Non-commutative normal form,spectrum and inverse problem

Abstract

We propose an approach to calculate asymptotic series for low lying eigenvalues of Schrödinger operator based on normal forms in the formal graded Weyl–Heisenberg algebra. The difference from a traditional scheme is that we don’t use any symbol map (Weyl, $p x$ , $x p$ , etc.). We show that our method may be useful for different reasons. Firstly, it enables to estimate the growth of the eigenvalues expansion coefficients, and secondly it may be efficient for practical calculations, e.g. for treating inverse problems. In particular, we prove that under some restrictions in the one-dimensional case the knowledge of asymptotic series for any pair of low lying eigenvalues is enough to recover the potential.

Keywords

Schrödinger operator spectrum Birkhoff normal form inverse problem

1. Introduction

It is well-known that asymptotic series for low lying eigenvalues of a Schrödinger operator (see [21]) can be effectively calculated by means of a quantized Birkhoff normal form. This idea was carried out in [22], where a sequence of unitary operators that sends higher and higher symbols of a pseudo-differential operator (PDO) to a form commuting with its quadratic part is presented.

Another way which seems to be more fit for practical calculations, is to work in the algebra of formal Weyl symbols in $p, x$ and ℏ (element ℏ is thought of as a degree two element). Let us call this algebra the formal graded Heisenberg algebra. A method for calculating the spectrum with help of a normal form in this algebra was implemented by [1,8,10], though without an accurate justification. A mathematically rigorous proof was given in [5].

In our work, we shall use a beautiful realization of the formal graded Heisenberg algebra (suggested in [23]) as an abstract non-commutative algebra with generators p and x subject to certain identities. This construction admits a global version within theory of non-commutative structures [24] in a spirit of [7] or [18]. The language of [23,24] allows one to prove non-commutative analogs of deep results from classical mechanics (e.g. Darboux theorem, Liouville integrability theorem, action-angle coordinates, etc.).

In this framework, the non-commutative analog of classical Birkhoff normal forms was studied in [2,3]. The analogy is quite thorough: it involves not only the formal theory (as in [1,8,10]) but also questions of convergence (analyticity) in various situations. For instance, it is well known that a normal form near a hyperbolic equilibrium of a Hamiltonian system with one degree of freedom and periodic dependence on time is convergent (see [19]). A non-commutative analog of this fact was proved in [2]. We also mention [4] and [20], where convergent quantum normal forms with the application to spectral problems were also studied but in a different context.

The goal of the present paper is to use non-commutative normal forms for calculating semi-classical power asymptotics for low lying eigenvalues (Theorems 3 and 4). The accurate proof is based on an idea to lift a normal form reduction procedure from the graded Heisenberg algebra to the PDO algebra of $\hat{x} \hat{p}$ -symbols.

It is important to note that the PDO algebra and symbol maps are used only on the stage of the proof. But unlike [1,8,10] and [5] these constructions are not required for concrete calculations of normal forms and eigenvalues. We shall see later that the employed abstract algebraic setting leads to aesthetic and symmetric combinatorial formulas, and may be convenient in practice.

To demonstrate the advantages of our approach, we consider two applications. In both cases we deal with a one-dimensional Schrödinger operator $\hat{H} = - \frac{h^{2}}{2} Δ + V (x)$ having a smooth potential $V (x) ⩾ 0$ raising at infinity as $x \to \infty$ , and such that $V (0) = 0$ is a non-degenerate minimum. Firstly, we show that our method enables to estimate the growth of the eigenvalues expansion coefficients. More precisely, let us write out the asymptotic power series for the low lying eigenvalues of $\hat{H}$ : $\begin{matrix} λ_{ν} (ℏ) \sim \frac{ℏ ω (2 ν + 1)}{2} + \sum_{k = 2}^{\infty} E_{k} ℏ^{k} . \end{matrix}$ (Here $ν \in Z_{+}$ is fixed, and $ω = \sqrt{V^{″} (0)} > 0$ is a harmonic oscillator frequency, $ℏ \to 0$ . The term ‘asymptotic series’ $f (x) \sim \sum_{k = 0}^{\infty} f_{k} x^{k}$ means that $f (x) - \sum_{k = 0}^{N} f_{k} x^{k} = O (x^{N + 1})$ as $x \to 0$ .) This series is usually divergent but one can control the growth of the expansion coefficients. Namely, we shall prove that $| E_{k} | ⩽ k! β γ^{k}$ , if $V (x)$ is analytic at $x = 0$ . Note that this result is quite natural in a context of resurgence theory and exact quantization (see [9,12]).

The proof essentially relies on a convergence of a non-commutative normal form in one-dimensional case. We do not know, if the methods from [1,8,10] and [5] can lead to the same result.

The second application is a semi-classical inverse spectral problem, i.e. a question of reconstructing $V (x)$ from the asymptotic expansions of eigenvalues of $\hat{H}$ . It was shown in [6] that the Taylor coefficients of $V (x)$ at $x = 0$ can be recovered uniquely up to a trivial $x \to - x$ symmetry from the asymptotic series for low lying eigenvalues, provided that $V^{(3)} (0) \neq 0$ . This result was partially generalized to many dimensions in [11,13].

However, in the one-dimensional case, the information about all the spectrum seems to be excessive. Indeed, suppose that $u_{3}, u_{4}, \dots$ are the Taylor coefficients of the potential. It is easy to see that $E_{2}$ depends only on $u_{3}$ and $u_{4}$ ; similarly, $E_{3}$ depends on $u_{3}, u_{4}, u_{5}$ and $u_{6}$ , etc. Therefore, it is natural to conjecture that the knowledge of asymptotic series for two eigenvalues is enough to recover the potential. We show that it is the case (provided that $V^{(3)} (0) \neq 0$ ).

Note that none of the mentioned inverse results was obtained by sheer manipulations with formal series the way we do. In particular, the results from [11,13] were established with help of wave trace invariants. They are related with much more difficult renowned Kac’s problem: ‘Can one hear the shape of a drum?’ (see [14,25]).

In the future, we are planning to turn to multi-dimensional problems. However, they are much more involved from a technical viewpoint.

The outline of our paper is as follows. In Section 2, we describe the graded Heisenberg algebra in the spirit of [23,24]. We mostly list constructions and results from the mentioned work, for proofs see [23].

In Section 3.1 we discuss normal forms in the graded Heisenberg algebra. In Section 3.2 we show how a normal form in the graded Heisenberg algebra can be lifted to a normal form in a PDO algebra. In Section 3.3 we deduce a formula for computing asymptotic series of low lying eigenvalues via a normal form.

In Section 4 we deal with two applications described above.

2. Formal graded Heisenberg algebra

2.1. Algebra $\hat{H}$

Fix the standard basis ${e_{1}, \dots, e_{2 n}}$ in $C^{2 n}$ , and let ${\hat{O}}_{k} = {(C^{2 n})}^{\otimes k}$ be kth tensor degree of $C^{2 n}$ . The tensor product endows direct product $\hat{O} = \prod_{k = 0}^{\infty} {\hat{O}}_{k}$ with an associative algebra structure. This is none else than the algebra $C ⟨ ⟨ e_{1}, \dots, e_{2 n} ⟩ ⟩$ of formal power series in non-commutative variables $e_{1}, \dots, e_{2 n}$ . (We avoid writing $⨁_{k = 0}^{\infty}$ because it lacks sense from an algebraic viewpoint.)

Let us define $\begin{matrix} r_{j} = e_{n + j} \otimes e_{j} - e_{j} \otimes e_{n + j}, j = 1, \dots, n . \end{matrix}$ Let $\hat{J} \subset \hat{O}$ be an ideal generated by $\begin{array}{l} e_{n + j} \otimes e_{k} - e_{k} \otimes e_{n + j}, 1 ⩽ j, k ⩽ n, j \neq k; \\ e_{j} \otimes e_{k} - e_{k} \otimes e_{j}, e_{n + j} \otimes e_{n + k} - e_{n + k} \otimes e_{n + j}, 1 ⩽ j < k ⩽ n; \\ e_{j} \otimes r_{j} - r_{j} \otimes e_{j}, e_{n + j} \otimes r_{j} - r_{j} \otimes e_{n + j}, 1 ⩽ j ⩽ n; \\ r_{j + 1} - r_{j}, 1 ⩽ j ⩽ n - 1 . \end{array}$ The formal graded Heisenberg algebra is defined as a quotient algebra $\hat{O} / \hat{J}$ , and denoted by $\hat{H}$ .

It can be easily seen that $\hat{J} = \prod_{k = 0}^{\infty} {\hat{J}}_{k}$ , where ${\hat{J}}_{k} = \hat{J} \cap {\hat{O}}_{k}$ . Clearly, $\begin{matrix} \hat{H} = \prod_{k = 0}^{\infty} {\hat{H}}_{k}, where {\hat{H}}_{k} = π ({\hat{O}}_{k}) = {\hat{O}}_{k} / {\hat{J}}_{k}, \end{matrix}$ and $π : \hat{O} \to \hat{H}$ is a natural projection.

Let us say that $deg F = k$ , if $F \in {\hat{H}}_{k}$ . Each element $F \in \hat{H}$ can be uniquely represented as a sum $F = \sum_{k = 0}^{\infty} F_{k}$ , where $deg F_{k} = k$ . We may introduce a topology on $\hat{H}$ by saying that a sequence in $\hat{H}$ converges, if sequences of all its components in vector spaces ${\hat{H}}_{k}$ do. Algebra $\hat{H}$ has a decreasing filtration $\begin{matrix} \hat{H} = {\hat{H}}^{(0)} \supset {\hat{H}}^{(1)} \supset \dots \supset {\hat{H}}^{(s)} \supset {\hat{H}}^{(s + 1)} \supset \dots, \end{matrix}$ where ${\hat{H}}^{(s)} = \prod_{k = s}^{\infty} {\hat{H}}_{k}, ⋂_{s = 0}^{\infty} {\hat{H}}^{(s)} = {0}$ .

Denote $\begin{matrix} x_{j} = π (e_{j}), p_{j} = π (e_{n + j}), j = 1, \dots, n; r = π (r_{1}) = \dots = π (r_{n}) . \end{matrix}$ Then $r = p_{j} x_{j} - x_{j} p_{j}$ , $deg p_{j} = deg x_{j} = 1$ , $deg r = 2$ .

In vector spaces ${\hat{H}}_{k}$ we may identify $p_{j}$ , $x_{j}$ and $r$ with operators $- i ℏ \frac{\partial}{\partial x_{j}}$ , $x_{j}$ and $- i ℏ$ respectively. Note that the element $r$ is not a scalar but a generator of the algebra’s center.

Fix $μ, ν \in Z_{+}^{n}$ . Define ${\hat{O}}_{(μ, ν)}$ , a vector space generated by monomials, where $p_{j}$ (resp. $x_{j}$ ) appears $μ_{j}$ (resp. $ν_{j}$ ) times. Define ${\hat{H}}_{(μ, ν)} = π {\hat{O}}_{(μ, ν)}$ . Note that if $n = 1$ , we have grading $\hat{H} = \prod_{(μ, ν) \in Z_{+}^{2}} {\hat{H}}_{(μ, ν)}$ . If $n > 1$ , it is possible that ${\hat{H}}_{(μ, ν)} \cap {\hat{H}}_{(α, β)} \neq {0}$ . For example, $r \in {\hat{H}}_{(1, 1; 0, 0)} \cap {\hat{H}}_{(0, 0; 1, 1)}$ .

An involution in $\hat{O}$ defined by a rule ${(c e_{j_{1}} \otimes \dots \otimes e_{j_{s}})}^{*} = \overline{c} e_{j_{s}} \otimes \dots \otimes e_{j_{1}}$ (here $\overline{c}$ means complex conjugate), is inherited by algebra $\hat{H}$ (since ${\hat{J}}^{*} = \hat{J}$ ). An element $F \in \hat{H}$ is said to be Hermitian, if $F^{*} = F$ .

Let $J_{0}$ be the ideal in $\hat{H}$ generated by $r$ . Then $O = \hat{H} / J_{0}$ can be identified with the algebra $O$ of usual (commutative) formal Taylor series in p and x. The natural projection $Π : \hat{H} \to O$ defines the correspondence between quantum and classical observables.

2.2. Commutator and Poisson bracket

For $F, G \in \hat{O}$ define the commutator $[F, G] = F G - G F$ . Since $Π [F, G] = 0$ , one has $[F, G] \in {\hat{J}}_{0}$ . Hence, $[F, G] = r H_{0}$ , where $H_{0}$ is determined uniquely, since $\hat{O}$ has no non-trivial zero divisors. Define the quantum Poisson bracket $F, G \mapsto [[F, G]] = H_{0}$ . It is easy to check that it satisfies the Leibniz and Jacobi identities: $\begin{array}{l} [[F, G]] H = F [[G, H]] + [[F, H]] G, [[[[F, G]], H]] + [[[[G, H]], F]] + [[[[H, F]], G]] = 0 . \end{array}$

The commutator introduces a structure of a Lie algebra on $\hat{H}$ .

The projection Π conserves the Poisson brackets: $\begin{matrix} Π [[F, G]] = {f, g} \equiv \sum_{j = 1}^{n} \frac{\partial f}{\partial p_{j}} \frac{\partial g}{\partial x_{j}} - \frac{\partial f}{\partial x_{j}} \frac{\partial g}{\partial x_{j}}, f = Π F, g = Π G . \end{matrix}$ Therefore, Π is a homomorphism of associative and Lie algebras.

2.3. Bases and identities

Suppose that $n = 1$ . There are some natural bases in ${\hat{H}}_{(μ, ν)}$ for $n = 1$ , e.g.:

$p x$ -basis: $p^{μ} x^{ν}, r p^{μ - 1} x^{ν - 1}, \dots$ .

$x p$ -basis: $x^{ν} p^{μ}, r x^{μ - 1} p^{ν - 1}, \dots$ .

Weyl basis the sum of the above elements.

Use of these bases helps one to identify

\hat{H}

with graded algebra of jets for Weyl symbols (as in [5,6], etc.) or with jets of

p x

x p

symbols (see [15–17], etc.). However, as pointed out earlier the choice of basis does not have to rely on expansions in powers of

r

Proposition 1.

The monomials $\begin{array}{l} {\hat{σ}}_{s}^{μ, ν} & = x^{ν - s} p^{μ} x^{s}, 0 ⩽ s ⩽ ν if ν ⩽ μ, \\ {\hat{ρ}}_{s}^{μ, ν} & = p^{μ - s} x^{ν} p^{s}, 0 ⩽ s ⩽ μ if μ ⩽ ν \end{array}$ form a basis in ${\hat{H}}_{(μ, ν)}$ .

For each monomial $z \in {\hat{O}}_{(μ, ν)}$ , the element $π z$ is a convex linear combination of ${\hat{σ}}_{s}^{μ, ν}$ or ${\hat{ρ}}_{s}^{μ, ν}$ .

Proof.
See [23]. □

2.4. Automorphisms of $\hat{H}$

A continuous unital homomorphism of algebras $A : \hat{H} \to \hat{H}$ such that $A r = r$ (in other words its restriction on the algebra’s center is identity) is said to be a canonical automorphism of $\hat{H}$ . This implies, in particular, that $A [[F, G]] = [[A F, A G]]$ . Obviously, $A$ is defined uniquely by images of the generators: $p_{1}, \dots, p_{n}, x_{1}, \dots, x_{n}$ . The set of all canonical automorphisms is a group [23]. A canonical automorphism is called Hermitian, if it sends Hermitian elements to Hermitian.

One of the examples of automorphisms are homogeneous ones (i.e. preserving ${\hat{H}}_{k}$ ). In particular, it sends ${\hat{H}}_{1}$ into ${\hat{H}}_{1}$ and can be defined in the basis $(p_{1}, \dots, p_{n}, x_{1}, \dots, x_{n})$ by a symplectic matrix $C \in Sp (2 n, C)$ . This automorphism is Hermitian, if and only if C is real.

A canonical automorphism $A_{0}$ is near-identity, if $A_{0} = Id + A_{1}$ , where $A_{1} {\hat{H}}^{(k)} \subset {\hat{H}}^{(k + 1)}$ for any k. Each canonical automorphism $A$ can be decomposed into a product of a homogeneous and a near-identity automorphisms. An important example of near-identity automorphism is an exponential automorphism. (In fact, one can show that each near-identity automorphism is exponential but we do not need this.)

Proposition 2.

For each $W \in {\hat{H}}^{(3)}$ , the map $\begin{matrix} (1) & F \mapsto e^{t {ad}_{W}} F \equiv \sum_{j = 0}^{\infty} \frac{t^{j}}{j!} {ad}_{W}^{j} F \equiv F + \frac{1}{1!} [[W, F]] t + \frac{1}{2!} [[W, [[W, F]]]] t^{2} + \dots \end{matrix}$ is a near-identity automorphism of $\hat{H}$ . The map is Hermitian, if W is.

If $W \in {\hat{H}}^{(3)}$ and $H \in \hat{H}$ , then function $H (t) = e^{t {ad}_{W}} H$ is a unique solution of Cauchy problem for Heisenberg equation: $\begin{matrix} \dot{H} (t) = [[W, H (t)]], H (0) = H . \end{matrix}$

Proof.
See [23]. □

The classical analog of an exponential map is a canonical transformation generated by a phase flow with Hamiltonian $Π W$ .
3. Normal forms and spectrum

In this section we present an algorithm for computing low lying eigenvalues of a Schrödinger operator by means of a normal form in algebra $\hat{H}$ .

Let us consider an operator in $L^{2} (R^{n})$ : $\begin{matrix} (2) & \hat{H} = \frac{{\hat{p}}^{2}}{2} + V (x), x = (x_{1}, \dots, x_{n}), \hat{p} \equiv (- i ℏ \frac{\partial}{\partial x_{1}}, \dots, - i ℏ \frac{\partial}{\partial x_{n}}) . \end{matrix}$ The real-valued function $V \in C^{\infty} (R^{n})$ is assumed to be bounded from below and raising to infinity with all derivatives not faster than some power of $| x |$ . The operator (2) (understood as a minimal operator, i.e. as a closure of an operator defined by the formula (2) on space $C_{0}^{\infty} (R^{n})$ ) is self-adjoint and semi-bounded from below in $L^{2} (R^{n})$ , and its spectrum is discrete. Let $x_{*} = 0$ be a non-degenerate minimum, $V (x_{*}) = 0$ .

The change of coordinates $x_{j} \sqrt{ω_{j}} \to x_{j}$ reduces the operator to the form $\hat{H} = \sum_{k = 2}^{\infty} {\hat{H}}_{k}$ , where ${\hat{H}}_{2} = \sum_{j = 1}^{n} ω_{j} \frac{{\hat{p}}_{j}^{2} + x_{j}^{2}}{2}$ .

In the next section, we shall implement the first step of the algorithm. Namely, we shall reduce the corresponding element $H \in \hat{H}$ of the formal graded Heisenberg algebra to a normal form.

3.1. Normal form in $\hat{H}$

Suppose we have a Hermitian element $H = \sum_{k = 2}^{\infty} H_{k} \in \hat{H}$ with the quadratic part as follows: $H_{2} = \sum_{j = 1}^{n} ω_{j} \frac{p_{j}^{2} + x_{j}^{2}}{2}$ .

Theorem 1.
For each $m = 3, 4, \dots$ there is a near-identity Hermitian canonical automorphism $A$ that maps H into a normal form, i.e. $\begin{matrix} A H = H_{2} + N + R, [[N, H_{2}]] = 0, N \in ⨁_{k = 3}^{m} {\hat{H}}_{k}, R \in {\hat{H}}^{(m + 1)} . \end{matrix}$
Proof.
See [3]. □

Now we show the normal form reduction procedure in detail for the one-dimensional case. Without loss of generality, we take $ω = 1$ .

Step 0. Use a homogeneous automorphism $A_{0}$ $\begin{matrix} P = \frac{p + i x}{\sqrt{2}}, X = \frac{i p + x}{\sqrt{2}} . \end{matrix}$

The inverse $A_{0}^{- 1}$ is given by formulas: $\begin{matrix} P = \frac{1}{\sqrt{2}} (p - i x), X = \frac{- i p + x}{\sqrt{2}} . \end{matrix}$

Note that if $p = - i ℏ \frac{d}{d x}$ and $x = x$ , then $\begin{matrix} a_{-} = \frac{1}{\sqrt{2 ℏ}} (p - i x), a_{+} = \frac{1}{\sqrt{2 ℏ}} (p + i x) \end{matrix}$ are annihilation and creation operators. So we have a connection: $\begin{matrix} P = \sqrt{ℏ} a_{-}, X = - i \sqrt{ℏ} a_{+} . \end{matrix}$ To make things clearer, we shall denote the generators by $a_{-}$ , $a_{+}$ instead of p, x.

The automorphism $A_{0}$ transforms H into $H^{0} = \sum_{k = 2}^{\infty} H_{k}^{0}$ , where $H_{2}^{0} = \frac{i}{2} (a_{-} a_{+} + a_{+} a_{-})$ .

As usual, on each step of the normal form reduction procedure, we need to solve a homological equation: $[[W, H_{2}]] + F = G$ , i.e. given F, choose W so that G takes the simplest possible form. If $W \in {\hat{H}}_{(μ, ν)}$ , then clearly $[[W, H_{2}]] = i (μ - ν) W$ .

Let us decompose $\hat{H}$ as follows: $\begin{matrix} \hat{H} = N \oplus N^{c}, N = \prod_{ν = 0}^{\infty} {\hat{H}}_{(ν, ν)}, N^{c} = \prod_{μ \neq ν}^{\infty} {\hat{H}}_{(μ, ν)}, \end{matrix}$ and let ${pr}_{N} : \hat{H} \to N$ be the natural projection. Then for each F, there is W such that $[[W, H_{2}]] + F = {pr}_{N} F$ .

Step 1. Choose $A_{3} = exp {ad}_{W_{3}}$ , where $W_{3} \in {\hat{H}}_{3}$ is such that $[[W_{3}, H_{2}^{0}]] + H_{3}^{0} = 0$ . Then $A_{3} H^{0} = H_{2}^{0} + H_{4}^{1} + O_{5} (a_{-}, a_{+})$ ( $R = O_{5} (a_{-}, a_{+})$ means that $R \in {\hat{H}}^{(5)}$ ), where $\begin{matrix} H_{4}^{1} = H_{4}^{0} + \frac{1}{2} [[W_{3}, [[W_{3}, H_{2}^{0}]]]] + [[W_{3}, H_{3}^{0}]] . \end{matrix}$

Step 2. An automorphism $A_{4} = exp {ad}_{W_{4}}$ , where $W_{4} \in {\hat{H}}_{4}$ is chosen so that $\begin{matrix} [[W_{4}, H_{2}^{0}]] + H_{4}^{1} = {pr}_{{\hat{H}}_{(2, 2)}} H_{4}^{1} = G_{4} . \end{matrix}$

Likewise, we go on with the procedure, and arrive at: $\begin{matrix} exp {ad}_{W_{2 N}} \dots exp {ad}_{W_{4}} exp {ad}_{W_{3}} H^{0} = H_{2}^{0} + G + O_{2 k + 1} (a_{-}, a_{+}), \end{matrix}$ where $G + \sum_{k = 2}^{N} G_{2 k}$ , and $G_{2 k} \in {\hat{H}}_{(k, k)}$ . Finally, acting with $A_{0}^{- 1}$ , we get the normal form from Theorem 1.

These steps may be shown on a commutative diagram: $\begin{matrix} \begin{matrix} \hat{H} (p, x) & \overset{B_{3}}{\to} & \hat{H} (p, x) & \overset{B_{4}}{\to} & \hat{H} (p, x) & \overset{}{\to} & \dots \\ ↓ A_{0} & ↓ A_{0} & ↓ A_{0} \\ \hat{H} (a_{-}, a_{+}) & \overset{e^{{ad}_{W_{3}}}}{\to} & \hat{H} (a_{-}, a_{+}) & \overset{e^{{ad}_{W_{4}}}}{\to} & \hat{H} (a_{-}, a_{+}) & \overset{}{\to} & \dots \end{matrix} \end{matrix}$

Here, leftmost downward arrow, gives step 0 of the procedure. The following steps are given by horizontal arrows. Going upwards on the kth step finishes the procedure, giving a Hermitian near-identity automorphism transforming H to a normal form.

Indeed, as shown in [3], $B_{k} = e^{{ad}_{A_{0}^{- 1} W_{k}}}$ , and $W_{k}$ may be chosen such that $A_{0}^{- 1} W_{k}$ is Hermitian. Therefore, $B_{k}$ is also Hermitian.
3.2. Lifting normal form to PDO algebra

Now we introduce a PDO algebra of $x p$ -symbols, and then lift the normal form reduction procedure from $\hat{H}$ to this algebra.

Let $S^{m} (ℏ)$ be a space of $C^{\infty}$ functions $f (p, x, ℏ)$ ( $f : C^{2 n} \times [0, 1] \to C$ ) such that $\begin{matrix} | f^{(α)} (z) | ⩽ C_{α} {(1 + | p | + | x |)}^{m}, | α | = 0, 1, 2, \dots, \end{matrix}$ and let $S^{\infty} (ℏ) = ⋃_{m \in Z} S^{m} (ℏ)$ . To each (symbol) $f \in S^{\infty} (ℏ)$ one can associate a smoothly depending on ℏ continuous operator $\hat{f} : S \to S$ in the Schwartz space $S = S (R_{x}^{n})$ of smooth rapidly decreasing at infinity functions $ψ (x)$ as follows $\begin{matrix} \hat{f} = f (\overset{2}{x}, - i ℏ \overset{1}{\frac{\partial}{\partial x}}, ℏ) \end{matrix}$ where the numbers above operators mean order of action [17]; let us also write $f = Smb (\hat{f})$ .

The space of such operators $Ψ^{\infty} (ℏ)$ is an associative algebra with involution: $\begin{array}{l} (3) & Smb (\hat{f} \hat{g}) = (f ⋆ g) (x, p, ℏ) \overset{def}{=} f (\overset{2}{x}, p - i ℏ \overset{1}{\frac{\partial}{\partial x}}) (g (x, p, ℏ)), \\ (4) & Smb ({(\hat{f})}^{*}) = f^{*} (x, p, ℏ) \overset{def}{=} \overline{f} (\overset{1}{x}, p - i ℏ \overset{2}{\frac{\partial}{\partial x}}) (1), \end{array}$ where $1$ is a constant unity function. Define also a quantum Poisson bracket: $\begin{matrix} [[f, g]] = \frac{i}{ℏ} (f ⋆ g - g ⋆ f) . \end{matrix}$

Let us denote by $Ψ^{\infty} [[- i ℏ]]$ a linear space of formal series $⋃_{m \in Z} S^{m} [[- i ℏ]]$ , where the product and involution are defined as follows: $\begin{array}{l} (5) & {(f ⋆ g)}_{k} (x, p) = \sum_{l + s + | α | = k} \frac{1}{α!} \frac{\partial^{α} f_{l}}{\partial p^{α}} (x, p) \frac{\partial^{α} g_{s}}{\partial x^{α}} (x, p), \\ (6) & {(f^{*})}_{k} (x, p) = \sum_{l + | α | = k} \frac{\partial^{2 | α |} f_{l}}{\partial p^{α} \partial x^{α}} (x, p) . \end{array}$

This is a quotient algebra of $Ψ^{\infty} (ℏ)$ modulo the ideal associated with symbols $f = O (ℏ^{\infty})$ . (The formulas (5) and (6) follow from (3) and (4) after a formal expansion into Taylor series.) Natural projection $π_{1} : Ψ^{\infty} (ℏ) \to Ψ^{\infty} [[- i ℏ]]$ maps Poisson bracket into the bracket as follows $\begin{matrix} {[[f, g]]}_{k} (x, p) = \sum_{\begin{array}{c} l + s + | α | = k + 1 \\ | α | > 0 \end{array}} \frac{1}{α!} [\frac{\partial^{α} f_{l}}{\partial p^{α}} (x, p) \frac{\partial^{α} g_{s}}{\partial x^{α}} (x, p) - \frac{\partial^{α} g_{s}}{\partial p^{α}} (x, p) \frac{\partial^{α} f_{l}}{\partial x^{α}} (x, p)] . \end{matrix}$

The above described algebra $\hat{H}$ is a quotient algebra of $Ψ^{\infty} [[- i ℏ]]$ modulo the ideal associated with symbols having zero Taylor coefficients in $p, x$ at zero. Let us denote natural projections as follows: $\begin{matrix} π_{2} : Ψ^{\infty} [[- i ℏ]] \to \hat{H}, \hat{π} = π_{2} \circ π_{1} : Ψ^{\infty} (ℏ) \to \hat{H} . \end{matrix}$ All these objects are well-defined thanks to Borel’s lemma.

Now we shall lift the normal form reduction procedure from $\hat{H}$ to $Ψ^{\infty} (ℏ)$ . Let us denote by $\hat{N}$ the (uniquely defined) self-adjoint differential operator such that $\hat{π} \hat{N} = N$ .

Theorem 2.
There is a unitary operator $U : Ψ^{\infty} (ℏ) \to Ψ^{\infty} (ℏ)$ such that $\begin{matrix} U^{} \hat{H} U = {\hat{H}}_{2} + \hat{N} + {\hat{R}}_{m + 1} + \hat{Z}, \end{matrix}$ where* $\hat{R}$ is a PDO with symbol $R_{m + 1} (x, p, ℏ)$ such that $\begin{matrix} \frac{\partial^{k + α + β} R_{m + 1}}{\partial ℏ^{k} \partial x^{α} \partial p^{β}} (0, 0, 0) = 0 if 2 k + | α | + | β | ⩽ m, \end{matrix}$ and $\begin{matrix} ‖ \hat{Z} ‖_{L_{2} (R^{n})} ⩽ C_{M} ℏ^{M} \forall M \in Z_{+} . \end{matrix}$
Proof.
See Appendix A. □

3.3. Calculating the spectrum

Let us denote by $\begin{matrix} λ_{ν, 2} (ℏ) = ℏ \sum_{j = 1}^{n} ω_{j} (ν_{j} + \frac{1}{2}), ν \in Z_{+}^{n} \end{matrix}$ the eigenvalues of the operator ${\hat{H}}_{2}$ . Let $V_{ν} (ℏ)$ be the corresponding eigenspaces.

In non-resonant case, i.e. when $ω_{j}$ are independent over $Z$ (particularly, in one-dimensional case), each space $V_{ν} (ℏ)$ is one-dimensional. Since the operator ${\hat{H}}_{2} + \hat{N}$ commutes with ${\hat{H}}_{2}$ , the space $V_{ν} (ℏ)$ is an eigenspace for ${\hat{H}}_{2} + \hat{N}$ as well. Denote its eigenvalues by $λ_{ν, m} (ℏ)$ .

Theorem 2 readily implies

Theorem 3.
Let $σ (\hat{H})$ be the spectrum of $\hat{H}$ , and dist the Euclidean distance from a point to a set on the complex plane. Then $\begin{matrix} dist (λ_{ν, m} (ℏ), σ (\hat{H})) = O (ℏ^{(m + 1) / 2}) . \end{matrix}$
Remark 1.
Note that if $x = 0$ is a global minimum, and the set $V^{- 1} (0)$ is finite, Theorem 3 applied to all the points of $V^{- 1} (0)$ describes all the eigenvalues lying in $[0, C ℏ]$ (see [21]). In case $inf V < 0$ , the procedure also works, but it only describes a spectral series associated with a local minimum (so-called quasi-stationary states).

In what follows we dwell on the one-dimensional case, and set $ω = 1$ . Theorem 3 shows that all the necessary information about spectral asymptotic series is contained in the normal form $H_{2} + N$ . However, for our needs it is more convenient to use $H_{2}^{0} (a_{-}, a_{+}) + G (a_{-}, a_{+})$ (see Section 3.1).

Let us agree to identify elements of ${\hat{H}}_{k}$ with differential operators. Expand all $G_{2 k}$ into primitive bases: $\begin{matrix} (7) & G_{2 k} = \sum_{s = 0}^{k} α_{s}^{k} {\hat{σ}}_{s}^{k, k}, α_{s}^{k} \in C . \end{matrix}$ We immediately find the asymptotic series for the eigenvalues $λ_{ν} (ℏ)$ of the operator $\hat{H}$ : Theorem 4.
The following expansion into asymptotic series is true: $\begin{matrix} (8) & λ_{ν} (ℏ) \sim \frac{ℏ}{2} (2 ν + 1) + \sum_{k = 2}^{\infty} ℏ^{k} E_{k}^{ν}, \end{matrix}$ where $\begin{matrix} (9) & E_{k}^{ν} = k! {(- i)}^{k} \sum_{s = 0}^{k} (\binom{ν + k - s}{k}) α_{s}^{2 k} . \end{matrix}$
Proof.
Indeed, let $ψ_{ν}$ be the eigenfunction of ${\hat{H}}_{2}$ associated with $λ_{ν, 2} (ℏ)$ . From the identities $\begin{matrix} a_{-} ψ_{ν} = - i ν \sqrt{2 ℏ} ψ_{ν - 1}, a_{+} ψ_{ν} = \frac{\sqrt{ℏ}}{\sqrt{2}} ψ_{ν + 1} \end{matrix}$ we easily find how elements of primitive basis act on $ψ_{ν}$ . □

4. Applications

Theorem 5.

If $V (x)$ is analytic at $x = 0$ , then in ( 8 ) $\begin{matrix} | E_{k}^{ν} | ⩽ β γ^{k} k!, for some β, γ > 0 . \end{matrix}$

Proof.

See Appendix B. □

Remark 2.

This fact shows that Theorem 4 may be useful in numerical calculations. Indeed, we do not face a factorial growth of coefficients while computing a normal form, the factor $k!$ appears only in the final formula.

Suppose ${\hat{H}}_{a} = - \frac{ℏ^{2} Δ}{2} + V_{a} (x)$ are two Schrödinger operators ( $a = 1, 2$ ) with potentials satisfying assumptions from Section 3.3 with (at least asymptotic) expansions: $V_{a} (x) \sim \frac{x^{2}}{2} + \sum_{j = 3}^{\infty} u_{a, j} x^{j}$ . Let $λ_{ν}^{a}$ be a series of the eigenvalues for ${\hat{H}}_{a}$ associated with the minimum $x = 0$ . Let us write down their expansions (8) denoting the coefficients $E_{k}^{ν, a}$ .

Here is the inverse spectral result:

Theorem 6.

Assume that $u_{1, 3} \neq 0$ . Fix $0 ⩽ μ < ν$ . Assume that $E_{k}^{μ, 1} = E_{k}^{μ, 2}$ and $E_{k}^{ν, 1} = E_{k}^{ν, 2}$ . Then either $u_{2, j} = u_{1, j}$ ( $\forall j ⩾ 3$ ) or $u_{2, j} = {(- 1)}^{j} u_{1, j}$ ( $\forall j ⩾ 3$ ). In particular, if $V_{a}$ are entire, then either $V_{2} (x) = V_{1} (x)$ or $V_{2} (x) = V_{1} (- x)$ .

We start by calculating a normal form coefficients via Taylor coefficients of a potential (from now on we omit the subscript a):

Lemma 1.

$\begin{matrix} α_{s}^{k} = u_{2 k} Ω_{s}^{k} + u_{2 k - 1} u_{3} Θ_{s}^{k} + {\tilde{α}}_{s}^{k} (u_{3}, u_{4}, \dots, u_{2 k - 2}), \end{matrix}$

where $\begin{matrix} Ω_{s}^{k} = \frac{i^{k} (2 k)!}{2^{2 k} {(k!)}^{2}} (\binom{k}{s}), Θ_{s}^{k} = - \frac{τ_{s}^{k} i^{k} (2 k)!}{3 k 2^{2 k} {(k!)}^{2}} (\binom{k}{s}), \end{matrix}$ and $τ_{s}^{k} = 2 k^{2} + 7 k + 16 s (k - s)$ .

Proof.

See Appendix C. □

Theorem 4 and Lemma 1 easily imply

Corollary 1.

Denote $γ_{k, s}^{ν} = (\binom{ν + k - s}{k}) (\binom{k}{s})$ , then $\begin{matrix} E_{k}^{ν} = \frac{(2 k)!}{2^{2 k} k!} [u_{2 k} \sum_{s = 0}^{k} γ_{k, s}^{ν} - (\sum_{s = 0}^{k} γ_{k, s}^{ν} τ_{s}^{k}) \frac{u_{3} u_{2 k - 1}}{3 k}] + {\tilde{E}}_{k}^{ν}, \end{matrix}$ where ${\tilde{E}}_{k}^{ν}$ depends only on $u_{3}, \dots, u_{2 k - 2}$ .

Define $\begin{matrix} Δ_{k}^{μ, ν} = |\begin{matrix} \sum_{s = 0}^{k} γ_{k, s}^{ν} & \sum_{s = 0}^{k} γ_{k, s}^{ν} τ_{s}^{k} \\ \sum_{s = 0}^{k} γ_{k, s}^{μ} & \sum_{s = 0}^{k} γ_{k, s}^{μ} τ_{s}^{k} \end{matrix}| . \end{matrix}$

Theorem 6 follows from

Lemma 2.

$Δ_{k}^{μ, ν} > 0$ for all $0 ⩽ μ < ν$ and $k ⩾ 2$ .

Let us write $Δ_{k}^{μ, ν}$ in the following form: $\begin{matrix} (10) & Δ_{k}^{μ, ν} = \sum_{0 ⩽ α < β ⩽ ν} ϑ_{α, β}^{k, μ, ν} τ_{α, β}^{k}, τ_{α, β}^{k} = τ_{α}^{k} - τ_{β}^{k}, \end{matrix}$ where $\begin{matrix} ϑ_{α, β}^{k, μ, ν} = γ_{k, α}^{μ} γ_{k, β}^{ν} - γ_{k, β}^{μ} γ_{k, α}^{ν} . \end{matrix}$ Define also $\begin{matrix} {\tilde{ϑ}}_{α, β}^{k, μ, ν} = \frac{ϑ_{α, β}^{k, μ, ν}}{(\binom{k}{α}) (\binom{k}{β})} = (\binom{μ + k - α}{k}) (\binom{ν + k - β}{k}) - (\binom{μ + k - β}{k}) (\binom{ν + k - α}{k}) . \end{matrix}$

Proposition 3.

${\tilde{ϑ}}_{α, β}^{k, μ, ν} > 0$ .

${\tilde{ϑ}}_{α + 1, β + 1}^{k, μ, ν} < {\tilde{ϑ}}_{α, β}^{k, μ, ν}$ .

Proof.

Let us define $\begin{array}{l} ϑ_{+}^{i} = (\binom{μ + k - i - α}{k}) (\binom{ν + k - i - β}{k}), \\ ϑ_{-}^{i} = (\binom{μ + k - i - β}{k}) (\binom{ν + k - i - α}{k}) \end{array}$ for $i = 0, 1$ , and also $κ_{1} = \frac{ϑ_{+}^{0}}{ϑ_{-}^{0}}$ . Then $\begin{array}{l} κ_{1} & = \prod_{r = 1}^{k} \frac{(μ + r - α) (ν + r - β)}{(μ + r - β) (ν + r - α)} = \prod_{r = 1}^{k} (1 + \frac{(β - α) (ν - μ)}{(μ + r - β) (ν + r - α)}) \\ (11) & ⩾ 1 + \frac{(β - α) (ν - μ)}{(μ + k - β) (ν + k - α)} . \end{array}$

In particular, $κ_{1} > 1$ implying 1).

Now $ϑ_{+}^{1} - ϑ_{-}^{1} = A ϑ_{+}^{0} - B ϑ_{-}^{0}$ , where $\begin{matrix} A = \frac{(μ - α) (ν - β)}{(μ + k - α) (ν + k - β)}, B = \frac{(μ - β) (ν - α)}{(μ + k - β) (ν + k - α)} . \end{matrix}$ Hence, $\begin{matrix} \frac{{\tilde{ϑ}}_{α + 1, β + 1}^{k, μ, ν}}{{\tilde{ϑ}}_{α, β}^{k, μ, ν}} = \frac{ϑ_{+}^{1} - ϑ_{-}^{1}}{ϑ_{+}^{0} - ϑ_{-}^{0}} = 1 - \frac{C ϑ_{+}^{0} - D ϑ_{-}^{0}}{ϑ_{+}^{0} - ϑ_{-}^{0}} \equiv 1 - κ_{1}, \end{matrix}$ where $C = {(μ + k - α)}^{- 1} {(ν + k - β)}^{- 1}$ and $D = {(μ + k - β)}^{- 1} {(ν + k - α)}^{- 1}$ . To prove our statement it suffices to show that $κ_{1} > 0$ . But $\begin{matrix} κ_{1} = C - \frac{D - C}{κ - 1} = C - \frac{(ν - μ) (β - α) C D}{κ - 1}, \end{matrix}$ and using the bound (11), we prove 2). □

Proof of Lemma 2.

Let us write $\begin{matrix} Δ = \sum_{α + β < k} ϑ_{α, β}^{k, μ, ν} (β - α) (k - α - β) - \sum_{α + β > k} ϑ_{α, β}^{k, μ, ν} (β - α) (α + β - k) \equiv Σ_{1} - Σ_{2} . \end{matrix}$ (Naturally, in both sums $0 ⩽ α < β ⩽ ν$ .) First, we note that statement 1) of Proposition 3 implies Lemma 2, if $k > μ + ν$ (since $Σ_{2} = 0$ ). Otherwise, we have three possibilities shown on Fig. 1. The values $α, β$ for which $ϑ_{α, β}^{k, μ, ν} \neq 0$ are contained in the domains bounded by the bold lines.

Denote by $D_{\pm}$ the domains containing $α, β$ such that $ϑ_{α, β}^{k, μ, ν}$ is positive (resp. negative). We see that in all cases $D_{+} \supset D_{+}^{0}$ , where $D_{+}^{0}$ is the image of $D_{-}$ under reflection with respect to the line $α + β = k$ (cut off $D_{-}$ by the broken line). It suffices to show that $\begin{matrix} (12) & \sum_{(α, β) \in D_{+}^{0}} ϑ_{α, β}^{k, μ, ν} (β - α) (k - α - β) > \sum_{(α, β) \in D_{-}} ϑ_{α, β}^{k, μ, ν} (β - α) (α + β - k) . \end{matrix}$

Indeed, let us take two points: $(α, β) \in D_{+}^{0}$ , and also the symmetric one $(k - β, k - α) \in D_{-}$ . Then the associated terms in (12) are $\begin{matrix} {\tilde{ϑ}}_{α + r, β + r}^{k, μ, ν} (\binom{k}{α}) (\binom{k}{β}) (β - α) (k - β - α), \end{matrix}$ where one should take $r = 0$ for the LHS, and $r = k - α - β$ for the RHS. Using statement 2) of Proposition 3, we establish (12) and conclude the proof. □

Fig. 1.

Signs of $ϑ_{α + r, β + r}^{k, μ, ν}$ in three cases. The line $α + β = k$ divides $D_{+}$ and $D_{-}$ , the broken curve cuts out $D_{+}^{0}$ .

Footnotes

Acknowledgements

The author thanks D.V. Treschev for many useful discussions and V.E. Nazaikinskii for suggesting the idea to prove Theorem . This work was supported by the Russian Foundation for Basic Research (Grant No. 15-01-03747) and the Program for Supporting Leading Scientific Schools (Grant No. NSh-2964.2014.1).

Proof of Theorem 2

Induction in m. Let us consider an operator and its projection onto $\hat{H}$ : $\begin{array}{l} \hat{H} = {\hat{H}}_{2} + \sum_{j = 3}^{m - 1} {\hat{N}}_{j} + {\hat{R}}_{m} + \hat{Z} \in Ψ^{\infty} (ℏ), [\hat{N_{j}}, {\hat{H}}_{2}] = 0, \\ π \hat{H} \equiv H = H_{2} + \sum_{j = 3}^{m - 1} N_{j} + R_{m}, [[N_{j}, H_{2}]] = 0, R_{m} = O_{m} (p, x), \end{array}$ and perform one step of the normal form reduction, i.e. find W such that $\begin{matrix} e^{{ad}_{W}} H = H_{2} + \sum_{j = 3}^{m} N_{j} + R_{m + 1}, [[N_{m}, H_{2}]] = 0, R_{m + 1} = O_{m + 1} (p, x) . \end{matrix}$

Now we choose any $W_{2} \in {\hat{π}}^{- 1} (W)$ , and take $\hat{W} = \frac{1}{2} ({\hat{W}}_{1} + {\hat{W}}_{1}^{*})$ , where $W_{1} (x, p, ℏ) = χ (x, p) W_{2}$ and $χ (x, p)$ is a cut-off function equal to unity near zero. Then $\hat{W}$ is a bounded self-adjoint operator in $L_{2}$ , and its symbol $W (x, p, ℏ)$ is compactly supported up to $O (ℏ^{\infty})$ . Hence, the operator $e^{\frac{i}{ℏ} t \hat{W}}$ preserves $S$ for all $t \in R$ .

Let us solve a Cauchy problem in algebra $Ψ^{\infty} [[- i ℏ]]$ $\begin{matrix} (13) & \dot{B} (t) = [[π_{1} (\hat{W}), B]], B (0) = π_{1} (\hat{H}) . \end{matrix}$ (In fact, this is equivalent to solving a sequence of ordinary linear equations along the trajectories of Hamiltonian system with compactly supported Hamiltonian $W (x, p, 0)$ .) Since $H (t) = e^{t {ad}_{W}} (H) \in \hat{H}$ is a solution of a Cauchy problem: $\begin{matrix} (14) & \dot{H} (t) = [[W, H (t)]], H (0) = H, \end{matrix}$ $π_{2}$ preserves the Poisson bracket, and both problems (13) and (14) have unique solutions, it follows that $π_{2} B (t) = H (t)$ for all $t \in R$ . Then any $\hat{G} (t) = π_{1}^{- 1} B (t) \in Ψ^{\infty} (ℏ)$ is a solution of $\begin{matrix} - i ℏ \frac{\partial}{\partial t} \hat{G} (t) = [\hat{W}, \hat{G} (t)] + Y (t), \hat{G} (0) = \hat{H}, \end{matrix}$ where $‖ Y (t) ‖_{L_{2}} ⩽ C_{M} ℏ^{M}$ for all $M \in Z_{+}$ (for brevity $Y = O (ℏ^{\infty})$ ). Then $\hat{G} (t) = e^{\frac{i}{ℏ} t \hat{W}} \hat{H} e^{- \frac{i}{ℏ} t \hat{W}} + Y_{1} (t)$ , where $Y_{1} = O (ℏ^{\infty})$ . On the other hand, since $\hat{π} (\hat{G} (t)) = H (t)$ , one has $\hat{G} (t) = {\hat{H}}_{2} + {\hat{N}}_{3} (t) + \dots + {\hat{N}}_{m} (t) + {\hat{R}}_{m + 1} (t) + Y_{2} (t)$ , where $R_{m + 1}$ is as required in the statement of theorem, and $Y_{2} = O (ℏ^{\infty})$ . Equating these two expressions finishes the proof.

Analyticity and proof of Theorem 5

At first, we shall need a definition of an analytic element in $\hat{H}$ .

Let us endow the finite-dimensional vector space ${\hat{H}}_{k}$ with a norm. For any element $F \in {\hat{O}}_{k} = \sum_{deg z = k} F_{z} z$ (where the sum is taken over monomials $z \in {\hat{O}}_{k}$ ) it is natural to put $‖ F ‖ : = \sum_{deg z = k} | F_{z} |$ . Further, for any $F \in {\hat{H}}_{k}$ , we say that $\begin{matrix} ‖ F ‖ : = inf_{G = π^{- 1} F} ‖ G ‖ . \end{matrix}$

It can be easily established that all the axioms of norm hold. In particular, if $z_{1}, \dots, z_{N}$ are monomials that form a basis in ${\hat{H}}_{k}$ , then $‖ z_{j} ‖ = 1$ , and also for any $F \in {\hat{H}}_{k}$ $\begin{matrix} (15) & ‖ F ‖ ⩽ \sum_{l = 1}^{N} | f_{l} |, where F = \sum_{l = 1}^{N} f_{l} z_{l} . \end{matrix}$

Before we define a concept of analyticity, we pay attention to one remarkable property of the primitive bases. Namely, it was shown in [23] that the primitive basis ${\hat{ρ}}_{s}^{μ, ν}$ or ${\hat{σ}}_{s}^{μ, ν}$ is optimal in a sense that the inequality (15) becomes an equality. (Actually, this is a simple consequence of Proposition 1.)

To illustrate this fact let us consider an example: the two-dimensional space ${\hat{H}}^{(2, 1)}$ . In a basis: $p^{2} x, x p^{2}$ (which is primitive) any monomial decomposes into a convex linear combination. (Actually, the only non-trivial expansion is $p x p = \frac{1}{2} p^{2} x + \frac{1}{2} x p^{2}$ .) On the other hand, for a different basis, say, $p^{2} x$ and $p x p$ there are non-convex expansions: $x p^{2} = 2 p x p - p^{2} x$ . Thus, the equality is implemented in (15) for the primitive bases, and is generally not for some other basis.

An element $F \in \hat{H}$ is said to be analytic, if there are constants $β, γ > 0$ such that $\begin{matrix} ‖ F_{k} ‖ ⩽ β γ^{k}, where F = \sum_{k = 0}^{\infty} F_{k}, F_{k} \in {\hat{H}}_{k} . \end{matrix}$ It is proved in [2] that (in case $n = 1$ ) the normal form of an analytic element $H = \sum_{k = 2}^{\infty} H_{k}$ is also analytic. This theorem is a non-commutative analog of the following fact from classical mechanics: a normal form and a transformation to a normal form for a one degree of freedom analytic Hamiltonian is also analytic (or convergent). The main reason for the convergence is the absence of small denominators for the case of one degree freedom.

Now we can readily prove Theorem 5. Since the normal form is analytic, its coefficients of expansion into primitive basis (7) satisfy the following estimate: $‖ G_{2 k} ‖ = \sum_{s = 0}^{k} | α_{s}^{k} | ⩽ β_{0} γ_{0}^{2 k}$ , and hence (9) can be estimated as follows: $\begin{matrix} | E_{k}^{ν} | ⩽ k! (\binom{ν + k}{k}) \sum_{s = 0}^{k} | α_{s}^{2 k} | ⩽ k! β_{1} k^{ν} β_{0} γ_{0}^{2 k} ⩽ k! β γ^{k} . \end{matrix}$

It is important to note that the usage of the primitive basis was essential for the argument. Indeed, assume that the normal form (a uniquely defined object) is expanded into a different basis, e.g. $p x$ basis (see Section 2.3), with coefficients ${\tilde{α}}_{s}^{k}$ . Then one has an inequality $‖ G_{2 k} ‖ ⩽ \sum_{s = 0}^{k} | {\tilde{α}}_{s}^{k} |$ , which does not imply the needed one: $\sum_{s = 0}^{k} | {\tilde{α}}_{s}^{k} | ⩽ β_{0} γ_{0}^{2 k}$ . Quite the contrary, easy calculations (for details see [23]) show that generically $\sum_{s = 0}^{k} | {\tilde{α}}_{s}^{k} | \sim k!$ .

Proof of Lemma 1

We shall need some technical statements.

We shall need the non-commutative analog of the binomial theorem:

Decompose a normal form term as follows $N_{2 k} = N_{2 k}^{0} + N_{2 k}^{1}$ , where $\begin{matrix} N_{2 k}^{0} = {pr}_{N} (H_{2 k} + [[W_{3}, H_{2 k - 1}]]), N_{2 k}^{1} = N_{2 k}^{1} (u_{3}, \dots, u_{2 k - 2}) . \end{matrix}$

Now, from Proposition 6 $\begin{matrix} {pr}_{N} H_{2 k} = \frac{u_{2 k}}{2^{k}} {pr}_{N} {(a_{+} + i a_{-})}^{2 k} = \frac{u_{2 k} i^{k}}{2^{2 k}} (\binom{2 k}{k}) \sum_{s = 0}^{k} (\binom{k}{s}) {\hat{σ}}_{s}^{k, k} \end{matrix}$ implying the needed formula for $Ω_{s}^{k}$ .

Further, using the expansion $H_{m} = \sum_{μ + ν = m} H_{μ, ν}$ , where $H_{μ, ν} \in {\hat{H}}_{(μ, ν)}$ and $W_{3} = \sum_{α + β = 3} \frac{1}{i (β - α)} H_{α, β}$ , we get $\begin{matrix} {pr}_{N} [[W_{3}, H_{2 k - 1}]] = \frac{u_{3} u_{2 k - 1} i^{k}}{2^{k + 1}} \sum_{α + β = 3} \frac{(\binom{2 k - 1}{k + 1 - α}) (\binom{3}{α})}{β - α} [[Σ_{α, β}, Σ_{k + 1 - α, k + 1 - β}]] \end{matrix}$ and then by Proposition 7 $\begin{matrix} (19) & \begin{matrix} {pr}_{N} [[W_{3}, H_{2 k - 1}]] = - \frac{u_{3} u_{2 k - 1} i^{k} (\binom{2 k - 1}{k - 1})}{2^{k}} {\tilde{N}}_{2 k}, \\ {\tilde{N}}_{2 k} = \frac{k - 1}{3 (k + 1)} [[Σ_{k + 1, k - 2}, Σ_{0, 3}]] + 3 [[Σ_{k, k - 1}, Σ_{1, 2}]] . \end{matrix} \end{matrix}$

Now we plug the following expansion from Proposition 6 $\begin{array}{l} [[Σ_{k + 1, k - 2}, Σ_{0, 3}]] = \frac{1}{2^{k - 2}} \sum_{s = 0}^{k - 2} (\binom{k - 2}{s}) [[{\hat{σ}}_{s}^{k + 1, k - 2}, a_{+}^{3}]], \\ [[Σ_{k, k - 1}, Σ_{1, 2}]] = \frac{1}{2^{k - 1}} \sum_{s = 0}^{k - 1} (\binom{k - 1}{s}) [[{\hat{σ}}_{s}^{k, k - 1}, a_{+} a_{-} a_{+}]] \end{array}$ into (19) and then use Proposition 5: $\begin{array}{l} {\tilde{N}}_{2 k} = & \frac{1}{2^{k - 1}} (\frac{2}{3} (k - 1) \sum_{s = 0}^{k - 2} (\binom{k - 2}{s}) ({\hat{σ}}_{s + 2}^{k, k} + {\hat{σ}}_{s + 1}^{k, k} + {\hat{σ}}_{s}^{k, k}) \\ (20) & + 3 \sum_{s = 0}^{k - 1} (\binom{k - 1}{s}) ((k - s) {\hat{σ}}_{s + 1}^{k, k} + (s + 1) {\hat{σ}}_{s}^{k, k})), \end{array}$ whence $\begin{matrix} Θ_{s}^{k} = - \frac{{\tilde{τ}}_{s}^{k} i^{k}}{2^{2 k - 1}} (\binom{2 k - 1}{k - 1}) = - \frac{{\tilde{τ}}_{s}^{k} i^{k} (2 k)!}{{((2 k)!!)}^{2}}, \end{matrix}$ where $\begin{array}{l} {\tilde{τ}}_{s}^{k} = & \frac{2}{3} (k - 1) ((\binom{k - 2}{s}) + (\binom{k - 2}{s - 1}) + (\binom{k - 2}{s - 2})) \\ + 3 ((k - s - 1) (\binom{k - 1}{s}) + (s + 1) (\binom{k - 1}{s - 1})) \\ = & (\binom{k}{s}) (\frac{2}{3} (k^{2} - k - s (k - s)) + 3 (k + 2 s (k - s))) = (\binom{k}{s}) \frac{τ_{s}^{k}}{3 k} . \end{array}$

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Non-commutative normal form,spectrum and inverse problem

Abstract

Keywords

1. Introduction

2. Formal graded Heisenberg algebra

2.1. Algebra H ˆ

2.2. Commutator and Poisson bracket

2.3. Bases and identities

3.1. Normal form in H ˆ

Footnotes

Acknowledgements

Proof of Theorem 2

Analyticity and proof of Theorem 5

Proof of Lemma 1

References

2.1. Algebra $\hat{H}$

3.1. Normal form in $\hat{H}$