}))  (37)

Under assumptions (14) the main contribution of (37) comes from the neighborhood of the origin. To make the proof more easy to follow we split the computation into several parts.

Tail estimate. In this part we will estimate the contribution of tail (a < 1)

∫_{a}^{∞} W (\frac{x}{ℏ^{β}})|Φ̃(x)|^{2}dx ≤ sup\{W (\frac{x}{ℏ^{β}}), x > a\} ∫_{a}^{∞} |Φ̃|^{2}
≤ C_{ℏ}^{−2} sup\{W (x), x > \frac{a}{ℏ^{β}} \}
≤ C_{ℏ}^{−2} sup\{< x >^{−r}, x > \frac{a}{ℏ^{β}} \}∥ < x >^{r} W ∥_{∞}
= C_{ℏ}^{−2} \frac{ℏ^{βr}}{(a^{2} + ℏ^{2β})^{\frac{r}{2}}} ∥ < x >^{r} W ∥_{∞}

We define normalization constant

C_{ℏ}^{−2} := ∫_{0}^{∞} |Φ̃|^{2}

So the result is

∫_{a}^{∞} W (\frac{x}{ℏ^{β}})|Φ̃(x)|^{2}dx ≤ \frac{C_{ℏ}^{−2}}{a^{r}} ℏ^{βr}∥ < x >^{r} W ∥_{∞}  (38)

Error contribution. We know that L^{2}(ℝ_{+}) solution of (18) can be expressed in the form (19), with error ε(ℏ, x) controlled by (20). Taking into account the definition (21) we will first of all use estimate (38) to get

< W (\frac{.}{ℏ^{β}}) Φ̃, Φ̃ > = ∫_{0}^{a} W (\frac{x}{ℏ^{β}})|Φ̃(x)|^{2}dx + ∫_{a}^{∞} W (\frac{x}{ℏ^{β}})|Φ̃(x)|^{2}dx
= ∫_{0}^{a} W (\frac{x}{ℏ^{β}})|Φ̃(x)|^{2}dx + O(ℏ^{βr})C_{ℏ}^{−2}

Now we will show that the influence of error to the quantity < W (\frac{.}{ℏ^{β}}) Φ̃, Φ̃) > is of order ℏ^{\frac{13}{6}} (ϕ is defined by (24)).

∫_{0}^{a} W (\frac{.}{ℏ^{β}})|Φ̃|^{2} = ∫_{0}^{a} W (\frac{.}{ℏ^{β}})|ϕ|^{2} + ∫_{0}^{a} W (\frac{.}{ℏ^{β}})(\frac{s̃}{V})^{\frac{1}{2}} |ε|^{2}

+2 ∫_{0}^{a} W (\frac{.}{ℏ^{β}})(\frac{s̃}{V})^{\frac{1}{2}} Re(\overline{Ai(ℏ^{− \frac{2}{3}} s̃)} ε)

Similarly to (34), using the boundedness of W

∫_{0}^{a} W (\frac{.}{ℏ^{β}})(\frac{s̃}{V})^{\frac{1}{2}} |ε|^{2} = O(ℏ^{\frac{7}{3}} ).

For the third integral the power \frac{7}{6} of ℏ in (35) is not sufficient. We will do the estimate more carefully

|∫_{0}^{a} W (\frac{.}{ℏ^{β}})(\frac{s̃}{V})^{\frac{1}{2}} Re(\overline{Ai(ℏ^{− \frac{2}{3}} s̃)} ε)| ≤ Kℏ^{\frac{7}{6}} ∫_{0}^{a} |s̃(x)|^{− \frac{1}{4}} | Ai(ℏ^{− \frac{2}{3}} s̃(x))|dx

= Kℏ^{\frac{7}{6}} ℏ^{\frac{1}{6}} ∫_{0}^{a} |s̃(x)|^{− \frac{1}{2}} |cos(\frac{1}{ℏ} ∫_{x}^{1} \sqrt{−V} − \frac{π}{4} )| dx (1 + O(ℏ)) = O(ℏ^{\frac{4}{3}} ).

So we obtain

∫_{0}^{a} W (\frac{.}{ℏ^{β}})|Φ̃(x)|^{2} = ∫_{0}^{a} W (\frac{.}{ℏ^{β}})|ϕ|^{2} + O(ℏ^{\frac{4}{3}} )  (39)

The leading term of < W (\frac{.}{ℏ^{β}}) Φ̃, Φ̃) >. Now we will concentrate on the most important term, from which the leading term will result. We will use the same tricks like in (29) - (32).

∫_{0}^{a} W (\frac{x}{ℏ^{β}})(\frac{s̃(x)}{V (x)})^{\frac{1}{2}} Ai^{2}(ℏ^{− \frac{2}{3}} s̃(x))dx

= \frac{ℏ^{\frac{1}{3}}}{π} ∫_{0}^{a} W (\frac{x}{ℏ^{β}})\frac{cos^{2}(\frac{1}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{4})}{\sqrt{1 − x^{α}}}(1 + O(ℏ))dx

= \frac{ℏ^{\frac{1}{3}}}{2π} ∫_{0}^{a} W (\frac{x}{ℏ^{β}})\frac{dx}{\sqrt{1 − x^{α}}}

+\frac{ℏ^{\frac{1}{3}}}{2π} ∫_{0}^{a} W (\frac{x}{ℏ^{β}})\frac{dx}{\sqrt{1 − x^{α}}} sin(\frac{2}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy) + O(ℏ^{\frac{4}{3}} ).  (40)

The first integral extends the window into whole line

∫_{0}^{a} W (\frac{x}{ℏ^{β}})\frac{dx}{\sqrt{1 − x^{α}}} = ℏ^{β} ∫_{0}^{∞} W − ∫_{a}^{∞} W (\frac{x}{ℏ^{β}})dx + ∫_{0}^{a} W (\frac{x}{ℏ^{β}}) (\frac{1}{\sqrt{1 − x^{α}}} − 1) dx

= ℏ^{β} ∫_{0}^{∞} W − K∥W < x >^{r} ∥_{∞} ℏ^{β r}

+K∥W < x >^{u} ∥_{∞} ∫_{0}^{a} \frac{x^{α}}{(1 + (ℏ^{−β}x)^{2})^{\frac{u}{2}}} \frac{dx}{(1 + \sqrt{1 − x^{α}})\sqrt{1 − x^{α}}}

= ℏ^{β} ∫_{0}^{∞} W (x)dx + O (ℏ^{β r)}) + O (ℏ^{β u}) = O (ℏ^{β u}) .  (41)

We chose

u := min\{r, 1 + \frac{α}{2}\}.  (42)

We will integrate the second term in (40) by parts

∫_{0}^{a} W (\frac{x}{ℏ^{β}})\frac{dx}{\sqrt{1 − x^{α}}} sin(\frac{2}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy) = ∫_{0}^{a} W (\frac{x}{ℏ^{β}}) sin(\frac{2}{ℏ}f(x))\frac{f′(x)}{(f′(x))^{2}} dx

= −\frac{ℏ}{2} [W (\frac{.}{ℏ^{β}})\frac{cos(\frac{2f}{ℏ})}{(f^{′2})}]_{0}^{a} + ℏ ∫_{0}^{a} cos(\frac{2}{ℏ}f(x))\frac{f′′(x)}{(f′(x))^{3}} W (\frac{x}{ℏ^{β}})dx

−ℏ^{−β} \frac{ℏ}{2} ∫_{0}^{a} cos(\frac{2}{ℏ}f(x))\frac{W′(\frac{x}{ℏ^{β}})}{(f′(x))^{2}} dx  (43)

The auxiliary function f is again defined by (31). We estimate roughly, (using (14))

|ℏ^{−β} ∫_{0}^{a} cos(\frac{2}{ℏ}f(x))\frac{W′(\frac{x}{ℏ^{β}})}{(f′(x))^{2}} dx| ≤ Kℏ^{−β} ∫_{0}^{a} |W′(\frac{x}{ℏ^{β}})| dx = K ∫_{0}^{aℏ^{−β}} |W′(x)| dx = K

Now because all functions in (43) are bounded on [0, a] we have

∫_{0}^{a} W (\frac{x}{ℏ^{β}})\frac{dx}{\sqrt{1 − x^{α}}} cos(\frac{2}{ℏ} ∫_{x}^{1} \sqrt{1 − y^{α}}dy − \frac{π}{4} ) = O(ℏ)  (44)

Putting (40), (41), (42) and (44) together yields

∫_{0}^{a} W (\frac{x}{ℏ^{β}})(\frac{s̃(x)}{V (x)})^{\frac{1}{2}} Ai^{2}(ℏ^{\frac{−2}{3}} s̃(x))dx

= \frac{ℏ^{\frac{1}{3}}}{2π} (ℏ^{β} ∫_{0}^{∞} W + O (ℏ^{β u})) + O(ℏ^{\frac{4}{3}} )

= \frac{ℏ^{β+ \frac{1}{3}}}{2π} (∫_{0}^{∞} W + O (ℏ^{min\{β(r−1), \frac{α}{α+2} \}}))  (45)

Proof of Theorem 2.6. Proof. If we collect all results (36), (38), (39) and (45), we obtain what we wanted to prove

< W (\frac{.}{ℏ^{β}}) Φ̃, Φ̃ > = ∫_{0}^{a} W (\frac{.}{ℏ^{β}}) |Φ̃|^{2} + ∫_{a}^{∞} W (\frac{.}{ℏ^{β}}) |Φ̃|^{2}
= ∫_{0}^{a} W (\frac{.}{ℏ^{β}})|ϕ|^{2} + O(ℏ^{\frac{4}{3}} ) + C_{ℏ}^{−2}O(ℏ^{βr})

= \frac{ℏ^{\frac{1}{3} +β}}{2π} (∫_{0}^{∞} W + O (ℏ^{min\{β(r−1), \frac{α}{2+α} \}}))

2.4.5. \underline{Proof of Theorem 2.2}. Now if we collect proven statements of Theorems 2.5 and 2.6 and use (22), (23), the relationship (17) between E and ℏ, we may conclude that Theorem 2.2 is proven.

2.5. Offdiagonal elements. For understanding the behavior of all elements in matrix our precise analysis of diagonal is not much important because the diagonal is only “a set of zero measure”. There are several main directions in our matrix, the first one is parallel to the diagonal, the second one is normal to the diagonal and the third one is supposed to go through columns with a fixed row. Our guess is that the behavior of elements close to the diagonal is more or less the same as on the diagonal. We will formulate this result with the help of Cauchy-Schwartz inequality. This estimate will be valid for all elements, but elements far from the diagonal decay much faster, so we will use other, direct method to improve the decay estimate for elements far from the diagonal.

2.5.1. \underline{Elements close to the diagonal}. By Cauchy-Schwartz inequality we mean

|F (a, b)|^{2} ≤ q_{F} (a) · q_{F} (b),

where F (. , .) is a sesquilinear form and q_{F} (a) := F (a, a) is its corresponding quadratic form which is supposed to be positive. If we apply this inequality to the result of diagonal case we obtain the estimate Theorem 2.7. Let α > 1, Ψ_{n}, Ψ_{l} are two L^{2}(ℝ_{+}) solutions of (1) with corresponding Energies E_{n} and E_{l} which depend on natural numbers n, l according to the prescription (12). Suppose moreover that a function W satisfies the decay properties (14). Then all matrix elements can be estimated by

W_{n, l} := \frac{< W Ψ_{n}, Ψ_{l} >}{\sqrt{< Ψ_{n}, Ψ_{n} >} \sqrt{< Ψ_{l}, Ψ_{l} >}} ≤ K(n l)^{− \frac{1}{α+2}}

2.5.2. \underline{Elements far from to the diagonal}. We will investigate elements for which

n > δ l , δ > 0.

directly, by estimating integral < W Ψ_{n}, Ψ_{l} >. Because of the presence of different Energies E_{n} and E_{l} is our scaling trick not possible, but we lose by doing it just a bit of comfort. Similarly to the part dealing with diagonal elements we will divide the area of ∫_{ℝ_{+}} W \overline{Ψ}_{n}Ψ_{l} into two parts, a window and a tail. By choosing a boundary of them between the turning points of functions Ψ_{n} , Ψ_{l} we get O(E_{n}^{−∞}) estimate of the tail! This is caused by exponential decay of the function Ψ_{l} above its turning point. Again the only mathematical tool we will use is integration by parts. Let us choose (without loss of generality) the boundary between the window and the tail by

a_{n} := (\frac{E_{n}}{2} )^{\frac{1}{α}}

and consider such n and l for which

E_{n} > 4E_{l}.  (46)

Of course, instead of 4 we can choose δ^{\frac{2α}{α+2}} . Again we will first of all formulate the result.

\underline{Result.} Theorem 2.8. Consider the case n > δ l for some positive number δ. α > 1. Assume that the perturbation W satisfies (14) and in addition that W′′ is defined and bounded almost everywhere on ℝ_{+} . Then matrix elements W_{n, l} can be estimated from above by

W_{n, l} := \frac{∫_{ℝ_{+}} W \overline{Ψ}_{n}Ψ_{l}}{\sqrt{∫_{ℝ_{+}} |Ψ_{n}|^{2}}\sqrt{∫_{ℝ_{+}} |Ψ_{l}|^{2}}} = O (n^{− \frac{α+1}{α+2}} l^{\frac{α−1}{α+2}} ) = O ((n l)^{− \frac{1}{α+2}} (\frac{l}{n})^{\frac{α}{α+2}} )

where Ψ_{n} is the Dirichlet L^{2}(ℝ_{+}) solution of equation

(−Δ + x^{α})Ψ_{n}(x) = E_{n}Ψ_{n}(x) .

So the relationship between the energy E_{n} and the natural number n is given by the Bohr-Sommerfeld quantization condition (12).

The proof of this theorem will be again given in the following text and finished in the very last part of this subsection. First of all we show that the main contribution comes from window.

Tail estimate. Set

ϕ_{n}(x) := (\frac{s_{n}(x)}{x^{α} − E_{n}})^{\frac{1}{4}} Ai(s_{n}(x)).

In this part we will show that the contribution of the tail

∫_{a_{n}}^{∞} W (x)ϕ_{n}(x)ϕ_{l}(x)dx

is negligible. Using (10) we obtain

∫_{a_{n}}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} Ai(s_{l}(x)) Ai(s_{n}(x))dx

= ∫_{a_{n}}^{∞} W (x) (\frac{s_{n}(x)}{x^{α} − E_{n}} )^{\frac{1}{4}} \frac{Ai(s_{n}(x))}{2\sqrt{π}\sqrt[4]{|E_{l} − x^{α}|}} exp (− ∫_{E \frac{1}{α} ; l}^{x} \sqrt{y^{α} − E_{l}} dy) dx

× (1 + O(E_{n}^{− \frac{α+2}{2α}}))
= O(E_{n}^{−∞})  (47)

because we can estimate with the help of (46)

∫_{E \frac{1}{α} ; l}^{a_{n}} \sqrt{y^{α} − E_{l}} dy = El^{\frac{α+2}{2α}} ∫_{1}^{( \frac{En}{2E_{l}})^{\frac{1}{α}}} \sqrt{y^{α} − 1} dy ≥ KE_{n}^{− \frac{α+2}{2α}}  (48)

Error estimate. The results (6), (7) will be often used in this part. The technique we will apply will be similar like in the tail estimate paragraph. We shall treat the integral

∫_{0}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} (ε̄_{l}(x) Ai(s_{n}(x)) + ε_{n}(x) Ai(s_{l}(x)) + ε̄_{l}(x)ε_{n}(x))dx

The last term is the smallest one

|∫_{0}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} ε̄_{l}(x)ε_{n}(x)dx|  (49)

≤ K(E_{l}E_{n})^{− \frac{α+2}{2α}} ∫_{0}^{∞} |W (x)|\frac{e^{− \frac{2}{3} s_{l+}(x)^{\frac{3}{2}}} e^{− \frac{2}{3} s_{n+}(x)^{\frac{3}{2}}}}{\sqrt[4]{|x^{α} − E_{l}||x^{α} − E_{n}|}} dx

≤ K(E_{l}E_{n})^{− \frac{α+2}{2α}} ∫_{0}^{2a_{l}} |W (x)|\frac{dx}{\sqrt[4]{|x^{α} − E_{l}||x^{α} − E_{n}|}} + O(E_{n}^{− \frac{3}{4} − \frac{1}{α}} E_{l}^{−∞})

≤ K∥W ∥_{∞}(E_{l}E_{n})^{− \frac{α+2}{2α}} E_{n}^{− \frac{1}{4}} ∫_{0}^{2a_{l}} \frac{dx}{\sqrt[4]{|x^{α} − E_{l}|}} + O(E_{n}^{− \frac{3}{4} − \frac{1}{α}} E_{l}^{−∞})

≤ K∥W ∥_{∞}E_{n}^{− \frac{3}{4} − \frac{1}{α}} E_{l}^{− \frac{1}{2} − \frac{1}{α} − \frac{1}{4} + \frac{1}{α}} ∫_{0}^{2} \frac{dx}{\sqrt[4]{|x^{α} − 1|}} + O(E_{n}^{− \frac{3}{4} − \frac{1}{α}} E_{l}^{−∞})

= O ((E_{n}E_{l})^{− \frac{3}{4}} E_{n}^{− \frac{1}{α}} )

The O(E^{−∞}) results are obtained using exponential decay of ε (see (6)) and Ai (see (10)) and with the help of (48). The second integral can be similarly estimated by

|∫_{0}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} ε_{n}(x) Ai(s_{l}(x))dx|

≤ KE_{n}^{− \frac{α+2}{2α}} ∫_{0}^{a_{n}} |W (x)| (\frac{s_{l}(x)}{(x^{α} − E_{l})|x^{α} − E_{n}|})^{\frac{1}{4}} | Ai(s_{l}(x))|dx + O(E_{n}^{−∞})

≤ KE_{n}^{− \frac{3}{4} − \frac{1}{α}} (∫_{0}^{a_{l}} \frac{|W (x)|dx}{\sqrt[4]{|x^{α} − E_{l}|}} + ∥ Ai ∥_{∞} sup_{[a_{l}, 2a_{l}]} |W (x)| ∫_{a_{l}}^{2a_{l}} (\frac{s_{l}(x)}{x^{α} − E_{l}} )^{\frac{1}{4}} dx + O(E_{l}^{−∞}))

+O(E_{n}^{−∞})

= E_{n}^{− \frac{3}{4} − \frac{1}{α}} (O (E_{l}^{− \frac{1}{4}} ) + O (E_{l}^{− \frac{r}{α} − \frac{1}{6} + \frac{7}{6α}} )) + O(E_{n}^{−∞}) ,  (50)

where we used the fact that

\frac{s_{l}(x)}{x^{α} − E_{l}} = E_{l}^{− \frac{2}{3} + \frac{2}{3α}} u(\frac{x}{El^{\frac{1}{α}}})

for a continuous function u. To obtain sufficient estimate of the first integral, we will have to integrate by parts, using again estimates (6), (7) and the relation s(x)s′^{2}(x) = x^{α} − E.

∫_{0}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} ε_{l}(x) Ai(s_{n}(x))dx

= − ∫_{0}^{a_{n}} W (x) (\frac{s_{l}(x)}{(x^{α} − E_{l})(E_{n} − x^{α})^{3}} )^{\frac{1}{4}} ε_{l}(x) cos(f(x))f′(x)dx + O(E_{n}^{−∞})

= − [\frac{W (x) sin(f(x))ε_{l}(x)}{\sqrt{s′_{l}(x)}\sqrt[4]{E_{n} − x^{α}}^{3}}]_{0}^{a_{n}} + ∫_{0}^{a_{n}} \frac{W′(x) sin(f(x))ε_{l}(x)}{\sqrt{s′_{l}(x)}\sqrt[4]{E_{n} − x^{α}}^{3}}dx

+ ∫_{0}^{a_{n}} \frac{W (x) sin(f(x))ε_{l}(x)}{\sqrt{s′_{l}(x)}\sqrt[4]{E_{n} − x^{α}}^{3}} (\frac{ε′_{l}(x)}{ε_{l}(x)} − \frac{s′′_{l}(x)}{2s′_{l}(x)} + \frac{3αx^{α−1}}{4(E_{n} − x^{α})}) dx + O(E_{n}^{−∞})

The auxiliary function f is now defined by

f(x) := ∫_{x}^{E \frac{1}{α} ; n} \sqrt{E_{n} − y^{α}}dy − \frac{π}{4} , f′(x) = −\sqrt{E_{n} − x^{α}}

Let us investigate all terms

− [\frac{W (x) sin(f(x))ε_{l}(x)}{\sqrt{s′_{l}(x)}\sqrt[4]{E_{n} − x^{α}}^{3}}]_{0}^{a_{n}} = \frac{W (0) sin(f(0))ε_{l}(0)}{\sqrt{s′_{l}(0)}En^{\frac{3}{4}}} + O(E_{n}^{−∞}) = O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{3}{4} − \frac{1}{α}}) , (51)

where we used again (6). The second term is estimated by (like in (49))

|∫_{0}^{a_{n}} \frac{W′(x) sin(f(x))ε_{l}(x)}{\sqrt{s′_{l}(x)}\sqrt[4]{E_{n} − x^{α}}^{3}}dx| ≤ E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{2} − \frac{1}{α}} (K ∫_{0}^{2a_{l}} \frac{|W′(x)|dx}{\sqrt[4]{|E_{l} − x^{α}|}} + O(E_{l}^{−∞}))

= O (E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{3}{4} − \frac{1}{2α}} )  (52)

Because of the Hölder inequality we may estimate

∫_{0}^{2a} |W′(x)g(\frac{x}{a})|dx ≤ \sqrt{∥W′∥_{∞}∥W′∥_{L^{1}(ℝ_{+})}}\sqrt{a ∫_{0}^{2} |g(x)|^{2}dx}

To treat the third integral it is useful to remark that (again with the help of s(x)s′^{2}(x) = x^{α} − E)

s(x) = E^{\frac{1}{3} + \frac{2}{3α}} s̃′(\frac{x}{E^{\frac{1}{α}}})

s′(x) = E^{\frac{1}{3} − \frac{1}{3α}} s̃′(\frac{x}{E^{\frac{1}{α}}})  (53)

s′′(x) = \frac{V′ − s′^{3}}{2ss′} = E^{\frac{1}{3} − \frac{4}{3α}} s̃′′(\frac{x}{E^{\frac{1}{α}}})

So using the same tricks and (7) we conclude again

∫_{0}^{a_{n}} \frac{W (x) sin(f(x))ε_{l}(x)}{\sqrt{s′_{l}(x)}\sqrt[4]{E_{n} − x^{α}}^{3}} (\frac{ε′_{l}(x)}{ε_{l}(x)} − \frac{s′′_{l}(x)}{2s′_{l}(x)} + \frac{3αx^{α−1}}{4(E_{n} − x^{α})}) dx

= E_{n}^{− \frac{3}{4}} (O(E_{l}^{− \frac{1}{4} − \frac{1}{α}}) + O(E_{l}^{− \frac{3}{4} − \frac{3}{2α}}) + O(E_{n}E_{l}^{\frac{1}{4} − \frac{3}{2α}})) = O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{4} − \frac{1}{α}}) (54)

If we collect these pieces of information (51), (52), (54) we obtain

∫_{0}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} ε_{l}(x) Ai(s_{n}(x))dx

= O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{3}{4} − \frac{1}{α}}) + O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{3}{4} − \frac{1}{2α}}) + O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{4} − \frac{1}{α}})

= O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{4} − \frac{1}{2α}})  (55)

Putting (49), (50), (55) together we see that under the assumptions W′′ ∈ L^{∞}(ℝ_{+}), (14) imposed on the perturbation W the contribution of the error may be estimated by

∫_{0}^{∞} W (x) (\frac{s_{l}(x)s_{n}(x)}{(x^{α} − E_{l})(x^{α} − E_{n})})^{\frac{1}{4}} (ε̄_{l}(x) Ai(s_{n}(x)) + ε_{n}(x) Ai(s_{l}(x)) + ε̄_{l}(x)ε_{n}(x))dx

= O (E_{n}^{− \frac{3}{4} − \frac{1}{α}} E_{l}^{− \frac{r}{α} − \frac{1}{6} + \frac{7}{6α}} ) + O (E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{4} − \frac{1}{2α}} )  (56)

\underline{Window.} We will compute the contribution of

∫_{0}^{a_{n}} W (x)ϕ_{n}(x)ϕ_{l}(x)dx =

∫_{0}^{a_{n}} W (x) (\frac{s_{l}(x)}{(E_{n} − x^{α})(x^{α} − E_{l})})^{\frac{1}{4}} cos (∫_{x}^{E \frac{1}{α} ; n} \sqrt{E_{n} − y^{α}}dy − \frac{π}{4} ) \frac{dx}{\sqrt{π}} (1 + O(E_{n}^{− \frac{α+2}{2α}}))

The main part is proportional to

∫_{0}^{a_{n}} W (x) (\frac{s_{l}(x)}{(E_{n} − x^{α})(x^{α} − E_{l})})^{\frac{1}{4}} cos (∫_{x}^{E \frac{1}{α} ; n} \sqrt{E_{n} − y^{α}}dy − \frac{π}{4} ) dx

= − [\frac{W (x) sin(f(x)) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}} (\frac{s_{l}(x)}{x^{α} − E_{l}} )^{\frac{1}{4}} ]_{0}^{a_{n}} + ∫_{0}^{a_{n}} \frac{W′(x) sin(f(x)) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}}dx

+ ∫_{0}^{a_{n}} W (x) sin(f(x)) (\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}})′ dx

= − [\frac{W (x) sin(f(x)) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}} (\frac{s_{l}(x)}{x^{α} − E_{l}} )^{\frac{1}{4}} ]_{0}^{a_{n}} + [\frac{W′(x) cos(f(x)) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{5}\sqrt{s′_{l}(x)}}]_{0}^{a_{n}}

+ ∫_{0}^{a_{n}} W (x) sin(f(x)) (\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}})′ dx

− ∫_{0}^{a_{n}} cos(f(x)) (\frac{W′(x) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{5}\sqrt{s′_{l}(x)}})′ dx,  (57)

where f is again defined by

f(x) := ∫_{x}^{E \frac{1}{α} ; n} \sqrt{E_{n} − y^{α}}dy − \frac{π}{4} , f′(x) = −\sqrt{E_{n} − x^{α}}

Let us investigate all terms individually. The first two of them are very small

[\frac{W (x) sin(f(x)) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}} (\frac{s_{l}(x)}{x^{α} − E_{l}} )^{\frac{1}{4}} ]_{0}^{a_{n}} = O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{3}{4} − \frac{1}{α}})  (58)

[\frac{W′(x) cos(f(x)) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{5}\sqrt{s′_{l}(x)}}]_{0}^{a_{n}} = O(E_{n}^{− \frac{7}{4} − \frac{1}{α}} E_{l}^{− \frac{3}{4} − \frac{1}{α}})  (59)

for the same reason like (51), namely with the help of (9), (11), (53). To investigate the third term

∫_{0}^{a_{n}} W (x) sin(f(x)) (\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}})′ dx  (60)

of (57) we firstly compute

(\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{k}\sqrt{s′_{l}(x)}})′ = (\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{k}\sqrt{s′_{l}(x)}}) [\frac{Ai′(s_{l}(x))}{Ai(s_{l}(x))}s′_{l}(x) − \frac{s′′_{l}(x)}{2s′_{l}(x)} + \frac{kαx^{α−1}}{4(E_{n} − x^{α})}]

Let us estimate the third expression (using similar tricks like in (50))

|∫_{0}^{a_{n}} W (x) sin(f(x))\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}} \frac{x^{α−1}}{E_{n} − x^{α}} dx|

≤ E_{n}^{− \frac{7}{4}} (∫_{0}^{a_{l}} \frac{|W (x)| x^{α−1}}{\sqrt[4]{E_{l} − x^{α}}} dx + K sup_{[a_{l}, 2a_{l}]} |W (x)| ∫_{a_{l}}^{2a_{l}} x^{α−1} (\frac{s_{l}(x)}{x^{α} − E_{l}} )^{\frac{1}{4}} dx + O(E_{l}^{−∞}))

= E_{n}^{− \frac{7}{4}} (O(E_{l}^{\frac{3}{4} − \frac{1}{α}}) + O(El^{\frac{5α+1−6r}{6α}}))  (61)

The second part of the integral above is

|∫_{0}^{a_{n}} \frac{W (x) sin(f(x)) Ai(s_{l}(x))s′′_{l}(x)}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}^{3}}dx| ≤ E_{n}^{− \frac{3}{4}} (K ∫_{0}^{2a_{l}} \frac{|W (x)s′′_{l}(x)|}{\sqrt{s′_{l}(x)}^{3}}dx + O(E_{l}^{−∞}))

= O (E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{6} − \frac{1}{3α}} )  (62)

The first part of the third term from (57) can be estimated with the help of (5)

∫_{0}^{a_{n}} \frac{W (x) sin(f(x)) Ai′(s_{l}(x))\sqrt{s′_{l}(x)}}{\sqrt[4]{E_{n} − x^{α}}^{3}}dx = O (E_{n}^{− \frac{3}{4}} El^{\frac{1}{4}} )  (63)

The results of (61), (62), (63) allow us to say that the third integral (60) in (57)

∫_{0}^{a_{n}} W (x) sin(f(x)) (\frac{Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{3}\sqrt{s′_{l}(x)}})′ dx = O (E_{n}^{− \frac{3}{4}} El^{\frac{1}{4}} )  (64)

The very last term in (57) is

∫_{0}^{a_{n}} cos(f(x)) (\frac{W′(x) Ai(s_{l}(x))}{\sqrt[4]{E_{n} − x^{α}}^{5}\sqrt{s′_{l}(x)}})′ dx = O (E_{n}^{− \frac{7}{4}} El^{\frac{1}{4}} )  (65)

by the same tricks like in the previous case (60).

The main result of this subsection is obtained by collecting (58), (59), (64) and (65).

∫_{0}^{a_{n}} W (x)ϕ_{n}(x)ϕ_{l}(x)dx =

= O(E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{3}{4} − \frac{1}{α}}) + O(E_{n}^{− \frac{7}{4} − \frac{1}{α}} E_{l}^{− \frac{3}{4} − \frac{1}{α}})

+O (E_{n}^{− \frac{3}{4}} El^{\frac{1}{4}} ) + O (E_{n}^{− \frac{7}{4}} El^{\frac{1}{4}} ) ,

which yields

∫_{0}^{a_{n}} W (x)ϕ_{n}(x)ϕ_{l}(x)dx = O (E_{n}^{− \frac{3}{4}} El^{\frac{1}{4}} )  (66)

Proof of the Theorem 2.8. Now we are ready to state upper bound of non-diagonal matrix elements. Proof. Taking (47), (56) and (66) together we get

∫_{ℝ_{+}} W Ψ_{n}Ψ_{l} = O(E_{n}^{−∞}) + O (E_{n}^{− \frac{3}{4} − \frac{1}{α}} E_{l}^{− \frac{r}{α} − \frac{1}{6} + \frac{7}{6α}} ) + O (E_{n}^{− \frac{3}{4}} E_{l}^{− \frac{1}{4} − \frac{1}{2α}} ) + O (E_{n}^{− \frac{3}{4}} El^{\frac{1}{4}} )

= O (E_{n}^{− \frac{3}{4}} El^{\frac{1}{4}} )

because E_{n} > 4E_{l}. We use the knowledge about normalization constant (13)

∫_{ℝ_{+}} Ψ_{n}^{2} = \frac{1}{2π} E_{n}^{− \frac{1}{2} + \frac{1}{α}} ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} + O (En^{\frac{16−10α}{18α}} )

to get

W_{n, l} := \frac{∫_{ℝ_{+}} W Ψ_{n}Ψ_{l}}{\sqrt{∫_{ℝ_{+}} Ψ_{n}^{2}}\sqrt{∫_{ℝ_{+}} Ψ_{l}^{2}}} = O (E_{n}^{− \frac{1}{2} − \frac{1}{2α}} E_{l}^{\frac{1}{2} − \frac{1}{2α}} ) ,

what is just the statement of Theorem 2.8.

2.6. Summary. If we sum up all results we can state the main theorem about asymptotic behavior of matrix elements. Theorem 2.9. Let Ψ_{n} is Dirichlet L^{2}(ℝ_{+}) solution of the equation

(−Δ + x^{α})Ψ_{n}(x) = E_{n}Ψ_{n}(x) ,

where α > 1. Dirichlet boundary condition implies that there is a relationship between energy E_{n} and a natural number n given by the Bohr-Sommerfeld quantization condition

E_{n} = (\frac{π}{∫_{0}^{1} \sqrt{1 − x^{α}}dx}(n − \frac{1}{4}))^{\frac{2α}{α+2}} + O ((\frac{1}{n})^{\frac{2α}{α+2}} ) .

Assume that a perturbation W satisfies the following decay properties (< x >= \sqrt{1 + x^{2}}).

W, W′ ∈ L^{1}(ℝ^{+}), < x >^{r} W ∈ L^{∞}(ℝ^{+}) , for some r > 1

and in addition that W′′ is defined and bounded almost everywhere on ℝ_{+} . Then for matrix elements W_{n, l} it holds true that

W_{n, n} := \frac{< W Ψ_{n}, Ψ_{n} >}{< Ψ_{n}, Ψ_{n} >} = (\frac{π(n − \frac{1}{4})}{∫_{0}^{1} \sqrt{1 − x^{α}}dx})^{− \frac{2}{α+2}} (∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{α}}} )^{−1} (∫_{ℝ_{+}} W + O (n^{−t})) ,

(where t := min\{\frac{1}{9}, \frac{2(r−1)}{α+2}\} > 0) for the diagonal elements. Other elements are estimated from above by

W_{n, l} := \frac{< W Ψ_{n}, Ψ_{l} >}{\sqrt{< Ψ_{n}, Ψ_{n} >} \sqrt{< Ψ_{l}, Ψ_{l} >}} ≤ K(n l)^{− \frac{1}{α+2}} ,

or if moreover n > δ l, for some δ > 0, by a better estimate

W_{n, l} ≤ K(n l)^{− \frac{1}{α+2}} (\frac{l}{n})^{\frac{α}{α+2}} .  (67)

3. Egorov theorem for time dependent case

3.1. Introduction. This section is dealing with Egorov theorem, mathematically rigorous tool for approximation of quantum time evolution with help of the classical one. It is proven that the difference between exact quantum evolution and approximation is proportional to ℏ^{2} in the sense of the statement of Theorem 3.7. Physical interpretation of such result is clear, when observables of given physical system with the same dimension as ℏ like action are much larger then numerical value of ℏ (i.e. is possible limit ℏ → 0), then quantum evolution of an observable tends to the classical one. The theory of this type uses so called Weyl-quantization, as powerful tool to construct quantum observables (i.e. symmetric or even essentially self-adjoint operators) according to the principal of correspondence. The result of this section, formulated in Theorem 3.7, is improvement of [Lev]. It is again based on [BR], but the generalization to time dependent systems is led directly using the trick (78). This is idea of Joachim Asch, I would like to thank him for it. The statement of Theorem 3.7 is improved with respect to [Lev] in several ways. The first is arbitrary dimension, then general time dependent Hamiltonian despite the Schrödinger one is included. The assumptions on the growth of H(τ) are also weaken.

3.2. Notation and conventions. Let us denote by X = ℝ^{n} the configuration space of a classical mechanical system with n degrees of freedom. The corresponding phase space Z is identified with ℝ^{2n} equipped with the symplectic form σ defined by

σ(z; z′) = ⟨Jz, z′⟩.  (68)

where ⟨ , ⟩ is the Euclidean scalar product and J is the 2n × 2n matrix

J = (\begin{matrix}0 & 1l \\ −1l & 0 \end{matrix}) .

A generic point in Z is denoted z and its coordinates by (q, p) where q, p ∈ ℝ^{n}. We will again use notation ⟨x⟩ = (1 + |x|^{2})^{1/2}. A classical time dependent Hamiltonian is a time dependent real function, defined on the phase space H : [s, t] × Z → ℝ . Basic example is H(τ, q, p) = \frac{∥p∥^{2}}{2m} + V (τ, q), (m > 0) where ∥p∥^{2} =< p, p > and V is time dependent potential. In what follows we put emphasis on the case X = ℝ^{n}. The motion of the classical system is determined by the system of Hamilton equations

\frac{dq}{dt} = \frac{∂H(t)}{∂p}(q, p), \frac{dp}{dt} = −\frac{∂H(t)}{∂q}(q, p).  (69)

We will suppose that the equations (69) generate a flow Φ(t, s) on the phase space Z, defined by

Φ(t, s, q, p) = (q(t, s), p(t, s)) , Φ(s, s) = 1l.

We define Poisson bracket by

{A, B} = ∂_{q}A · ∂_{p}B − ∂_{p}A · ∂_{q}B.

Here we have used the notation ∂_{q} = \frac{∂}{∂q}.

3.3. Weyl quantization. First of all we define a class of classical observables, which are quantizable by Weyl quantization, introduced after.

Definition 3.1. A ∈ O(m), m ∈ ℝ, if and only if Z →^{A} ℂ is C^{∞} in Z and for every multi-index γ ∈ ℕ^{2n} there exists C > 0 such that

|∂_{z}^{γ}A(z)| ≤ C⟨z⟩^{m}, ∀z ∈ Z.

Definition 3.2. Let A ∈ O(m). We will define Â as operator from S(X) to itself, called ℏ-Weyl quantization of A, by the following formula, with ψ ∈ S(X),

Âψ(x) = (2πℏ)^{−n} ∫_{X} (∫_{X} A (\frac{x + y}{2} , p) e^{iℏ^{−1}⟨x−y,p⟩}ψ(y)dy) dp.

Example: Let us focus for example to observable A := x.p. To compute Â we will use

p exp(\frac{i}{ℏ}(x − y)p) = −\frac{ℏ}{i}∂_{y} exp(\frac{i}{ℏ}(x − y)p)

and after integration by parts and using inverse Fourier transform we get Â = \frac{1}{2}(p̂q̂ + q̂p̂). For A ∈ S(Z) we will compute, (with the help of integral kernel) inverse prescription of Weyl symbol if operator is given as an integral operator.

(Âψ)(x) = (2πℏ)^{−n} ∫ ∫_{Z} A (\frac{x + y}{2} , p) e^{iℏ^{−1}⟨x−y,p⟩}ψ(y)dydp = ∫_{ℝ^{n}} K_{A}(x, y)ψ(y)dy

K_{A}(x, y) := (2πℏ)^{−n} ∫_{ℝ^{n}} A (\frac{x + y}{2} , p) e^{iℏ^{−1}⟨x−y,p⟩}dp

After inverse Fourier transform.

A(x, ξ) = ∫_{ℝ^{n}} K_{A} (x + \frac{v}{2}, x − \frac{v}{2}) e^{− \frac{i}{ℏ}v·ξ}dv  (70)

The class O(m) has nice properties. For example, if A ∈ O(m) is a real-valued function, then Â is symmetric operator in L^{2}(X), with domain S(X). Further there exists a product formula. For the proof one can see [Ro1], [Fo] or [Ho].

3.4. Product formula. Theorem 3.3. Let A ∈ O(m), B ∈ O(p), then there exists C ∈ O(m + p), such that

Â · B̂ = Ĉ

And there exists following asymptotic expansion of Weyl symbol of this operator

C(q, p) = exp (\frac{iℏ}{2}σ(D_{q}, D_{p}; D_{q′}, D_{p′})) A(q, p)B(q′, p′)|_{(q,p)=(q′,p′)},

where σ is the symplectic bilinear form (68) and D = i^{−1}∇. By expanding the exponential term, we get

C(q, p) = ∑_{j≥0} \frac{ℏ^{j}}{j!} (\frac{i}{2}σ(D_{q}, D_{p}; D_{q′}, D_{p′}))^{j}A(q, p)B(q′, p′)|_{(q,p)=(q′,p′)}.

So that C(q, p) is a formal power series in ℏ with coefficients given by

C_{j}(q, p) = \frac{1}{2^{j}} ∑_{|α+β|=j} \frac{(−1)^{|β|}}{α!β!} (D_{q}^{β}∂_{p}^{α}A).(D_{q}^{α}∂_{p}^{β}B)(q, p).  (71)

Proof. We will show this for A, B ∈ S(Z)

(Â · B̂ψ) (x) = ∫_{ℝ^{n}} K_{A·B}(x, y)ψ(y)dy

where

K_{A·B}(x, y) = \frac{1}{(2πℏ)^{2n}} ∫_{ℝ^{3n}} A (\frac{x + z}{2} , p) B (\frac{z + y}{2} , η) e^{− \frac{i}{ℏ}⟨x−z,p⟩+⟨z−y,η⟩}dzdpdη

Let’s define

Ĉ := Â · B̂

then Weyl symbol of Ĉ can be computed with help of formula (70).

C(x, ξ) = ∫_{ℝ^{n}} K_{A·B} (x + \frac{v}{2}, x − \frac{v}{2}) e^{− \frac{i}{ℏ}v·ξ}dv
= \frac{1}{(2πℏ)^{2n}} ∫_{ℝ^{4n}} A (\frac{x + z − \frac{v}{2}}{2} , p) B (\frac{z + y − \frac{v}{2}}{2} , η) ×
×e^{− \frac{i}{ℏ}(⟨x+ \frac{v}{2} −z,p⟩+⟨z−x+ \frac{v}{2} ,η⟩−⟨v,ξ⟩)}dzdpdηdv

We apply the change of variables

s := \frac{1}{2}(x + z) + \frac{1}{4}v, t := \frac{1}{2}(x + z) − \frac{1}{4}v ⇒ z = s + t − x, v = 2(s − t), dzdv = 4^{n}dsdt

to get

C(x, ξ) = \frac{1}{(πℏ)^{2n}} ∫_{ℝ^{4n}} A (s, p) B (t, η) e^{− \frac{2i}{ℏ}(⟨x−s,ξ−η⟩−⟨x−t,ξ−p⟩)}dsdpdtdη

Now using Taylor formula

A(s, p)B(t, η) = ∑_{|α+β|<N} ∂_{x}^{α}A(x, p) ∂_{x}^{β}B(x, η)\frac{(s − x)^{α}}{α!} \frac{(t − x)^{β}}{β!}

+ ∑_{|α+β|=N} ∫_{0}^{1}(1 − σ)^{N}∂_{x}^{α}A(x + σ(s − x), p) ∂_{x}^{β}B(x + σ(t − x), η)\frac{(s − x)^{α}}{α!} \frac{(t − x)^{β}}{β!} dσ

we obtain

C(x, ξ) = \frac{1}{(πℏ)^{2n}} ∑_{|α+β|<N} ∫_{ℝ^{4n}} ∂_{x}^{α}A(x, p) ∂_{x}^{β}B(x, η)\frac{(s − x)^{α}}{α!} \frac{(t − x)^{β}}{β!} ×
×e^{^{\frac{2i}{ℏ}}(⟨x−s,ξ−η⟩−⟨x−t,ξ−p⟩)}dsdpdtdη + R_{N−1}(A, B; z; ℏ),

where the remainder is defined by:

C(z) − ∑_{0≤j≤N} ℏ^{j}C_{j}(z) =: R_{N}(A, B; z; ℏ).  (72)
z = (x, ξ) ∈ Z

Using

(s − x)^{α}(t − x)^{β}e^{^{\frac{2i}{ℏ}}(⟨x−s,ξ−η⟩−⟨x−t,ξ−p⟩)} = (\frac{ℏ}{2i})^{|α|+|β|} (−1)^{|β|} ∂_{η}^{α}∂_{p}^{β} e^{^{\frac{2i}{ℏ}}(⟨x−s,ξ−η⟩−⟨x−t,ξ−p⟩)}

and integrating by parts we get C(x, ξ) =

\frac{1}{(πℏ)^{2n}} ∑_{|α+β|<N} (\frac{ℏ}{2i})^{|α|+|β|} (−1)^{|α|} ∫_{ℝ^{4n}} \frac{∂_{p}^{β}∂_{x}^{α}A(x, p)}{α!} \frac{∂_{η}^{α}∂_{x}^{β}B(x, η)}{β!} ×

× e^{^{\frac{2i}{ℏ}}(⟨x−s,ξ−η⟩−⟨x−t,ξ−p⟩)}dsdpdtdη + R_{N−1}(A, B; z; ℏ).

Finally

C(x, ξ) = ∑_{|α+β|<N} (\frac{ℏ}{2i})^{|α|+|β|} (−1)^{|α|} \frac{∂_{ξ}^{β}∂_{x}^{α}A(x, ξ)}{α!} \frac{∂_{ξ}^{α}∂_{x}^{β}B(x, ξ)}{β!} +
+ R_{N−1}(A, B; z; ℏ).

Remainder (72) can be also expressed as R_{N−1}(A, B; z; ℏ) :=

\frac{1}{(πℏ)^{2n}} ∑_{|α+β|=N} ∫_{0}^{1}(1 − σ)^{N}∂_{x}^{α}A(x + σ(s − x), p) ∂_{x}^{β}B(x + σ(t − x), η) ×

×\frac{(s − x)^{α}}{α!} \frac{(t − x)^{β}}{β!} e^{^{\frac{2i}{ℏ}}(⟨x−s,ξ−η⟩−⟨x−t,ξ−p⟩)}dσdsdpdtdη

After the change of variables

σ(s − x) =: s′, σ(t − x) =: t′, η − ξ =: η′, p − ξ =: p′

and using the same trick with integrating by parts (and in what follows omitting primes) we finally get

R_{N−1}(A, B; z; ℏ) :=

(\frac{ℏ}{2i})^{N} ∫_{0}^{1} \frac{(1 − σ)^{N}}{(πℏσ)^{2n}} ∑_{|α+β|=N} \frac{(−1)^{α}}{α!β!} ∂_{p}^{β}∂_{x}^{α}A(x + s, p + ξ) ∂_{η}^{α}∂_{x}^{β}B(x + t, η + ξ) ×

× exp (\frac{2i}{ℏσ} (⟨s, η⟩ − ⟨t, p⟩)) dσdsdpdtdη

= (\frac{ℏ}{2i})^{N} ∫_{0}^{1} \frac{(1 − τ)^{N}}{(πℏτ)^{2n}} ∫_{Z×Z} exp(−\frac{2i}{τℏ}σ(u, v))σ^{N}(D_{u}, D_{v})A(u + z)B(v + z)dudvdτ.

Now we estimate this remainder and his derivatives. Theorem 3.4. Suppose that λ > 1. For every m ∈ ℕ, m > 2n, for every s > 4n there exists a constant ρ_{n,m,s} such that for every N ≥ 1, for every multi-index γ, there exists K_{n,N,|γ|} > 0, such that the following estimate holds, for every A, B ∈ S(Z), z ∈ Z,

|∂_{z}^{γ} (C(z) − ∑_{0≤j≤N} ℏ^{j}C_{j}(z)) | ≤ ℏ^{N+1}ρ_{n,m,s}K_{n,N,|γ|} ×

× sup_{(∗)} [(1 + u^{2} + v^{2})^{(s−m)/2}|∂_{u}^{(α,β)+μ}A(u + z)||∂_{v}^{(β,α)+ν}B(v + z)|]  (73)

where sup_{(∗)} means that the supremum holds under the conditions

u, v ∈ Z, |μ| + |ν| ≤ m + |γ|, |α| + |β| = N + 1 (μ, ν ∈ ℕ^{2n}, α, β ∈ ℕ^{n}).

Proof : We shall use the following lemma to estimate R_{N}(A, B; z, ℏ). Lemma 3.5. Let us consider F ∈ S(Z × Z) and the integral (λ > 1)

I(λ) = λ^{2n} ∫_{Z×Z} exp[−iλσ(u, v)]F (u, v)dudv.

Then for every real number s > 4n and every integer m > 2n there exists κ(n, s, m) > 0 depending only on n, s, m (but independent of F) such that the following estimate holds

|I(λ)| ≤ κ(n, s, m) sup_{{u,v∈Z, \atop |μ|+|ν|≤m}}(1 + u^{2} + v^{2})^{(s−m)/2}|∂_{u}^{μ}∂_{v}^{ν}F (u, v)|.

Proof : Let us introduce a cut-off χ_{0}, C^{∞} on ℝ, χ_{0}(x) = 1 for |x| ≤ 1/2 and χ_{0}(x) = 0 for |x| ≥ 1. We split I(λ) into three pieces

I_{0}(λ) = λ^{2n} ∫_{Z×Z} exp[−iλσ(u, v)]χ_{0}(u^{2} + v^{2})χ_{0}(λ(u^{2} + v^{2}))F (u, v)dudv,
I_{1}(λ) = λ^{2n} ∫_{Z×Z} exp[−iλσ(u, v)] (1 − χ_{0}(λ(u^{2} + v^{2}))) χ_{0}(u^{2} + v^{2})F (u, v)dudv,
I_{2}(λ) = λ^{2n} ∫_{Z×Z} exp[−iλσ(u, v)](1 − χ_{0}(u^{2} + v^{2}))F (u, v)dudv.

For I_{0}(λ), we easily have

|I_{0}(λ)| ≤ ω_{4n} sup_{u^{2}+v^{2}≤1} |F (u, v)|,

where ω_{4n} is the volume of the unit ball in Z^{2}. For I_{1}(λ) and I_{2}(λ), we integrate by parts with the differential operator

L = \frac{i}{u^{2} + v^{2}} (Ju\frac{∂}{∂v} − Jv \frac{∂}{∂u}) ,

where J is the matrix associated to the symplectic form (σ(u, v) =< Ju, v >). It holds true that

L (exp[−iλσ(u, v)]) = λ exp[−iλσ(u, v)].

Performing 2n integrations by parts, we can see that it exists a constant c_{n} such that

|I_{1}(λ)| ≤ c_{n} sup_{{u^{2}+v^{2}≤1 \atop |μ|+|ν|≤4n}} |∂_{u}^{μ}∂_{v}^{ν}F (u, v)|.

Similarly, performing m integrations by parts,

|I_{2}(λ)| ≤ | ∫_{Z×Z} exp[−iσ(u, v)] (1 − χ_{0}(\frac{1}{λ}(u^{2} + v^{2})) F (\frac{u}{\sqrt{λ}}, \frac{v}{\sqrt{λ}}) dudv|
≤ c_{m} ∫_{Z×Z}(\frac{|u| + |v|}{u^{2} + v^{2}})^{m}|∂_{u}^{μ}∂_{v}^{ν} ((1 − χ_{0}(\frac{1}{λ}(u^{2} + v^{2}))F (\frac{u}{\sqrt{λ}}, \frac{v}{\sqrt{λ}})) |dudv.

Using the existence of integral

∫_{Z×Z}(\frac{|u| + |v|}{u^{2} + v^{2}})^{m}(1 + u^{2} + v^{2})^{\frac{m−s}{2}} dudv

we get for a constant c(n, s, m),

|I_{2}(λ)| ≤ c(n, s, m) sup_{{u,v∈Z \atop |μ|+|ν|≤m}}(1 + u^{2} + v^{2})^{(s−m)/2}|∂_{u}^{μ}∂_{v}^{ν}F (u, v)|.

Now we can complete the proof of the theorem by using Lemma 3.5, the Leibniz formula and the following elementary estimate, using in Z the coordinates u = (x, ξ), v = (y, η),

|σ^{N}(∂_{x}, ∂_{ξ}; ∂_{y}, ∂_{η}))A(x, ξ)B(y, η)| ≤ (2n)^{N} sup_{|α|+|β|=N} |∂_{x}^{α}∂_{ξ}^{β}A(x, ξ)∂_{y}^{β}∂_{η}^{α}B(y, η)|.

Remark 3.6. We can easily extend the estimate (73) for observables A, B with polynomial growth at infinity, by choosing m large enough to get a finite r.h.s. Let us assume that A ∈ O(μ_{A}), B ∈ O(μ_{B}), where μ_{A}, μ_{B} ∈ ℝ. Then we can apply (73) to A_{ε}(u) = e^{−εu^{2}}A(u) and B_{ε}(v) = e^{−εv^{2}}B(v) for ε > 0 and pass to the limit ε → 0 with m − s ≥ μ_{A} + μ_{B}.

3.5. Main theorem. Theorem 3.7. Let [t, s] be a compact interval. Let us consider time dependent Hamiltonian function H(τ) ∈ O(c) and an observable A ∈ O(d) for some c, d, satisfying

|∂_{z}^{γ}H(τ, z)| ≤ C_{γ}, for 3 ≤ |γ| ≤ 4n + [\frac{n}{2}] + 5 , τ ∈ [s, t]
|∂_{z}^{γ}A(z)| ≤ C_{γ}, for 1 ≤ |γ| ≤ 4n + [\frac{n}{2}] + 5  (74)
|∂_{z}^{γ}Φ(t, τ, z)| ≤ C_{γ}, for 1 ≤ |γ| ≤ 4n + [\frac{n}{2}] + 5, τ ∈ [s, t],

where the classical flow Φ(t, s) is given by classical Hamilton equations (69) and is assumed to be unique. We know that { \atop H(τ)}, Â are symmetric operators on Schwartz space ∀τ ∈ [s, t]. We assume moreover that A is essentially self-adjoint operator in L^{2}(X), with core S(X). Finally we suppose that quantum evolution given by Schrödinger equation is defined on Schwartz space by the propagator U

iℏ d_{τ}U(τ, s)ψ = { \atop H(τ)}U(τ, s)ψ,  (75)

which conserves Schwartz space and which satisfies initial condition

U(τ, τ) = 1l , ∀τ ∈ [s, t]  (76)

valid also on Schwartz space. Then the quantum evolution of observable Â can be approximated by the operator with the Weyl symbol

A_{0}(t, s, z) = A(Φ(t, s, z))

in the following sense:

∥Â(t, s) − { \atop A_{0}(t, s)}∥_{L^{2}(X)} ≤ ℏ^{2}C_{n}(t, s)

Proof. For any B(t) ∈ O(m), ψ ∈ S(X) it holds true that

\frac{d}{dt}B̂(t)ψ = { \atop (\frac{d}{dt}B(t))}ψ  (77)

On Schwartz space holds equation

U(s, t)ÂU(t, s) − Â_{0}(t, s) = ∫_{s}^{t} \frac{d}{dτ} (U(s, τ){ \atop A ∘ Φ(t, τ)}U(τ, s)) dτ

= ∫_{s}^{t} U(s, τ) (\frac{i}{ℏ}[{ \atop H(τ)}, { \atop A ∘ Φ(t, τ)}] + { \atop \frac{d}{dτ} A ∘ Φ(t, τ)}) U(τ, s) dτ

= ∫_{s}^{t} U(s, τ) (\frac{i}{ℏ}[{ \atop H(τ)}, { \atop A ∘ Φ(t, τ)}] − { \atop \{H(τ), A ∘ Φ(t, τ)\}}) U(τ, s) dτ, (78)

where we used (77), Schrödinger equation (75), with initial condition (76). Differentiating a basic property of the propagator, U(τ, s)U(s, τ)ψ = ψ, yields

d_{τ}U(s, τ) = −U(s, τ)(d_{τ}U(τ, s))U(s, τ).

And finally it was used

\frac{d}{dτ} A ∘ Φ(t, τ) = −\{H(τ), A ∘ Φ(t, τ)\},

which can be obtained by differentiating Φ(t, τ, y) = y with respect to t (see (69)) and then putting y := Φ(t, τ, x). With the help of (71) we compute that the Weyl symbol of the expression

Ĉ := \frac{i}{ℏ}[Ĥ, Â]

is

C = 0 + \{H, A\} + 0 + 2\frac{i}{ℏ}R_{2}(H, A; z, ℏ) ,

where R_{2}(H, A; z, ℏ) =

(\frac{ℏ}{2i})^{3} ∫_{0}^{1} \frac{(1 − τ)^{3}}{(πℏτ)^{2n}} ∫_{Z×Z} exp(−\frac{2i}{τℏ}σ(u, v))σ^{3}(D_{u}, D_{v})H(u + z)A(v + z)dudvdτ.

Applying L^{2} norm to (78) we obtain

∥U(s, t)Â U(t, s) − { \atop A_{0}(t, s)}∥_{L^{2}} ≤ 2\frac{|t − s|}{ℏ}∥R̂_{2}(H, A; z, ℏ)∥_{L^{2}}.

But we can apply Calderon-Vaillancourt Theorem with improvements by A. Boulkhemair [Bo], which estimates the norm of the Weyl-quantized operator

∥B̂∥_{L^{2}} ≤ γ_{n} sup_{{|α|,|β|≤[n/2]+1 \atop z∈Z}} |∂_{z}^{α,β}B(z)|.

So we need to estimate

sup_{{|α|,|β|≤[n/2]+1 \atop z∈Z}} |∂_{z}^{α,β}R_{2}(H, A; z, ℏ)|.

This is exactly what we have prepared in Theorem 3.4. Using (74) we deduce that for s = m := 4n + 1 there exists K_{n}(t, s), such that:

K_{n}(t, s) := sup_{(∗)} [|∂_{u}^{(α,β)+μ}H(τ, u + z)||∂_{v}^{(β,α)+ν}A(Φ(t, τ, u + z))|] < ∞

where sup_{(∗)} means that the supremum holds under the conditions

u, v, z ∈ Z, τ ∈ [s, t], |μ| + |ν| ≤ 5n + [n/2] + 1, |α| + |β| = 3 (μ, ν ∈ ℕ^{2n}, α, β ∈ ℕ^{n}).

We have also used the following estimate on derivatives of composition of functions

|∂^{α}(f ∘ g)| ≤ K_{n,|α|} sup_{{β,γ≠0 \atop |β|,|γ|≤|α|}} |∂^{β}f||∂^{γ}g|

Combining all these facts we may finish the proof.

4. Weakly regular Floquet Hamiltonians with pure point spectrum

4.1. Introduction. The problem we address in this part concerns spectral analysis of so called Floquet Hamiltonians. The study of stability of non autonomous quantum dynamical systems is an effective tool to understand most of quantum problems which involve a small number of particles. When these systems are periodic the spectral analysis of the evolution operator over one period can give a fairly good information on this stability, see e.g. [EV]. In fact this type of result generalizes the celebrated RAGE theorem concerned with time-independent systems (one can consult [RS] for a summary). As shown in [How] and [Ya] the spectral analysis of the evolution operator over one period (so called monodromy operator or Floquet operator) is equivalent to the spectral analysis of the corresponding Floquet Hamiltonian (also called operator of quasi-energy). This is also what we are aiming for in this part. More precisely, we analyze periodic quantum systems which are weakly regular in time and ”space” in the sense of an appropriately chosen norm, and give sufficient conditions to insure that the Floquet Hamiltonians has a pure point spectrum. Another generalization is that in the present result (Theorem 4.1) we allow degenerate eigenvalues of the unperturbed Hamilton operator (denoted H in what follows). The degeneracy of eigenvalues h_{m} of H can grow arbitrarily fast with m provided the time-dependent perturbation is sufficiently regular. To our knowledge this is a new feature in this context. Previously two conditions were usually imposed, namely bounded degeneracy and a growing gap condition on eigenvalues h_{m}, reducing this way the scope of applications of this theory to one dimensional confined systems. Owing to the generalization to degenerate eigenvalues we are able to consider also some models in higher dimensions, for example the N-dimensional quantum top, i.e., the N-dimensional version of the pulsed rotor. A short description of this model is also given in subsection 4.3. Because the proof of Main Theorem 4.1 is quite long we will address reader to [D2]. Further generalization to unbounded perturbations is treated in [D3].

4.2. Main theorem. The central object we wish to study in this part is a self-adjoint operator of the form K + V acting in the Hilbert space

K = L^{2}([ 0, T ], dt) ⊗ H ≅ L^{2}([ 0, T ], H, dt)

where T = 2π/ω, ω is a positive number (a frequency) and H is a fixed separable Hilbert space. The operator K is self-adjoint and has the form

K = −i ∂_{t} ⊗ 1 + 1 ⊗ H

where the differential operator −i∂_{t} acts in L^{2}([ 0, T ], dt) and represents the self-adjoint operator characterised by periodic boundary conditions. This means that the eigenvalues of −i∂_{t} are kω, k ∈ ℤ, and the corresponding normalised eigenvectors are χ_{k}(t) = T^{−1/2} exp(ikωt). H is a self-adjoint operator in H and is supposed to have a discrete spectrum. Finally, V is a bounded Hermitian operator in K determined by a measurable operator-valued function t ↦ V (ωt) ∈ B(H) such that sup_{t∈ℝ} ∥V (t)∥ < ∞, V (t) is 2π-periodic, and for almost all t ∈ ℝ, V (t)^{∗} = V (t). Naturally, (Vψ)(t) = V (ωt)ψ(t) in K ≅ L^{2}([ 0, T ], H, dt). Let

∑_{k∈ℤ}kω P_{k}

be the spectral decomposition of −i∂_{t} in L^{2}([ 0, T ], dt) and let

H = ∑_{m∈ℕ}h_{m} Q_{m}

be the spectral decomposition of H in H. Thus we can write

H = \begin{matrix}∑ & ⊕ \\ m∈ℕ \end{matrix} H_{m}

where H_{m} = Ran Q_{m} are the eigenspaces. We suppose that the multiplicities are finite,

M_{m} = dim H_{m} < ∞, ∀m ∈ ℕ.

Hence the spectrum of K is pure point and its spectral decomposition reads

K = ∑_{k∈ℤ} ∑_{m∈ℕ}(kω + h_{m}) P_{k} ⊗ Q_{m},

implying a decomposition of K into a direct sum,

K =\begin{matrix}∑ & ⊕ \\ (k,m)∈ℤ×ℕ \end{matrix}Ran(P_{k} ⊗ Q_{m}) .

Here is some additional notation. Set

V_{knm} = \frac{1}{T} ∫_{0}^{T} e^{−ikωt} Q_{n}V (ωt) Q_{m} dt = \frac{1}{2π} ∫_{0}^{2π} e^{−ikt} Q_{n}V (t) Q_{m} dt ∈ B(H_{m}, H_{n}) . (79)

Further,

Δ_{mn} = h_{m} − h_{n},

and

Δ_{0} = inf_{m≠n} |Δ_{mn}|.

Finally we set

μ_{mn} = M_{m}M_{n}.

Now we are able to formulate our main result. Though not indicated explicitly in the notation the operator K + V is considered as depending on the parameter ω. Theorem 4.1. Fix J > 0 and set Ω_{0} := [ \frac{8}{9}J, \frac{9}{8}J ]. Assume that Δ_{0} > 0 and that there exists σ > 0 such that

Δ_{σ}(J) := J^{σ} ∑_{\begin{matrix}m,n∈ℕ \\ Δ_{mn}>J/2 \end{matrix}} \frac{μ_{mn}}{(Δ_{mn})^{σ}} < ∞ .

Then for every r > σ + \frac{1}{2} there exist positive constants (depending, as indicated, on σ, r, Δ_{0} and J but independent of V), ϵ_{⋆}(r, Δ_{0}, J) and δ_{⋆}(σ, r, J), with the property: if

ϵ_{V} := sup_{n∈ℕ} ∑_{k∈ℤ} ∑_{m∈ℕ}∥V_{knm}∥ max\{|k|^{r}, 1\} < min \{ϵ_{⋆}(r, Δ_{0}, J), \frac{|Ω_{0}|}{δ_{⋆}(σ, r, J)}\}

(here |Ω_{∗}| stands for the Lebesgue measure of Ω_{∗}) then there exists a measurable subset Ω_{∞} ⊂ Ω_{0} such that

|Ω_{∞}| ≥ |Ω_{0}| − δ_{⋆}(σ, r, J) ϵ_{V}  (80)

and the operator K + V has a pure point spectrum for all ω ∈ Ω_{∞} Remark.

1) It is sometimes necessary to consider potentials V which depend on the frequency ω in a more elaborate way. Suppose that V : ℝ × ℝ_{+} → B(H) is a bounded measurable function, which is 2π periodic with respect to the first variable and such that for almost all t ∈ ℝ and ω ∈ ℝ_{+}, V (t, ω)^{⋆} = V (t, ω); then (Vψ)(t) = V (ωt, ω)ψ(t) defines an operator family which is uniformly bounded on K with respect to the variable ω. Let now

ϵ_{V} := sup_{ω,ω′∈Ω_{0}} sup_{n∈ℕ} ∑_{k∈ℤ,m∈ N} (∥V_{knm}(ω)∥ + J∥∂V_{knm}(ω, ω′)∥) max{|k|^{r}, 1}

where

V_{knm}(ω) := \frac{1}{2π} ∫_{0}^{2π} e^{−ikt}Q_{n}V (t, ω)Q_{m}dt

and where the symbol ∂ is the discrete derivative in ω,

∂V_{knm}(ω, ω′) := \frac{V_{knm}(ω) − V_{knm}(ω′)}{ω − ω′}.

It is not difficult to check that Theorem 1 applies in exactly the same conditions.

2) In the course of the proof is shown even more. Namely, for all ω ∈ Ω_{∞} and any eigenvalue of K + V the corresponding eigen-projector P belongs to the Banach algebra with the norm

∥P ∥ = sup_{n∈ℕ} ∑_{k∈ℤ} ∑_{m∈ℕ}∥P_{knm}∥ max\{|k|^{r−σ− \frac{1}{2}} , 1\}.

This shows that P is (r−σ −1/2)-differentiable as a map from [ 0, T ] to the space of bounded operators in H.

3) The constants ϵ_{⋆}(r, Δ_{0}, J) and δ_{⋆}(σ, r, J) are in fact known quite explicitly .

ϵ_{⋆}(r, Δ_{0}, J) = \frac{min \{4Δ_{0}, J\}}{270 e^{3}},

and

δ_{⋆}(σ, r, J) = 720π e^{5}2^{σ} (\frac{2σ + 1}{(1 − e^{− \frac{2}{r}} ) e})^{σ+ \frac{1}{2}} (∑_{s=1}^{∞} s^{2}e^{− \frac{2}{r} (r−σ− \frac{1}{2} )s}) Δ_{σ}(J)

= 720π (\frac{2σ + 1}{(1 − e^{− \frac{2}{r}} ) e})^{σ+ \frac{1}{2}} 2^{σ}e^{3+ \frac{2}{r}(σ+ \frac{1}{2})} \frac{1 + e^{−2+ \frac{2}{r}(σ+ \frac{1}{2})}}{(1 − e^{−2+ \frac{2}{r}(σ+ \frac{1}{2})})^{3}} Δ_{σ}(J)

where we used the identity

∑_{s=1}^{∞} s^{2}e^{−2xs} = \frac{cosh(x)}{4 sinh(x)^{3}}

4.3. Two models. We conclude this subsection with a brief description of two models illustrating the effectiveness of Theorem 4.1. In the first model we set H = L^{2}([ 0, 1 ], dx), H = −∂_{x}^{2} with Dirichlet boundary conditions, and V (t) = z(t)x^{2} where z(t) is a sufficiently regular 2π-periodic function. As shown in [Š] the spectral analysis of this simple model is essentially equivalent to the analysis of the so called quantum Fermi accelerator. The particularity of the latter model is that the underlying Hilbert space itself is time-dependent, H_{t} = L^{2}([ 0, a(t) ], dx) where a(t) is a strictly positive periodic function. The time-dependent Hamiltonian is −∂_{x}^{2} with Dirichlet boundary conditions. Using a convenient transformation one can pass from the Fermi accelerator to the former model getting the function z(t) expressed in terms of a(t), a′(t) and a′′(t). But let us return to the analysis of our model. Eigenvalues of H are nondegenerate, h_{m} = m^{2}π^{2} for m ∈ ℕ, with normalised eigenfunctions equal to \sqrt{2} sin(mπx). A straightforward calculation gives

V_{knm} = z_{k} × \{ \begin{matrix}\frac{8(−1)^{m+n}mn}{(m^{2}−n^{2})^{2}π^{2}} if m ≠ n, \\ \frac{1}{3} − \frac{1}{2m^{2}π^{2}} if m = n, \end{matrix}

where z_{k} = \frac{1}{2π} ∫_{0}^{2π} e^{−ikt}z(t) dt is the Fourier coefficient of z(t). Hence one derives that

ϵ_{V} = sup_{n∈ℕ}(\frac{1}{3} + \frac{2}{n^{2}π^{2}} + \frac{4}{π^{2}} ∑_{j=1}^{n−1} \frac{1}{j^{2}} ) ∑_{k∈ℤ}|z_{k}| max\{|k|^{r}, 1\} = ∑_{k∈ℤ}|z_{k}| max\{|k|^{r}, 1\}.

For any J > 0, Δ_{σ}(J) is finite if and only if σ > 1. On the other hand, to have ϵ_{V} finite it is sufficient that z(t) ∈ C^{s} where s > r + 1 > σ + \frac{1}{2} + 1 > \frac{5}{2}. So z(t) ∈ C^{3} suffices for the theory to be applicable. This may be compared to an older result in [D1], §4.2, giving a much worse condition, namely z(t) ∈ C^{17}.

The second model is the pulsed rotator in N dimensions. In this case H = L^{2}(S^{N}, dμ), with S^{N} ⊂ ℝ^{N+1} being the N-dimensional unit sphere with the standard (rotationally invariant) Riemann metric and the induced normalised measure dμ, and H = −Δ_{LB} is the Laplace-Beltrami operator on S^{N}. The spectrum of H is well known, [Ne]: Spec(H) = {h_{m}}_{m=0}^{∞}, where

h_{m} = m(m + N − 1)

and the multiplicities are

M_{m} = \binom{m + N}{N} − \binom{m + N − 2}{N}.

The time-dependent operator V (t) in H acts via multiplication, (V (t)φ)(x) = v(t, x)φ(x), where v(t, x) is a real measurable bounded function on ℝ × S^{N} which is 2π-periodic in the variable t. Consequently, K ≅ L^{2}([ 0, T ] × S^{N}, dt dμ) and (Vψ)(t, x) = v(ωt, x)ψ(t, x). Note that the asymptotic behavior of the eigenvalues and the multiplicities, as m → ∞, is h_{m} ∼ m^{2}, M_{m} ∼ (2/(N − 1)!) m^{N−1}. So Δ_{σ}(J) is finite, for any J > 0, if and only if

∑_{m^{2}−n^{2}>J/2} \frac{(mn)^{N−1}}{(m^{2} − n^{2})^{σ}} < ∞ .

To ensure this condition we require that σ > 2(N − 1) + 1. Let us assume that there exist s, u ∈ ℤ_{+} such that, for any system of local (smooth) coordinates (y_{1}, …, y_{N}) on S^{N}, the derivatives ∂_{t}^{α}∂_{y_{1}}^{β_{1}} … ∂_{y_{N}}^{β_{N}} v(t, y_{1}, …, y_{N}) exist and are continuous for all α, β, α ≤ s and β_{1} + … + β_{N} ≤ u. If u ≥ 4 then [ H, [ H, V (t) ]] is a well defined second order differential operator with continuous coefficient functions and the operator [ H, [ H, V (t) ]](1 + H)^{−1} is bounded. Clearly,

\frac{(h_{m} − h_{n})^{2}}{1 + h_{m}} Q_{n}V (t)Q_{m} = Q_{n}[ H, [ H, V (t) ]](1 + H)^{−1}Q_{m}.

Using this relation one derives an estimate on V_{knm},

∥V_{knm}∥ ≤ const \frac{1 + min\{h_{n}, h_{m}\}}{|k|^{s}(h_{m} − h_{n})^{2}} ,

valid for k ≠ 0 and m ≠ n. The number

sup_{n∈ℤ_{+}} ∑_{m∈ℤ_{+}, m≠n} \frac{1 + min\{h_{n}, h_{m}\}}{(h_{m} − h_{n})^{2}}

is finite. To see it one can employ the asymptotics of h_{m} and the fact that the sequence

a_{n} = ∑_{m∈ℤ_{+}, m≠n} \frac{1 + min\{n^{2}, m^{2}\}}{(m^{2} − n^{2})^{2}} = (1 + \frac{1}{n^{2}} ) \frac{π^{2}}{12} − \frac{3}{16n^{2}} + \frac{5}{16n^{4}} − \frac{1}{2n} ∑_{m=1}^{2n−1} \frac{1}{m} ,

n = 1, 2, 3, …, is bounded. It follows that the norm ϵ_{V} is finite if s > r + 1 > σ + \frac{1}{2} + 1 > 2(N − 1) + 1 + \frac{3}{2} = 2N + \frac{1}{2}. Thus the theory is applicable provided u ≥ 4 and s > 2N + \frac{1}{2}. The same example has also been treated by adiabatic methods in [Ne]. In that case the assumptions are weaker. It suffices that v(t, x) be (N + 1)-times differentiable in t with all derivatives ∂_{t}^{α}v(t, x), 0 ≤ α ≤ N + 1, uniformly bounded. However the conclusion is somewhat weaker as well. Under this assumption K + V has no absolutely continuous spectrum but nothing is claimed about the singular continuous spectrum.

Contents

1. Preface 1 2. Behavior of matrix elements investigated with WKB 3 2.1. Introduction 3 2.2. Results due to F.W.J. Olver 4 2.3. Bohr-Sommerfeld 5 2.4. Normalization constant and diagonal elements 6 2.5. Offdiagonal elements 15 2.6. Summary 21 3. Egorov theorem for time dependent case 23 3.1. Introduction 23 3.2. Notation and conventions 23 3.3. Weyl quantization 24 3.4. Product formula 24 3.5. Main theorem 28 4. Weakly regular Floquet Hamiltonians with pure point spectrum 31 4.1. Introduction 31 4.2. Main theorem 31 4.3. Two models 33 References 37

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