Spectral asymptotics for the Schrödinger operator with a non-decaying potential

Abstract

We study Schrödinger operators $H (h) = - h^{2} Δ + V (x)$ acting in $L^{2} (R^{n})$ for non-decaying potentials V. We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold $max W$ and Mourre estimates in the range of W except at its critical values.

Keywords

Stark Schrödinger operator spectral shift function asymptotic expansions

1. Introduction

The main purpose of this paper is to study the spectral shift function (SSF for short) for the Schrödinger operators in $L^{2} (R^{n})$ $\begin{array}{l} (1.1) & H_{j} (h) = - h^{2} Δ + V_{j} (x), j = 1, 2, \end{array}$ where h is a small parameter and $V_{1}$ , $V_{2}$ are real-valued bounded smooth potentials with difference $V_{2} (x) - V_{1} (x)$ of order $O (| x |^{- ρ})$ as $| x | \to \infty$ for some $ρ > n$ . The SSF $ξ (λ, h)$ is defined as a distribution on $R_{λ}$ by the relation $\begin{array}{l} (1.2) & ⟨ ξ^{'} (\cdot, h), f ⟩ : = - tr (f (H_{2} (h)) - f (H_{1} (h))), \forall f \in C_{0}^{\infty} (R; R), \end{array}$ with a normalization condition $ξ (λ, h) = 0$ for $λ < inf (σ (H_{1} (h)) \cup σ (H_{2} (h)))$ . The SSF plays an important role in perturbation theory for self-adjoint operators. When $f (H_{1} (h)) = 0$ , it coincides with the eigenvalue counting function of $H_{2}$ for $λ \in supp f$ . It was introduced in a special case by I. Lifshitz [20] and generalized by M. Krein in [2, 19]. The background of the SSF theory can be found in [33].

In the last thirty years, the asymptotic behavior of the SSF of the Schrödinger operator with a long-range or short-range potential has intensively been studied in different aspects. In the semi-classical regime, $h ↘ 0$ , the Weyl type asymptotics of $ξ (\cdot, h)$ with sharp remainder estimate has been obtained (see [6, 7, 30] and the references given therein). On the other hand, a complete asymptotic expansion in powers of h of $ξ (\cdot, h)$ has been obtained for non-trapping energies λ (see [3–5, 30, 31]) i.e. for energies at which any hamiltonian flow of the underlying classical mechanics tends to ∞ as time tends to $\pm \infty$ . Similar results are well-known for the SSF in the high energy regime, $h = 1$ and $λ \to \infty$ (see [6, 7, 23, 26, 29–31]). In [29, 31], it was established that the leading terms of the asymptotic behavior of $ξ (λ, 1)$ as $λ \to + \infty$ only depends on the average value of $V_{2} - V_{1}$ . The proof of all the above results follows from a beautiful local trace formula in the configuration space due to D. Robert (see Theorem 1.10 in [31]). However, the proof of this local trace formula, based on the construction of a long time parametrix for time-dependent Schrödinger equation, involves the decay assumptions for both potentials $V_{1}$ and $V_{2}$ .

The relation between the asymptotics of the SSF and resonances was first investigated by R. Melrose [22], and then by many authors with successive extensions (see [24] and the references given therein). All these works use the scattering theory. In [32], J. Sjöstrand proposed a new approach based on the complex scaling of operators. The scattering determinant is replaced by $D (z, h) = det (I + K (z, h))$ , where $K (z, h)$ is a trace class operator whose zeros are the resonances (see Section 4). Applying this approach, V. Bruneau and V. Petkov established in [4] a representation of the derivative of SSF as a sum of harmonic measures related to resonances.

There are only few works treating the SSF of the Schrödinger operator with non-decaying potentials, such as those homogeneous of degree zero, periodic or even of logarithmic decay.

In [8], the first author established a trace formula relating the SSF and the resonances of the periodic Schrödinger operator with slowly varying perturbation $W (h x)$ . Using the Peierls substitution method, he reduced the spectral study of the perturbed operator to the study of the semiclassical operator $E (h D_{x}) + W (x)$ for a band function $E (k)$ describing the Floquet spectrum of the non-perturbed operator. Unfortunately, this method fails at high energy, since the band functions are not smooth due to the degeneracy of the Floquet eigenvalues.

The spectral and scattering theory for Schrödinger operator with a homogeneous potential of degree zero was investigated in [12] (see also [13] and the references there). The asymptotics of the number of eigenvalues for a perturbation of such an operator below the essectial spectrum was studied in [27]. To our best knowledge, the SSF has not been treated.

The aim of this paper is to fill this gap. We consider Schrödinger operators with non decaying potentials including in particular homogeneous ones of degree zero.

In the first sections, we study the high energy asymptotics of the SSF. In Section 2, we compute the trace formulas (Theorem 2.1) and the explicit coefficients of all order of the weak asymptotic expansion in powers of $λ^{- 1}$ of $ξ^{'} (λ, 1)$ as $λ \to \infty$ (Corollary 2.2). The k-th coefficient is given by the integral of the difference of a polynomial of degree k with respect to the potential and its derivatives as suggested in [29, 31]. For this we only use a standard pseudodifferential calculus combined with some commutator formulas for $H_{0} = - Δ$ (see Section 6 for the proof, and also [21, 25]).

In Section 3, we give a strong sense to this expansion for potentials homogeneous of degree zero, or those analytic and bounded in a complex sector at infinity (Corollary 3.3). For such a potential V, say for homogeneous one, the operator $H = - Δ + V (x)$ is unitarily equivalent to $H_{θ} : = U_{θ} H U_{- θ} = - e^{- 2 θ} Δ + V (x)$ , where $U (θ) f (x) = e^{n θ / 2} f (e^{θ} x)$ is a dilation operator in $L^{2} (R^{n})$ for real θ. The operator $- e^{- 2 θ} Δ + V (x)$ is analytic for $θ \in C$ . The uniqueness of analytic continuation implies the invariance of the SSF under complex dilation. Since the resolvent is continued analytically to the lower half plane after a complex dilation with $ℑ θ > 0$ , a representation formula of $ξ^{'}$ in terms of the resolvent (Lemma 3.4) enables us to show the analyticity and a polynomial estimate of $ξ^{'}$ as well as its derivatives (Theorem 3.2). It is now classical to deduce the strong full asymptotic expansion from the weak one using these estimates and Lemma 3.5.

Next we study the semiclassical asymptotics. We will restrict our attention to potentials V homogeneous of degree zero at infinity (i.e., there exists W independent of $| x |$ such that $| V (x) - W (x) | \to 0$ as $| x | \to \infty$ ). The essential spectrum of $U_{θ} H (h) U_{- θ}$ is the union of the semi-axis $t + e^{- 2 θ} \overline{R_{+}}$ over t in the range $W (S^{n - 1})$ , a band in the lower half plane intersecting with $R$ on $W (S^{n - 1})$ when $ℑ θ > 0$ . In Section 4, we consider the SSF for λ above the range $W (S^{n - 1})$ , and generalize the result of V. Bruneau and V. Petkov [3] proving a representation of $ξ^{'} (λ, h)$ in terms of the resonances (Theorem 4.1). We apply this result to establish a Weyl-type formula for the SSF with optimal remainder estimate $O (h^{1 - n})$ . Moreover, under resonance free domain condition, we give a complete asymptotic expansion of $ξ^{'} (λ, h)$ .

Finally, in Section 5, we consider λ in the range $W (S^{n - 1})$ and prove a semi-classical Mourre estimate away from critical values of $W_{| S^{n - 1}}$ . This is a semiclassical version of S. Agmon, J. C.-Sampedro and I. Herbst [1] (Appendix C). The proof is based on a construction of an escape function for Schrödinger operators adapted to a homogeneous potential.

Notations: Throughout this paper, h is an asymptotic positive parameter going to zero. We use $f_{h} = O (h^{N})$ to denote an h-dependent function that is bounded in magnitude by an expression $C_{N} h^{N}$ , where the implied constant $C_{N}$ is independent of h but may depend on parameters independent of h. Similarly, we use $f_{h} = O (h^{\infty})$ or $f_{h} \equiv 0$ to denote the estimate $| f | ⩽ C_{N} h^{N}$ for every N. For any quantity $a_{j}$ defined for each $j = 1, 2$ concerning $H_{j}$ , we sometimes denote their difference $a_{2} - a_{1}$ by ${[a_{\cdot}]}_{1}^{2}$ .

2. Weak asymptotics

In this and the next sections, we study the high-energy asymptotics of $ξ (λ) = ξ (λ, 1)$ . Let $H_{1}, H_{2}$ denote the operators (1.1) with $h = 1$ with potentials satisfying

$V_{j}$ are real-valued smooth functions and there exists $ρ > n$ such that for all $α \in N^{n}$ $\begin{array}{l} (2.1) & \partial_{x}^{α} V_{j} (x) = O (1), j = 1, 2, \partial_{x}^{α} (V_{2} (x) - V_{1} (x)) = O (| x |^{- ρ}) as | x | \to \infty . \end{array}$

Theorem 2.1.
Assume (A1). Then the following full asymptotic expansion holds as $h ↘ 0$ : $\begin{array}{l} (2.2) & tr {[f (h^{2} H_{\cdot})]}_{1}^{2} \sim \sum_{k = 1}^{\infty} c_{k} (f) h^{2 k - n}, \end{array}$ for every $f \in C_{0}^{\infty} (] 0, + \infty [; R)$ , with $\begin{array}{l} (2.3) & c_{k} (f) = \frac{n κ_{0}}{k! {(2 π)}^{n}} \int_{0}^{\infty} f^{(k)} (r^{2}) r^{n - 1} d r \int_{R^{n}} P_{k} (x) d x . \end{array}$ Here $κ_{0}$ is the measure of the unit ball in $R^{n}$ , and $\begin{array}{l} P_{1} (x) = {[V_{\cdot}]}_{1}^{2} : = V_{2} - V_{1}, P_{k} (x) = {[P_{k} ({D^{α} V_{\cdot}}_{| α | ⩽ 2 k - 4})]}_{1}^{2} for k ⩾ 2, \end{array}$ where $P_{j}$ is a universal polynomial of degree j. In particular, $\begin{array}{l} (2.4) & P_{2} = V^{2}, P_{3} = V^{3} - \frac{1}{2} V Δ V, \\ (2.5) & P_{4} = V^{4} + \frac{3}{5} V Δ^{2} (V) + \frac{4}{5} V^{2} (Δ V) - \frac{2}{5} V Δ (V^{2}) + \frac{2}{5} V | \nabla V |^{2} . \end{array}$

This theorem will be proved in Section 6.

For f in $C_{0}^{\infty} (] 0, + \infty [)$ , a change of variable and integration by parts yield $\begin{array}{l} (2.6) & \int_{0}^{\infty} f^{(k)} (r^{2}) r^{n - 1} d r = \frac{{(- 1)}^{k}}{2} \frac{Γ (\frac{n}{2})}{Γ (\frac{n}{2} - k)} \int_{- \infty}^{\infty} f (λ) λ^{\frac{n}{2} - k - 1} d λ, \end{array}$ with the convention that $Γ {(- m)}^{- 1} = 0$ for $m \in N : = {0, 1, \dots}$ , and hence $\begin{array}{l} (2.7) & c_{k} (f) = - a_{k} ⟨ λ^{\frac{n}{2} - k - 1}, f ⟩, \end{array}$ where $\begin{array}{l} (2.8) & a_{k} = ω_{k} \int_{R^{n}} P_{k} (x) d x, ω_{k} = \frac{{(- 1)}^{k + 1}}{2} \frac{n κ_{0}}{k! {(2 π)}^{n}} \frac{Γ (\frac{n}{2})}{Γ (\frac{n}{2} - k)} . \end{array}$ On the other hand, from (1.2) we have $\begin{array}{l} tr {[f (h^{2} H_{\cdot})]}_{1}^{2} = - \int_{- \infty}^{+ \infty} ξ^{'} (λ, H_{2}, H_{1}) f (h^{2} λ) d λ = - ⟨ \frac{1}{h^{2}} ξ^{'} (\frac{\cdot}{h^{2}}), f (\cdot) ⟩ . \end{array}$ As a consequence of Theorem 2.1 and (2.7), we have the weak asymptotics of $ξ^{'} (λ)$ as $λ \to + \infty$ . Corollary 2.2.
For λ large enough, the asymptotic expansion $\begin{array}{l} (2.9) & ξ^{'} (λ) \sim \sum_{k = 1}^{\infty} a_{k} λ^{\frac{n}{2} - k - 1} \end{array}$ holds in the sense of distribution, where $a_{k}$ are given by ( 2.8 ). In particular, modulo $O (λ^{- \infty})$ , $ξ^{'} (λ)$ is a polynomial of degree $\frac{n}{2} - 2$ when $n ⩾ 4$ is even.

3. High energy asymptotics

In this section, we suppose the following analyticity condition in addition to (A1) that

There exist $c, C > 0$ such that the functions: $] - c, c [\times {| x | ⩾ C} ∋ (θ, x) \mapsto V_{j} (e^{θ} x)$ , $j = 1, 2$ have an analytic extention on θ to a complex disk $D (c) : = {θ \in C, | θ | ⩽ c}$ , and the estimate (2.1) holds for $x \mapsto V_{j} (e^{θ} x)$ uniformly for all $θ \in D (c)$ .

Remark 3.1.
Potentials $V_{j} \in C^{\infty} (R^{n}; R)$ which extend analytically in a complex angular domain ${| ℜ x | ⩾ C, | ℑ x | ⩽ δ | x |}$ with estimates $V_{j} (x) = O (1)$ , $V_{2} (x) - V_{1} (x) = O ({⟨ x ⟩}^{- ρ})$ there satisfy (A1) and (A1).

Our main results of this section are the followings: Theorem 3.2.
Under (A1) and (A1), there exists $λ_{0}$ such that $ξ (λ)$ is an analytic function in $] λ_{0}, + \infty [$ and for every $N \in N$ there exists $C_{N}$ such that for $m > n / 2$ we have $\begin{array}{l} (3.1) & | ξ^{(N + 1)} (λ) | ⩽ C_{N} λ^{m - N - 1}, \end{array}$ uniformly for $λ \in [λ_{0}, + \infty [$ .
Corollary 3.3.
Under (A1) and (A1), we have for every integer N $\begin{array}{l} (3.2) & lim_{λ \to + \infty} λ^{N + 1 - \frac{n}{2}} [ξ^{'} (λ) - \sum_{k = 1}^{N} a_{k} λ^{\frac{n}{2} - k - 1}] = 0, \end{array}$ where the coefficients $a_{k}$ are given by ( 2.8 ).

3.1. Proof of Theorem 3.2

Let $H_{j}, j = 1, 2$ be two operators satisfying (A2). From now on we fix $z_{0} ≪ inf (σ (H_{1}) \cup σ (H_{2}))$ independent on h.

Fix an integer $m > n / 2$ so that the operator $G (z) : = {[{(z - H_{j})}^{- 1} {(H_{j} - z_{0})}^{- m}]}_{1}^{2}$ is of trace class (we recall the notation ${[a_{j}]}_{1}^{2} = a_{2} - a_{1}$ ). To see this, we write $\begin{array}{l} (3.3) & G (z) = ({(z - H_{2})}^{- 1} - {(z - H_{1})}^{- 1}) {(H_{2} - z_{0})}^{- m} + {(z - H_{1})}^{- 1} {[{(H_{j} - z_{0})}^{- m}]}_{1}^{2} = I + II . \end{array}$ By Theorem A.5, $H_{j} - z_{0}$ is elliptic, hence the symbol of its inverse is in $S (R^{2 n}, (1 + | ξ |^{2}))$ . Combining this with (2.1) and Theorem A.3 we deduce that $(V_{2} - V_{1}) {(H_{2} - z_{0})}^{- m}$ is of trace class for $m > n / 2$ . Therefore, $I = {(z - H_{1})}^{- 1} (V_{2} - V_{1}) {(H_{2} - z_{0})}^{- m} {(z - H_{2})}^{- 1}$ is of trace class. Now, the $m - 1$ -th derivatives of the resolvent identity implies that ${(z_{0} - H_{1})}^{- m} - {(z_{0} - H_{2})}^{- m}$ is a linear combination of terms of the form ${(z_{0} - H_{1})}^{- j} (V_{2} - V_{1}) {(z_{0} - H_{2})}^{- (m + 1 - j)}$ with $1 ⩽ j ⩽ m$ . This shows that II is also of trace class.

Let $z_{0}$ be in $ρ (H_{1}) \cap ρ (H_{2}) \cap R$ and introduce the function $\begin{array}{l} (3.4) & σ_{\pm} (z) = {(z - z_{0})}^{m} tr {[{(z - H_{\cdot})}^{- 1} {(H_{\cdot} - z_{0})}^{- m}]}_{1}^{2}, \pm ℑ z > 0 . \end{array}$ First, we give a representation formula of $ξ^{'} (λ)$ in terms of $σ_{+}$ .

Lemma 3.4.
In the sense of distribution, we have $\begin{array}{l} ξ^{'} (λ) = \frac{1}{π} ℑ σ_{+} (λ + i 0) . \end{array}$ More precisely, for all $f \in C_{0}^{\infty} (R)$ , we have $\begin{array}{l} ⟨ ξ^{'}, f ⟩ = lim_{ϵ ↘ 0} \frac{1}{π} \int f (λ) ℑ σ_{+} (λ + i ϵ) d λ, \end{array}$ where the limit is taken in the sense of distribution.
Proof.
Let $f \in C_{0}^{\infty} (R)$ and let $\tilde{f} \in C_{0}^{\infty} (C)$ be an almost analytic extension of f. According to the formula (6.2), we have $\begin{array}{l} (3.5) & tr {[f (H_{\cdot})]}_{1}^{2} = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) {(z - z_{0})}^{m} \times tr {[{(H_{\cdot} - z_{0})}^{- m} {(z - H_{\cdot})}^{- 1}]}_{1}^{2} L (d z) . \end{array}$ Since we have $σ_{\pm} (z) = O (| ℑ z |^{- 1})$ and ${\bar{\partial}}_{z} \tilde{f} = O (| ℑ z |^{\infty})$ , we may write the right hand side of the above identity as $\begin{array}{l} ⟨ ξ^{'}, f ⟩ & = - tr {[f (H_{\cdot})]}_{1}^{2} \\ = lim_{ϵ ↘ 0} \frac{1}{π} (\int_{ℑ z > 0} {\bar{\partial}}_{z} \tilde{f} (z) σ_{+} (z + i ϵ) L (d z) + \int_{ℑ z < 0} {\bar{\partial}}_{z} \tilde{f} (z) σ_{-} (z - i ϵ) L (d z)) . \end{array}$ The function $σ_{+} (z + i ϵ)$ (resp. $σ_{-} (z - i ϵ)$ ) is holomorphic on the complex domain ${z \in C : ℑ z > 0}$ (resp. ${z \in C : ℑ z < 0}$ ). Thus applying the Green’s formula we obtain $\begin{array}{l} ⟨ ξ^{'}, f ⟩ = lim_{ϵ ↘ 0} \frac{1}{2 π i} \int f (λ) (σ_{+} (λ + i ϵ) - σ_{-} (λ - i ϵ)) d λ . \end{array}$ Using the above formula and the fact that $σ_{-} (λ - i ϵ) = \overline{σ_{+} (λ + i ϵ)}$ we get the lemma. □

Now we prove Theorem 3.2. For $θ \in R$ set for $j = 1, 2$ , $\begin{array}{l} H_{j, θ} = - e^{- 2 θ} Δ + V_{j} (e^{θ} x) . \end{array}$ The operator ${(z - H_{j})}^{- 1} {(H_{j} - z_{0})}^{- m}$ is unitarily equivalent to ${(z - H_{j, θ})}^{- 1} {(H_{j, θ} - z_{0})}^{- m}$ for real θ. Consequently, the cyclicity of the trace yields $\begin{array}{l} (3.6) & σ_{+} (z) = {(z - z_{0})}^{m} tr {[{(z - H_{., θ})}^{- 1} {(H_{., θ} - z_{0})}^{- m}]}_{1}^{2}, \end{array}$ for all $z \in C_{+} = {z \in C; ℑ z > 0}$ and $θ \in D (c) \cap R$ .

Fix $δ > 0$ , and let $z \in C_{δ} = {z \in C; ℑ z ⩾ δ}$ . Since $H_{j, θ}$ extends to an analytic type A family of operators on $D (c)$ and $z \in C_{δ}$ , the right hand side of (3.6) extends by analytic continuation in θ to the disc $D (c^{'})$ for small enough $c^{'} > 0$ . For $θ \in D (c^{'})$ with $ℑ θ > 0$ , both terms of (3.6) are analytic on $C_{+}$ and consequently (3.6) remains true for all z in $C_{+}$ .

From now on, we fix $θ = i η$ , $η > 0$ in $D (c^{'})$ . Set $A_{a, A} : = {z \in C; ℜ z > A, ℑ z > - a}$ for positive numbers a and A. The following estimate holds uniformly on $A_{a, 1}$ for some positive constant a: $\begin{array}{l} ‖ {(- e^{- 2 θ} Δ - z)}^{- 1} ‖ ⩽ sup_{ξ \in R^{n}} ({| e^{- 2 θ} | ξ |^{2} - z |}^{- 1}) ⩽ C η^{- 1} {(ℜ z)}^{- 1} . \end{array}$ Using (A1) and the above estimate, we see that $\begin{array}{l} H_{j, θ} - z = (- e^{- 2 θ} Δ - z) (I + {(- e^{- 2 θ} Δ - z)}^{- 1} V_{j} (e^{θ} x)), \end{array}$ is invertible for $z \in A_{a, A}$ with sufficiently large A. Moreover, uniformly on $z \in A_{a, A}$ , $\begin{array}{l} (3.7) & A_{a, A} ∋ z \to {(H_{j, θ} - z)}^{- 1} is holomorphic, and ‖ {(H_{j, θ} - z)}^{- 1} ‖ = O ({(ℜ z)}^{- 1}) . \end{array}$ On the other hand, for the same reason as in the beginning of this section, we see from (A2) that ${[V_{\cdot, θ}]}_{1}^{2} {(H_{j, θ} - z_{0})}^{- m}$ and ${[{(H_{\cdot, θ} - z_{0})}^{- m}]}_{1}^{2}$ are of trace class.

Next, we write $σ_{+} (z) = σ_{+}^{1} (z) + σ_{+}^{2} (z)$ , where $\begin{array}{l} σ_{+}^{1} (z) & = tr ({(z - z_{0})}^{m} {(z - H_{1, θ})}^{- 1} {[{(H_{\cdot, θ} - z_{0})}^{- m}]}_{1}^{2}), \\ σ_{+}^{2} (z) & = tr ({(z - z_{0})}^{m} {[{(z - H_{\cdot, θ})}^{- 1}]}_{1}^{2} {(H_{2, θ} - z_{0})}^{- m}) \\ = tr [{(z - z_{0})}^{m} {(z - H_{1, θ})}^{- 1} {[V_{\cdot, θ}]}_{1}^{2} {(H_{2, θ} - z_{0})}^{- m} {(z - H_{2, θ})}^{- 1}] . \end{array}$ Then from (3.7), we deduce that the RHS are holomorphic in $A_{a, A}$ which implies that $ξ^{'} (λ) = \frac{1}{π} ℑ (σ_{+}^{1} (λ + i 0) + σ_{+}^{2} (λ + i 0))$ is analytic in $] λ_{0}, + \infty [$ for a large constant $λ_{0}$ , and that $| σ_{+}^{1} (λ + i ε) |, | σ_{+}^{2} (λ + i ε) | = O (λ^{m - 1})$ , uniformly for $λ > λ_{0} ≫ 1$ and $ε \in [0, ε_{0} [$ for some $ε_{0}$ sufficiently small. Consequently, $\begin{array}{l} ξ^{'} (λ) = \frac{1}{π} ℑ σ_{+} (λ + i 0) = \frac{1}{π} ℑ (σ_{+}^{1} (λ + i 0) + σ_{+}^{2} (λ + i 0)) = O (λ^{m - 1}) . \end{array}$

This ends the proof of Theorem 3.2 for $N = 0$ . For $N ⩾ 1$ we take derivatives of $σ_{+}^{\cdot} (z)$ with respect to z and repeat the same arguments as above.
3.2. Proof of Corollary 3.3

The proof of Corollary 3.3 is a simple consequence of Theorem 3.2 and the following lemma. Let $F_{h}^{- 1} ψ$ be the semiclassical Fourier transform: $\begin{array}{l} F_{h}^{- 1} ψ (μ) = \frac{1}{2 π h} \int_{R} e^{i t μ / h} ψ (t) d t . \end{array}$

Lemma 3.5.
Let $ψ \in C_{0}^{\infty} (R)$ , and let $f_{h}$ be a $C^{\infty}$ function in $R$ , depending on a parameter $h \in (0, 1]$ . We suppose that, there exist $m \in R$ and $δ \in [0, 1 [$ such that for all $k \in N$ , $\begin{array}{l} (3.8) & {(\frac{d}{d μ})}^{k} f_{h} (μ) = O (h^{- m - k δ}) as h \to + 0 uniformly for μ \in R . \end{array}$ Then for all $N \in N$ , there exists $h_{N} > 0$ such that $\begin{array}{l} (3.9) & F_{h}^{1} ψ * f_{h} (μ) = \sum_{k = 0}^{N} \frac{{(- i h)}^{k}}{k!} ψ^{(k)} (0) {(\frac{d}{d μ})}^{k} f_{h} (μ) + O (h^{N (1 - δ) + m}), \end{array}$ uniformly for $μ \in R$ and $h \in (0, h_{N}]$ . In particular, if $ψ = 1$ near zero, then $\begin{array}{l} (3.10) & F_{h}^{- 1} ψ * f_{h} (μ) = f_{h} (μ) + O (h^{\infty}) . \end{array}$
Proof.
By a change of variable, we have $\begin{array}{l} (3.11) & F_{h}^{- 1} ψ * f_{h} (μ) = \int_{R} F_{1}^{- 1} ψ (ν) f_{h} (μ - h ν) d ν . \end{array}$ Applying Taylor’s formula to the function $ν ⟼ f_{h} (μ - h ν)$ at $ν = 0$ , and using (3.8), we get $\begin{array}{l} (3.12) & f_{h} (μ - h ν) = \sum_{k = 0}^{N - 1} f_{h}^{(k)} (μ) \frac{{(- ν h)}^{k}}{k!} + O (h^{N (1 - δ) - m} ν^{N}) . \end{array}$ Inserting the above equality in (3.11) and using the fact that $\int_{R} {(- i t)}^{k} F_{1}^{- 1} ψ (ν) d ν = ψ^{(k)} (0)$ we obtain (3.9). □

Now we pass to the prove of Corollary 3.3. Let $g \in C_{0}^{\infty} (] \frac{1}{2}, \frac{3}{2} [)$ be equal to 1 near one. For $h > 0$ , we set $f_{h} (μ) : = g (μ) ξ^{'} (\frac{μ}{h^{2}})$ . Using Theorem 3.2, we see that the function $f_{h}$ satisfies all the assumptions in Lemma 3.5 with $δ = 0$ . Let $ψ \in C_{0}^{\infty} (R)$ be as in Lemma 3.5 with $ψ = 1$ near zero. According to Lemma 3.5, we have $\begin{array}{l} (3.13) & F_{h}^{- 1} ψ * f_{h} (μ) = f_{h} (μ) + O (h^{\infty}) . \end{array}$ On the other hand, a simple calculation shows that $\begin{array}{l} h^{- 2} F_{h}^{- 1} ψ * f_{h} (μ) & = h^{- 2} \int_{R} F_{h}^{- 1} ψ (μ - t) g (ν) ξ^{'} (ν / h^{2}) d ν \\ = \int_{R} F_{h}^{- 1} ψ_{h} (μ - ν h^{2}) g (h^{2} ν) ξ^{'} (ν) d ν \\ = ⟨ ξ^{'}, F_{h}^{- 1} ψ (μ - h^{2} \cdot) g (h^{2} \cdot) ⟩ \\ (3.14) & = tr {[F_{h}^{- 1} ψ (μ - h^{2} H_{.}) g (h^{2} H_{.})]}_{1}^{2} . \end{array}$ For $0 < h ≪ 1, h^{2} H_{j}$ is an h-pseudodifferential operator with principal symbol $| ξ |^{2}$ . Hence any positive energy is non-trapping. According to [29, 31] (see also chapters 10-12 in [9]), the right hand side of the last equality has a complete asymptotic expansion in powers of $h^{2}$ . 1
¹
As in Theorem 2.1, the proof of (3.15) is based on the functional calculus of h-pseudodifferential operators. Nevertheless, to treat (3.15) one requires a non critical assumption for the principal symbol of $B : = h^{2} H_{j}$ . In both cases we investigate the asymptotic behavior as $h ↘ 0$ of the trace of $F (B)$ where F is real valued function (known as weak asymptotics). However, in Theorem 2.1, $F (x) = f (x)$ is h independent, while in (3.14), $F (x) = F_{h}^{- 1} ψ (μ - x) g (x)$ depends on h, and then the proof of (3.15) is more technical and similar to those in [29, 31]. The second one is more precise since it allows in some case to provide an estimate of the SSF in an interval of length $O (h)$ . For more details about these asymptotics and their applications, we refer to chapters 9, 10 and 12 in [9]. See also the comments in the beginning of the proof of Theorem 4.4.

Combining this with (3.10), we get $\begin{array}{l} (3.15) & h^{- 2} F_{h}^{- 1} ψ * f_{h} (μ) = h^{- 2} f_{h} (μ) + O (h^{\infty}) = h^{- n} \sum_{j = 1}^{\infty} a_{j} (μ) h^{2 j} + O (h^{\infty}) . \end{array}$ Taking $μ = 1$ and $λ = h^{- 2}$ we obtain $\begin{array}{l} ξ^{'} (λ) = λ^{\frac{n}{2}} \sum_{j = 1}^{\infty} a_{j} (1) λ^{- j - 1} + O (λ^{- \infty}) . \end{array}$ We recall that $f_{h} (μ) = g (μ) ξ^{'} (μ / h^{2})$ and $g (1) = 1$ . This ends the proof of Corollary 3.3. The explicit formula of $a_{j}$ is given by (2.8) in Theorem 2.1. Remark 3.6.
Theorem 3.2 remains true if $V_{1}$ is a homogeneous potential of degree zero and smooth on $R^{n} ∖ {0}$ (i.e. a pure homogeneous potential of degree zero). In fact, according to the proof of Theorem 3.2, one only needs that the operator $L^{2} (R^{n}) ∋ u \to V_{1, θ} (x) u (x) \in L^{2} (R^{n})$ is analytic with respect to $θ \in D (c)$ and uniformly bounded.

4. Semiclassical asymptotics

In this section we consider the semiclassical Schrödinger operators $H_{1} (h)$ and $H_{2} (h)$ given in (1.1). To simplify the presentation, let us assume, throughout this section, that the potentials are homogeneous of degree zero at infinity. More precisely, in addition to the conditions (A1), (A1) we suppose

There exists a homogeneous function W of degree zero such that $\begin{array}{l} (4.1) & lim_{| x | \to \infty} (V_{1} (x) - W (x)) = 0 . \end{array}$

To state the main result of this section, let us recall the notion of resonances. For $θ \in R$ , we introduce $\begin{array}{l} H_{j, θ} (h) = - h^{2} e^{- 2 θ} Δ + V_{j} (e^{θ} x) . \end{array}$ By assumption (A1), the right hand side of the above equality extends by analytic continuation to θ in the disc $D (c)$ with $c > 0$ small enough. Moreover, for $θ \in D (c)$ , we have $\begin{array}{l} (4.2) & σ_{ess} (H_{1, θ} (h)) = σ_{ess} (H_{2, θ} (h)) = S_{θ} : = {e^{- 2 θ} s + t; s ⩾ 0 and t \in W (S^{n - 1})} . \end{array}$ In fact, we easily see $σ_{ess} (- e^{- 2 θ} h^{2} Δ + W (x)) = S_{θ}$ by Weyl’s criterion, and (4.2) follows from (4.1) and Theorem 5.35 in [18].2

²
Theorem 5.35 [18]: Let T be a closed operator on a Hilbert space $H$ and let A be a relatively T-compact operator. Then $σ_{ess} (T) = σ_{ess} (T + A)$ .

In particular,

\begin{array}{l} σ_{ess} (H_{1} (h)) = σ_{ess} (H_{2} (h)) = [{min}_{ω \in S^{n - 1}} W (ω), + \infty), \end{array}

and for

ℑ θ \neq 0

\begin{array}{l} σ_{ess} (H_{j, θ} (h)) \cap ({sup}_{ω \in S^{n - 1}} W (ω), + \infty) = \emptyset . \end{array}

Therefore, for

ℑ θ \neq 0

, the operator

H_{j, θ} (h)

may have discrete eigenvalues with finite multiplicities near

] {sup}_{ω \in S^{n - 1}} W (ω), + \infty)

. For

ℑ θ > 0

, these eigenvalues (called resonances of

H_{j} (h)

) are in the lower half plane

C^{-}

. The background of the theory of resonances can be found in [15]).

If $z_{0}$ is a resonance, we define its multiplicity by $m (z_{0}) = rank (Π_{z_{0}})$ with $\begin{array}{l} (4.3) & Π_{z_{0}} = \frac{1}{2 π i} \int_{γ} {(z - H_{j, θ} (h))}^{- 1} d z \end{array}$ where the integral is over a circle γ containing no other pole of ${(z - H_{j, θ} (h))}^{- 1}$ than $z_{0}$ .

Fix an interval $J \subset R$ with $inf J > {max}_{ω \in S^{n - 1}} W (ω)$ . Let $Res H_{j} (h)$ denote the set of resonances of $H_{j} (h)$ .

Theorem 4.1.

Under (A1)–(A1), there exist an h-independent open complex neighborhood Ω of J and a holomorphic function $r (z, h)$ in Ω satisfying $\begin{array}{l} (4.4) & | r (z, h) | ⩽ C h^{- n} \end{array}$ such that for h small enough and $λ \in J$ , it holds that $\begin{array}{l} (4.5) & ξ^{'} (λ, h) = ℑ r (λ, h) + {[\sum_{\begin{array}{c} ω \in Res H_{\cdot} (h) \cap Ω \\ ℑ ω \neq 0 \end{array}} \frac{- ℑ ω}{| λ - ω |^{2}} + \sum_{ω \in σ_{p p} (H_{\cdot} (h)) \cap Ω} δ (λ - ω)]}_{1}^{2} . \end{array}$

By Lemma 3.4, we have $\begin{array}{l} (4.6) & ξ^{'} (λ, h) = \frac{1}{π} ℑ σ_{+} (λ + i 0, h), \end{array}$ where $\begin{array}{l} σ_{+} (z, h) = {(z - z_{0})}^{m} tr {[{(z - H_{\cdot} (h))}^{- 1} {(H_{\cdot} (h) - z_{0})}^{- m}]}_{1}^{2}, ℑ z > 0, \end{array}$ with a fixed $z_{0} < inf [σ (H_{1} (h)) \cup σ (H_{2} (h))]$ independent of h. As in the proof of (3.6), analytic continuation argument shows that, there exists $θ_{0}$ small enough such that for any $θ \in D (θ_{0})$ we have $\begin{array}{l} (4.7) & σ_{+} (z, h) = {(z - z_{0})}^{m} tr {[{(z - H_{\cdot, θ} (h))}^{- 1} {(H_{\cdot, θ} (h) - z_{0})}^{- m}]}_{1}^{2}, ℑ z > 0 . \end{array}$

From now on, we fix $θ = i η$ with $η > 0$ , and we let Ω be a bounded simply connected complex neighborhood of J such that $Ω \cap σ_{ess} (H_{j, i η} (h)) = \emptyset$ for $j = 1, 2$ .

The proof of Theorem 4.1 consists in the construction of a finite rank operator $K_{j}$ satisfying the following properties: Proposition 4.2.

There exist finite rank operators $K_{j} = K_{j} (z, h)$ , $j = 1, 2$ , on $L^{2} (R^{n})$ with the following two properties:

1) $rank K_{j} = O (h^{- n})$ , $‖ K_{j} ‖_{tr} = O (h^{- n})$ and $\begin{array}{l} (4.8) & σ_{+} (z, h) = {[tr ({(Id + K_{\cdot})}^{- 1} \partial_{z} K_{\cdot})]}_{1}^{2} + k, \forall z \in Ω_{+}, \end{array}$ where $k = k (z, h)$ is a holomorphic function in Ω satisfying the estimate $| k (z, h) | = O (h^{- n})$ .

2) The resonances of $H_{j} (h)$ in Ω are precisely the zeros of the holomorphic function $\begin{array}{l} D_{j} (z, h) : = det (Id + K_{j} (z, h)), \end{array}$ and their multiplicities agree.

Proof.

Let $f \in C_{0}^{\infty} (] - 3 R, 3 R [; [0, 1])$ with $f = 1$ on $[- 2 R, 2 R]$ , and let $χ, \tilde{χ} \in C_{0}^{\infty} (] - 2 R, 2 R [; [0, 1]))$ be equal to one on $[- R, R]$ , where R will be chosen sufficiently large. For $j = 1, 2$ , we define $\begin{array}{l} (4.9) & {\hat{H}}_{j, R} (h) : = H_{j, θ} (h) - K (h), \end{array}$ where $\begin{array}{l} K (h) : = i M f (- h^{2} Δ + | x |^{2}) \tilde{χ} (- h^{2} Δ) χ (| x |^{2}) \tilde{χ} (- h^{2} Δ) f (- h^{2} Δ + | x |^{2}) . \end{array}$ We claim the following estimate. Lemma 4.3.

For R and M large enough, one has, for h sufficiently small, $\begin{array}{l} (4.10) & ‖ {(z - {\hat{H}}_{j, R} (h))}^{- 1} ‖ = O (1), \end{array}$ uniformly for $z \in Ω$ .

Proof.

Notice that $\begin{array}{l} (4.11) & K (h) = i M {Op}_{h} (f {(| ξ |^{2} + | x |^{2})}^{2} \tilde{χ} {(| ξ |^{2})}^{2} χ (| x |^{2})) + O (h), \end{array}$ in $L (L^{2} (R^{n}), L^{2} (R^{n}))$ norm.3

Functional calculus for pseudodifferential operator (see Theorems A.1 and A.5) shows that $\begin{array}{l} K (h) = i M {Op}_{h} (f {(| ξ |^{2} + | x |^{2})}^{2} \tilde{χ} {(| ξ |^{2})}^{2} χ (| x |^{2})) + h {Op}_{h} (r (x, ξ, h)), \end{array}$ with $r (x, ξ, h)$ bounded with all its derivatives with respect to $(x, ξ)$ . Therefore $‖ {Op}_{h} (r (x, ξ, h)) ‖_{L (L^{2} (R^{n}))} = O (1)$ by Theorem A.2.

Hence, to prove (4.10) we may assume that

\begin{array}{l} K (h) = i M {Op}_{h} (f {(| ξ |^{2} + | x |^{2})}^{2} \tilde{χ} {(| ξ |^{2})}^{2} χ (| x |^{2})) . \end{array}

Set, for

j = 1, 2

\begin{array}{l} (4.12) & {\hat{H}}_{j, R} (h) : = H_{j, θ} (h) - K (h) . \end{array}

The symbol of

{\hat{H}}_{j, R} (h)

is given by

\begin{array}{l} {\hat{H}}_{j, R} (x, ξ) = e^{- 2 θ} | ξ |^{2} + V_{j, θ} (x) - i M f {(| ξ |^{2} + | x |^{2})}^{2} \tilde{χ} {(| ξ |^{2})}^{2} χ (| x |^{2}), \end{array}

and for

z \in Ω

| {\hat{H}}_{j, R} (x, ξ) - z |^{2} = R_{1} + R_{2}

with

\begin{array}{l} R_{1} & = {[| ξ |^{2} cos (2 η) + ℜ (V_{j, θ} - z)]}^{2}, \\ R_{2} & = {[| ξ |^{2} sin (2 η) + M f {(| ξ |^{2} + | x |^{2})}^{2} \tilde{χ} {(| ξ |^{2})}^{2} χ (| x |^{2}) + ℑ (z - V_{j, θ})]}^{2} . \end{array}

Choose R large enough so that

R > 2 {sup}_{x \in R^{n}, z \in Ω} | ℜ (V_{j, θ} (x) - z) |

. It follows that

\begin{array}{l} (4.13) & R_{1} ⩾ C {(1 + | ξ |^{2})}^{2} for | ξ |^{2} ⩾ R, x \in R^{n}, z \in Ω . \end{array}

Next, we choose M large enough so that

M > {sup}_{x \in R^{n}, z \in Ω} | ℑ (z - V_{j, θ} (x)) |

. Then

\begin{array}{l} (4.14) & R_{2} ⩾ {(M + ℑ z - ℑ V_{j, θ} (x))}^{2} ⩾ c > 0 for | ξ |^{2} ⩽ R, | x |^{2} ⩽ R, z \in Ω . \end{array}

It remains to esgtimate

| {\hat{H}}_{j, R} (x, ξ) - z |

for

| ξ |^{2} ⩽ R

and

| x |^{2} ⩾ R

. From (2.1) and the assumptions (A1), (A1), we have for

j = 1, 2

\begin{array}{l} ℜ V_{j, θ} (x) = W (\frac{x}{| x |}) + o_{R} (1), ℑ V_{j, θ} (x) = o_{R} (1) . \end{array}

Since

α : = {inf}_{z \in Ω} ℜ z > {sup}_{x \in S^{n - 1}} W (x) = : β

, it follows that for θ small enough

\begin{array}{l} (4.15) & R_{1} ⩾ {(ℜ z - W (\frac{x}{| x |}) + o_{R} (1) - | ξ |^{2} cos (2 η))}^{2} ⩾ \tilde{c} > 0 for | ξ |^{2} ⩽ \frac{α - β}{2 cos (2 η)} . \end{array}

On the other hand, for

| ξ |^{2} ⩾ \frac{α - β}{2 cos (2 η)}

and

| x |^{2} ⩾ R

, we have

\begin{array}{l} (4.16) & R_{2} ⩾ {(| ξ |^{2} sin (2 η) + M f (| ξ |^{2} + | x |^{2}) {\tilde{χ}}^{2} (| ξ |^{2}) χ (| x |^{2}) + ℑ z + o_{R} (1))}^{2} ⩾ c^{'} > 0, \end{array}

uniformly for

z \in Ω

provided that

Ω \subset {z \in C, ℑ z ⩾ - κ η}

with

0 < κ ≪ 1

From (4.13), (4.14), (4.15) and (4.16) we deduce that, uniformly for $(x, ξ) \in R^{2 n}$ and $z \in Ω$ , $\begin{array}{l} (4.17) & | {\hat{H}}_{j, R} (x, ξ) - z | ⩾ C (1 + | ξ |^{2}) . \end{array}$ Hence, for h small enough, the operator ${\hat{H}}_{j, R} (h) - z$ is elliptic for $z \in Ω$ . We conclude from Theorem A.4 that $z - {\hat{H}}_{j, R} (h)$ is invertible for h small enough, and that (4.10) holds uniformly for $z \in Ω$ (see Theorem A.2). □

Now we return to the proof of Proposition 4.2. By construction, we have $\begin{array}{l} z - H_{j, θ} (h) = (Id + K_{j} (z, h)) (z - {\hat{H}}_{j, R} (h)), \end{array}$ where $\begin{array}{l} (4.18) & K_{j} (z, h) = - K (h) {(z - {\hat{H}}_{j, R} (h))}^{- 1} . \end{array}$ Since $z - {\hat{H}}_{j, R} (h)$ is invertible, it follows that $z - H_{j, θ} (h)$ is invertible if and only if $Id + K_{j} (z, h)$ is invertible. Moreover, for $z \notin σ (H_{j, θ} (h))$ , we have $\begin{array}{l} (4.19) & {(z - H_{j, θ} (h))}^{- 1} = {(z - {\hat{H}}_{j, R} (h))}^{- 1} {(Id + K_{j} (z, h))}^{- 1} . \end{array}$

We can now decompose the right hand side of (4.7) as $\begin{array}{l} (4.20) & σ_{+} (z, h) = {[I_{\cdot}]}_{1}^{2} + I, z \in Ω_{+}, \end{array}$ where $\begin{array}{l} I_{j} & = {(z - z_{0})}^{m} tr [({(z - H_{j, θ} (h))}^{- 1} - {(z - {\hat{H}}_{j, R} (h))}^{- 1}) {(H_{j, θ} (h) - z_{0})}^{- m}] \\ = {(z - z_{0})}^{m} tr [{(z - H_{j, θ} (h))}^{- 1} K (h) {(z - {\hat{H}}_{j, R} (h))}^{- 1} {(H_{j, θ} (h) - z_{0})}^{- m}] \\ = {(z - z_{0})}^{m} tr [K (h) {(z - {\hat{H}}_{j, R} (h))}^{- 1} {(z - H_{j, θ} (h))}^{- 1} {(H_{j, θ} (h) - z_{0})}^{- m}] \\ I & = {(z - z_{0})}^{m} tr {[{(z - {\hat{H}}_{\cdot, R} (h))}^{- 1} {(H_{\cdot, θ} (h) - z_{0})}^{- m}]}_{1}^{2} . \end{array}$ Clearly, I is analytic on Ω and $I = O (h^{- n})$ . To deal with $I_{j}$ , we use the resolvent equation $\begin{array}{l} {(z - z_{0})}^{m} {(z - H_{j, θ} (h))}^{- 1} {(H_{\cdot, θ} (h) - z_{0})}^{- m} = {(z - H_{j, θ} (h))}^{- 1} - \sum_{k = 0}^{m - 1} {(z_{0} - z)}^{k} {(z_{0} - H_{j, θ} (h))}^{- k - 1} . \end{array}$ Using the above equality and the cyclicity of the trace we deduce that $\begin{array}{l} I_{j} = tr [{(z - H_{j, θ} (h))}^{- 1} K (h) {(z - {\hat{H}}_{j, R} (h))}^{- 1}] + g_{j} (z, h), \end{array}$ where $z \to g_{j} (z, h)$ is analytic on Ω and $| g_{j} (z, h) | = O (h^{- n})$ uniformly for $z \in Ω$ . Here we have used the fact that ${(z_{0} - H_{j, θ} (h))}^{- k - 1}$ and ${(z - {\hat{H}}_{j, R} (h))}^{- 1}$ are h-pseudo-differential operators uniformly for $z \in Ω$ (see Theorem A.4).

Inserting the right hand side of (4.19) in the above equality and using again the cyclicity of the trace as well as the fact that $\begin{array}{l} (4.21) & \partial_{z} K_{j} (z, h) = - K_{j} (z, h) {(z - {\hat{H}}_{j, R} (h))}^{- 1}, \end{array}$ we obtain $\begin{array}{l} I_{j} = tr [{(Id + K_{j} (z, h))}^{- 1} \partial_{z} K_{j} (z, h)] + g_{j} (z, h) . \end{array}$ Summing up, we have proved (4.8). From the definition of $K (h)$ , we have $\begin{array}{l} (4.22) & rank K_{j} (z, h) ⩽ rank f (- h^{2} Δ + | x |^{2}) = O (h^{- n}) . \end{array}$

By Theorem A.3, the operatpr $K (h)$ is of trace class and $‖ K (h) ‖ = O (h^{- n})$ . Combining this with (4.10) we get $\begin{array}{l} (4.23) & {‖ K_{j} (z, h) ‖}_{tr} = O (h^{- n}) . \end{array}$ It remains to prove the second assertion of Proposition 4.2.

Let $z_{*}$ be an eigenvalue of $H_{j, θ} (h)$ and let γ be a small positively oriented circle centered at $z_{*}$ such that $z_{*}$ is the unique pole of ${(z - H_{j, θ} (h))}^{- 1}$ inside γ. Using that $z \to {(z - {\hat{H}}_{j, R} (h))}^{- 1}$ is holomorphic, we obtain $\begin{array}{l} (4.24) & Π_{z_{*}} = \frac{1}{2 π i} \int_{γ} {(z - H_{j, θ} (h))}^{- 1} d z = \frac{1}{2 π i} \int_{γ} [{(z - H_{j, θ} (h))}^{- 1} - {(z - {\hat{H}}_{j, R} (h))}^{- 1}] d z . \end{array}$ From (4.19), we have $\begin{array}{l} {(z - H_{j, θ} (h))}^{- 1} - {(z - {\hat{H}}_{j, R} (h))}^{- 1} & = {(z - {\hat{H}}_{j, R} (h))}^{- 1} ({(Id + K_{j} (z, h))}^{- 1} - Id) \\ = - {(z - {\hat{H}}_{j, R} (h))}^{- 1} {(Id + K_{j} (z, h))}^{- 1} K_{j} (z, h) . \end{array}$ Since $Π_{z_{*}}$ is a projection, it follows that $m (z_{*}) : = rank (Π_{z_{*}}) = tr (Π_{z_{*}})$ . Inserting the above equality in the right hand side of (4.24), and using the cyclicity of the trace and (4.21), we get $\begin{array}{l} (4.25) & m (z_{*}) = \frac{1}{2 π i} \int_{γ} tr ({(Id + K_{j} (z, h))}^{- 1} \partial_{z} K_{j} (z, h)) d z . \end{array}$

According to Liouville formula (see [10]), we have $\begin{array}{l} (4.26) & tr ({(Id + K_{j} (z, h))}^{- 1} \partial_{z} K_{j} (z, h)) = \frac{\partial_{z} D_{j} (z, h)}{D_{j} (z, h)}, \end{array}$ which together with (4.3), and (4.25) yields the second assertion of Proposition 4.2. □

Proof of Theorem 4.1.

This follows from a routine application of Proposition 4.2. For the reader’s convenience we give the main steps of the proof and we refer to [32] for the details (see also [4, 8]).

Step 1. Let $Ω_{0} ⋐ Ω$ , and let $N_{j} : = N (H_{j, θ} (h), Ω_{0}; h)$ be the number of eigenvalues of $H_{j, θ} (h)$ in $Ω_{0}$ counted with their multiplicity. We claim that $\begin{array}{l} (4.27) & N_{j} = O (h^{- n}) . \end{array}$ Let $Γ \subset Ω$ be a simple closed curve which avoids the resonances of $H_{j} (h)$ and contains $Ω_{0}$ inside. From (4.3), (4.25) and (4.28) we deduce that $\begin{array}{l} N_{j} = \sum_{z \in Res (H_{j} (h)) \cap Ω_{0}} m (z) ⩽ \frac{1}{2 π i} \int_{Γ} \frac{\partial_{z} D_{j} (z, h)}{D_{j} (z, h)} d z . \end{array}$ Applying Theorem A.6 to the right hand side of the above inequality we obtain $\begin{array}{l} (4.28) & N_{j} ⩽ C {sup}_{z \in Ω} log | D_{j} (z) | - C log | D_{j} ({\tilde{z}}_{0}) |, \end{array}$ where ${\tilde{z}}_{0} \in Ω$ is not a resonance of $H_{j} (h)$ which will be fixed later.

From Proposition 4.2, we obtain, using Theorem A.7 for the trace class operator $A = K_{j}$ , $\begin{array}{l} (4.29) & log | D_{j} (z) | = O (h^{- n}), \end{array}$ uniformly for $z \in Ω$ .

We choose ${\tilde{z}}_{0}$ with $ℑ {\tilde{z}}_{0} > 0$ such that ${\tilde{z}}_{0} - H_{j, θ}$ is elliptic and satisfies (4.10).4

⁴

The symbol of $H_{j, θ} (h) - z$ is $e^{- 2 θ} | ξ |^{2} + V_{j, θ} (x) - z$ . If $ℑ z > {sup}_{x \in R^{n}} | ℑ V_{j, θ} (x) | + c = : ε$ then $\begin{array}{l} | e^{- 2 θ} | ξ |^{2} + V_{j, θ} (x) - z | > | ℑ (e^{- 2 θ} | ξ |^{2} + V_{j, θ} (x) - z) | > | ξ |^{2} sin (2 η) + c . \end{array}$ Hence $H_{j, θ} (h) - z$ is elliptic and satisfies (4.10) by Theorems A.2–A.4 uniformy for $ℑ z ⩾ ε$ .

Using (4.12) and (4.19) we get for

z = {\tilde{z}}_{0}

\begin{array}{l} {(Id + K_{j} ({\tilde{z}}_{0}, h))}^{- 1} & = ({\tilde{z}}_{0} - {\hat{H}}_{j, R} (h)) {({\tilde{z}}_{0} - H_{j, θ} (h))}^{- 1} \\ = Id + K_{j} ({\tilde{z}}_{0}, h) {({\tilde{z}}_{0} - H_{j, θ} (h))}^{- 1} = : Id + \tilde{K_{j}} ({\tilde{z}}_{0}, h) . \end{array}

Again using Theorem A.7 for the trace class operator

\tilde{K_{j}} ({\tilde{z}}_{0}, h)

, we get

\begin{array}{l} log | det (Id + {\tilde{K}}_{j} ({\tilde{z}}_{0}, h)) | ⩽ C h^{- n}, \end{array}

which yields

\begin{array}{l} (4.30) & log | D_{j} ({\tilde{z}}_{0}, h) | = - log | det (Id + {\tilde{K}}_{j} ({\tilde{z}}_{0}, h)) | ⩾ - C h^{- n} . \end{array}

Putting together (4.28), (4.29) and (4.30) we obtain (4.27).

Step 2. For each $j = 1, 2$ , let $z_{j l}, l = 1, \dots, N_{j}$ be the resonances of $H_{j} (h)$ in Ω repeated according to their multiplicity and put $\begin{array}{l} S_{j} (z, h) : = \prod_{l = 1}^{N_{j}} (z - z_{j l}) . \end{array}$ We factorise $D_{j}$ : $\begin{array}{l} (4.31) & D_{j} (z, h) = G_{j} (z, h) S_{j} (z, h), \end{array}$ where $G_{j}$ and $\frac{1}{G_{j}}$ are holomorphic in Ω. Repeating the proof of formula (8.45) in [32], we get (taking Ω smaller if necessary): $\begin{array}{l} (4.32) & | \frac{G_{j}^{'} (z, h)}{G_{j} (z, h)} | = O (h^{- n}), uniformly for z \in Ω . \end{array}$

The proof of the above estimate uses the first step, Proposition 4.2 and Harnack’s inequality for non-negative harmonic functions. We omit the details and we refer to [32].

Step 3. From (4.8), (4.28) and (4.31), we have $\begin{array}{l} (4.33) & σ_{+} (z, h) = r (z, h) + \sum_{l = 1}^{N_{2}} \frac{1}{z - z_{2 l}} - \sum_{l = 1}^{N_{1}} \frac{1}{z - z_{1 l}} \end{array}$ where $\begin{array}{l} r (z, h) = {[\frac{G_{\cdot}^{'} (z, h)}{G_{\cdot} (z, h)}]}_{1}^{2} + k (z, h) . \end{array}$ Recall that $ℑ z_{j l} ⩽ 0$ , and notice that for $ℑ z_{j l} = 0$ $\begin{array}{l} (4.34) & \frac{i}{2 π} lim_{ϵ ↘ 0} (\frac{1}{λ + i ϵ - z_{j l}} - \frac{1}{λ - i ϵ - z_{j l}}) = δ (λ - z_{j l}), \end{array}$ while for $ℑ z_{j l} < 0$ $\begin{array}{l} (4.35) & \frac{i}{2 π} lim_{ϵ ↘ 0} (\frac{1}{λ + i ϵ - z_{j l}} - \frac{1}{λ - i ϵ - \overline{z_{j l}}}) = - \frac{1}{π} \frac{ℑ z_{j l}}{| λ - z_{j l} |^{2}} . \end{array}$ Clearly, $Ω ∋ z \to r (z, h)$ is holomorphic and satisfies (4.4) due to Proposition 4.2 and (4.32). Formula (4.5) follows from (4.6), (4.33), (4.34) and (4.35). This completes the proof of Theorem 4.1. □

Theorem 4.4.

Fix a compact interval J with $inf J > {max}_{ω \in S^{n - 1}} W (ω)$ and assume (A1)–(A1). We suppose that $p_{j} (x, ξ) : = | ξ |^{2} + V_{j} (x)$ is non critical for all $λ \in J$ (i.e., $\nabla_{x} V_{j} (x) \neq 0$ for all $x \in V^{- 1} (J)$ ). Then we have $\begin{array}{l} (4.36) & ξ (λ, h) = {(2 π h)}^{- n} c_{0} (λ) + O (h^{- n + 1}), \end{array}$ uniformly for $λ \in J$ , where $\begin{array}{l} c_{0} (λ) = κ_{0} \int {[{(λ - V_{\cdot} (x))}_{+}^{\frac{n}{2}}]}_{1}^{2} d x . \end{array}$ Moreover, if there exists $δ < 1$ such that $\begin{array}{l} (4.37) & Res (H_{j} (h)) \cap (J - i [0, h^{δ}]) = \emptyset, j = 1, 2, \end{array}$ then $ξ^{'} (λ, h)$ has a complete asymptotic expansion with smooth coefficients $\begin{array}{l} (4.38) & ξ^{'} (λ, h) \sim \sum_{k = 0}^{\infty} b_{k} (λ) h^{k - n}, \end{array}$ as $h ↘ 0$ uniformly for $λ \in J$ . In particular ${(2 π)}^{n} b_{0} (λ) = c_{0}^{'} (λ)$ .

Proof.

Here we use the same notation as in Theorem 4.1. Let g and Ψ be smooth functions with supports in small neighborhoods of J and zero respectively, with $Ψ = 1$ near zero, $F_{h}^{- 1} Ψ ⩾ 0$ and $g = 1$ near J. From the definition (1.2), we have $\begin{array}{l} - tr ({[(F_{h}^{- 1} Ψ) (λ - H_{j}) g (H_{j})]}_{1}^{2}) = (F_{h}^{- 1} Ψ) * (g ξ^{'}) (λ, h) . \end{array}$ According to Theorem 12.2 in [9] (see also [11, 16, 17, 30, 31]), the right hand side of the above equality has a full asymptotic expansion in powers of h: $\begin{array}{l} (4.39) & (F_{h}^{- 1} Ψ) * (g ξ^{'}) (λ, h) \sim \sum_{k = 0}^{\infty} b_{k} (λ) h^{k - n}, \end{array}$ provided that λ is not a critical value of $p_{j} (x, ξ)$ . If ξ were a monotone function as in the case of eigenvalue counting function, the Weyl asymptotis (4.36) would follow simply from this formula by a Tauberian argument. However, it is not the case for the SSF. To overcome this difficulty, we use (4.5).

The proof is rather similar to the one in [4, 8] and for this reason we omit the details. Without any loss of generality, we may assume that $H_{1} (h)$ has no resonances near J.5

⁵

As indicated in Remark 4.5, if $H_{1} (h) = \tilde{H} (h) = - h^{2} Δ + W (x)$ then $H_{1} (h)$ has no resonances near J. Next, since $ξ (λ, h) = ξ_{1} (λ, H_{1} (h), \tilde{H} (h)) - ξ_{2} (λ, H_{2} (h), \tilde{H} (h))$ , where $ξ_{j} (λ, H_{j} (h), \tilde{H} (h))$ is the SSF corresponding to the pair $(H_{j} (h), \tilde{H} (h))$ , it suffices to prove (4.36) to $ξ_{1}$ and $ξ_{2}$ .

On the other hand, by covering the compact interval J by a finite number of small intervals we may assume that

J = [τ - ϵ, τ + ϵ]

where τ is not a critical value of

p_{j}

. Let

φ_{1}, φ_{2} \in C_{0}^{\infty} (R)

be such that:

\begin{array}{l} supp (φ_{1}) \subset] - \infty, τ - ϵ [, supp (φ_{2}) \subset] τ - 2 ϵ, τ - + 2 ϵ [, \\ φ_{1} + φ_{2} = 1 near [z_{0}, τ + ϵ] . \end{array}

The constant

z_{0} \in R

is given by (4.7). The assumption of the above theorem allows us to use (4.5). Therefore, according to Lemma 3 in [8], we have

\begin{array}{l} (4.40) & ξ (λ, h) = tr ({[φ_{1} (H_{j} (h))]}_{1}^{2}) + M_{φ_{2}} (λ, h) + G_{φ_{2}} (λ, h), \end{array}

where

\begin{array}{l} (4.41) & G_{φ_{2}} (λ, h) = \int_{- \infty}^{λ} φ_{2} (t) ℑ r (t, h) \frac{d t}{π} \\ (4.42) & M_{φ_{2}} (λ, h) = \sum_{\begin{array}{c} ω \in Res H_{2} (h) \cap Ω, \\ ℑ ω < 0 \end{array}} \int_{- \infty}^{λ} φ_{2} (t) \frac{- ℑ ω}{| t - ω |^{2}} d t + \sum_{ω \in Res (H_{2} (h)) \cap [τ - 2 ϵ, λ]} φ_{2} (ω) . \end{array}

Second, Lemma 4 in [8] yields $\begin{array}{l} (4.43) & F_{h}^{- 1} Ψ * G_{φ_{2}} (λ, h) = G_{φ_{2}} (λ, h) + O (h^{- n + 1}) \\ (4.44) & F_{h}^{- 1} Ψ * M_{φ_{2}} (λ, h) = M_{φ_{2}} (λ, h) + O (h^{- n + 1}) \\ (4.45) & G_{φ_{2}} (λ, h) + M_{φ_{2}} (λ, h) = b_{0} (λ) + O (h^{- n + 1}), \end{array}$ where $b_{0} (λ) = \int_{\infty}^{λ} φ_{2} (t) c_{0}^{'} (t) d t$ . Since $λ \to M_{φ_{2}} (λ, h)$ is monotone, the proof of (4.44) is based on Tauberian argument. For the proof of (4.43) we use Lemma 3.5 and the estimate (4.48) which follows from (4.4). Finally, the proof of (4.45) is based on the h-pseudifferential calculus as in the proof of Theorem 2.1 and (4.39).

As in Theorem 2.1, by the functional calculus of h-pseudodiffential opetrator, the first term in the right hand side of (4.40) has complete asymptotic expansion in powers of h. Combining this with (4.45), we obtain (4.36).

The second assertion of the above theorem can be established combining Lemma 3.5 and Theorem 4.1. Formula (4.5) and the condition (4.37) yield $\begin{array}{l} (4.46) & ξ^{'} (λ, h) = ℑ r (λ, h) + {[\sum_{\begin{array}{c} ω \in Res H_{\cdot} (h) \cap Ω, \\ ℑ ω ⩽ - h^{δ} \end{array}} \frac{- ℑ ω}{| λ - ω |^{2}}]}_{1}^{2} . \end{array}$

For $ω \in Ω$ with $ℑ ω ⩽ - h^{δ}$ , consider the function $\begin{array}{l} f_{ω} (λ) : = - g (λ) \frac{ℑ ω}{| λ - ω |^{2}} . \end{array}$ For every $p \in N$ , we have as $h \to 0$ , $\begin{array}{l} \partial_{λ}^{p} f_{ω} (λ) = O (h^{- p δ - δ}) . \end{array}$ Applying Lemma 3.5 to $f_{h} = f_{ω}$ and using the above estimate as well as the fact that $Ψ = 1$ near zero, we deduce that $\begin{array}{l} (4.47) & (F_{h}^{- 1} Ψ) * f_{ω} (λ) = f_{ω} (λ) + O (h^{\infty}) . \end{array}$ Next, since $r (z, h)$ is holomorphic, it follows from Cauchy’s inequality and (4.4) that $\begin{array}{l} (4.48) & \partial_{λ}^{p} r (λ, h) = O (h^{- n}), \forall p \in N, \end{array}$ uniformly for $λ \in J$ . Applying again Lemma 3.5 to $ℑ r (λ, h)$ we get $\begin{array}{l} (4.49) & F_{h}^{- 1} Ψ * ℑ r (λ, h) = ℑ r (λ, h) + O (h^{\infty}) . \end{array}$ Notice that the sum in (4.46) is of order $O (h^{- n})$ , since the number of resonances is $O (h^{- n})$ . Combining this with (4.46), (4.47) and (4.49) we obtain $\begin{array}{l} F_{h}^{- 1} Ψ * (g ξ^{'}) (λ, h) = g (λ) ξ^{'} (λ, h) + O (h^{\infty}), \end{array}$ which together with (4.39) yields (4.38). The explicit formula for $b_{0} (λ)$ and $c_{0} (λ)$ follows from the weak asymptotics given by (4.39). □

Remark 4.5.

Notice that if $V_{1} = W$ is a purely homogeneous potential of degree zero, then the operator $H_{1} (h) = - h^{2} Δ + W (x)$ has no eigenvalues (see [12]). On the other hand, the arguments in the proof of Theorem 3.2 and Proposition 4.2 show that $H_{1} (h) = - h^{2} Δ + W (x)$ has no resonances near λ for $λ > sup W$ . In this case we may write (4.5) as $\begin{array}{l} ξ^{'} (λ, h) = ℑ r (λ, h) + \sum_{ω \in Res H_{2} (h) \cap Ω} \frac{- ℑ ω}{| λ - ω |^{2}} + \sum_{ω \in σ_{p p} (H_{2} (h)) \cap Ω} δ (λ - ω) . \end{array}$ In general one cannot exclude the existence of embedded eigenvalues of a perturbation of a purely homogeneous potential (see [1, 13]). Nevertheless, as will be shown in the next section, the only possible threshold energies of $H_{j} (h)$ are the critical values of W.

5. Semi-classical Mourre estimate

In this section, we prove a semi-classical Mourre estimate for $H (h) = - h^{2} Δ + V (x)$ where the potential $V (x)$ is homogeneous of degree zero at infinity:

$V \in C^{\infty} (R^{n}; R)$ , and there exists a homogeneous function $W \in C^{4} (R^{n}; R)$ of degree zero such that $\begin{array}{l} (5.1) & lim_{| x | \to \infty} | x |^{2} \partial_{x}^{α} (V (x) - W (x)) = 0 for all | α | ⩽ 4 . \end{array}$

We introduce a function

F (x)

with a positive parameter β and a differential operator

A_{F}

\begin{array}{l} (5.2) & F (x) = \frac{1}{2} (1 - 2 β V (x)) | x |^{2}, 2 A_{F} = \nabla F \cdot h D_{x} + h D_{x} \cdot \nabla F . \end{array}

The following theorem shows that the only possible threshhold energies6

⁶
We define threshold energies as those energies at which the Mourre estimate (5.3) fails to hold (see Section 3 in [14]).

H (h)

to which eigenvalues may accumulate are the critical values of W:

\begin{array}{l} C_{cr} = {λ \in R; \exists ω \in S^{n - 1} such that W (ω) = λ and \nabla W (ω) = 0} . \end{array}

In fact, if

{ψ_{j}}

were an infinite sequence of orthonormal eigenfunctions corresponding to the eigenvalues

{λ_{j}} \notin C_{cr}

H (h)

, then (5.3) would yield with virial Theorem

\begin{array}{l} 0 = ⟨ ψ_{j}, f (H) [H, A_{F}] f (H) ψ_{j} ⟩ ⩾ C h f {(λ_{j})}^{2} + ⟨ ψ_{j}, K ψ_{j} ⟩ . \end{array}

Since K is compact,

⟨ ψ_{j}, K ψ_{j} ⟩

tends to 0, which is a contradiction for f which does not vanish at the limit of

{λ_{j}}

. Theorem 5.1.

For $λ \notin C_{cr}$ , there exist small positive constants $ϵ, h_{0}$ and compact operators K , such that, for $β > 0$ small enough and $f \in C_{0}^{\infty} (] λ - ϵ, λ + ϵ [; R)$ we have $\begin{array}{l} (5.3) & f (H) [H, A_{F}] f (H) ⩾ C h f {(H)}^{2} + K, \end{array}$ uniformly for $h \in] 0, h_{0}]$ .

Proof.

A straightforward calculation shows that $\begin{array}{l} i [H, A_{F}] = 2 h (h D, (\nabla \otimes \nabla F) h D) + β h | x |^{2} | \nabla V |^{2} + r (x, h) \end{array}$ where $\nabla \otimes \nabla F = Id - β \nabla \otimes \nabla (| x |^{2} V (x))$ is the Hessian matrix and $\begin{array}{l} (5.4) & r (x, h) = - \frac{h^{3}}{2} Δ^{2} F (x) - h (1 - 2 β V (x)) x \cdot \nabla V (x) . \end{array}$ It follows from the assumption (5.1) that $\nabla \otimes \nabla (| x |^{2} V (x))$ is a bounded symmetric matrix. Hence, for β small enough, we have $\begin{array}{l} 2 h (h D, (\nabla \otimes \nabla F) h D) ⩾ - h^{2} Δ . \end{array}$

The condition (5.1) also implies that $r (x, h)$ is a continuous function decaying at infinity. By Rellich’s theorem, $r (x, h) {(H + i)}^{- 1}$ is compact, and $r (x, h) f (H) = r (x, h) {(H + i)}^{- 1} (H + i) f (H)$ is also compact for all $f \in C_{0}^{\infty} (R)$ , since $(H + i) f (H)$ is bounded. Therefore, there exists a compact operator $K_{1}$ such that for β small enough, we have $\begin{array}{l} (5.5) & f (H) i [H, A_{F}] f (H) ⩾ h f (H) H_{β} f (H) + K_{1}, H_{β} : = - h^{2} Δ + β | x |^{2} | \nabla V |^{2} . \end{array}$

Now we fix $λ \notin C_{r}$ . Then there exist $ϵ, κ > 0$ such that $\begin{array}{l} (5.6) & | \nabla W (ω) | ⩾ κ, for all ω \in S^{n - 1} with W (ω) \in] λ - 3 ϵ, λ + 3 ϵ [. \end{array}$ We divide the unit sphere $S^{n - 1}$ into three open subsets $S^{n - 1} = O_{1} \cup O_{2} \cup O_{3}$ , where $\begin{array}{l} O_{1} = {λ - 3 ϵ < W (ω) < λ + 3 ϵ}, O_{2} = {W (ω) < λ - 2 ϵ}, O_{3} = {λ + 2 ϵ < W (ω)} . \end{array}$ Let ${χ_{0}, χ_{1}, χ_{2}, χ_{3}}$ be a smooth partition of unity of $R^{n}$ , $\sum_{k = 0}^{3} χ_{k}^{2} (x) = 1$ , satisfying $\begin{array}{l} supp χ_{0} \subset {x \in R^{n}; | x | ⩽ R}, and χ_{0} (x) = 1 for | x | < R / 2, \\ supp χ_{k} \subset {x \in R^{n}; | x | > R / 2, \frac{x}{| x |} \in O_{k}}, and x \cdot \nabla χ_{k} (x) = 0 for | x | > 3 R / 4 . \end{array}$

We choose R large enough such that for $| x | > \frac{R}{2}$ one has $\begin{array}{l} (5.7) & | | x |^{2} {| \nabla V (x) |}^{2} - | x |^{2} {| \nabla W (x) |}^{2} | ⩽ \frac{κ}{2}, | V (x) - W (x) | ⩽ \frac{ϵ}{2} . \end{array}$

By the so-called IMS localization formula, i.e., $\begin{array}{l} H_{β} = \sum_{k = 0}^{3} χ_{k} H_{β} χ_{k} - h^{2} \sum_{k = 0}^{3} {(\nabla χ_{k})}^{2}, \end{array}$ it follows from (5.5) that $\begin{array}{l} (5.8) & f (H) i [H, A_{F}] f (H) ⩾ h \sum_{k = 1}^{3} I_{k} + K_{2}, I_{k} = f (H) χ_{k} H_{β} χ_{k} f (H), \end{array}$ where $\begin{array}{l} K_{2} = K_{1} + h f (H) χ_{0} H_{β} χ_{0} f (H) - h^{3} \sum_{k = 0}^{3} f (H) {(\nabla χ_{k})}^{2} f (H) \end{array}$ is a compact operator for the same reason as $K_{1}$ , since $χ_{0}$ has a compact support and $\sum_{k = 0}^{3} {(\nabla χ_{k})}^{2}$ tends to zero as $| x | \to \infty$ by the homogeneity of $χ_{k}$ .

Let us prove (5.3) for $I_{1}, I_{2}$ and $I_{3}$ . We begin with $I_{1}$ . On the support of $χ_{1}$ , we have by the homogeneity of W, $\begin{array}{l} | x |^{2} {| \nabla W (x) |}^{2} ⩾ κ . \end{array}$ Combining this with (5.7), we obtain $χ_{1} H_{β} χ_{1} ⩾ \frac{κ β}{2} χ_{1}^{2}$ and hence $\begin{array}{l} (5.9) & I_{1} ⩾ \frac{h κ β}{2} f (H) χ_{1}^{2} f (H) . \end{array}$

Next, we study $I_{2}$ . From now on we restrict the support of f to $] λ - ϵ, λ + ϵ [$ . This implies $f (t) (t - λ) ⩾ - ϵ f (t)$ , and hence, by the spectral theorem, $\begin{array}{l} (5.10) & f (H) (H - λ) ⩾ - ϵ f (H) . \end{array}$ On the support of $χ_{2}$ , we have $\begin{array}{l} I_{2} ⩾ h f (H) χ_{2} (- h^{2} Δ) χ_{2} f (H) = h (I_{2, 1} + I_{2, 2}), \\ I_{2, 1} = f (H) χ_{2} (H - λ) χ_{2} f (H), I_{2, 2} = f (H) χ_{2} (λ - V) χ_{2} f (H) . \end{array}$ Since $I_{2, 1} = f (H) (H - λ) χ_{2}^{2} f (H) + h^{2} f (H) (Δ χ_{2}) χ_{2} f (H) + 2 h f (H) \nabla χ_{2} \cdot h \nabla f (H)$ , the estimate (5.10) and the fact that $f (H) Δ χ_{2}, f (H) \nabla χ_{2}$ are compact operators as well as that $h \nabla f (H)$ is bounded lead us to the estimate $\begin{array}{l} I_{2, 1} ⩾ - ϵ f (H) χ_{2}^{2} f (H) + K_{3}, \end{array}$ where $K_{3}$ is a compact operator.

On the other hand, on the support of $χ_{2}$ we have $λ - V ⩾ \frac{3}{2} ϵ$ . Therefore $\begin{array}{l} I_{2, 2} ⩾ \frac{3}{2} ϵ f (H) χ_{2}^{2} f (H) . \end{array}$ Summing these estimates about $I_{2, 1}$ and $I_{2, 2}$ , we get $\begin{array}{l} (5.11) & I_{2} ⩾ \frac{1}{2} ϵ f (H) χ_{2}^{2} f (H) + K_{3} . \end{array}$

Finally, we show that $I_{3}$ is a compact operator. On the support of $χ_{3}$ , one has $V (x) ⩾ λ + \frac{3 ϵ}{2}$ . Thus, the support of $χ_{3}$ is contained in the classically forbidden region of the operator $f (H)$ . In particular, by the semiclassical Weyl calculus, one has $f (H) χ_{3} = O (h^{\infty} {⟨ ξ ⟩}^{- \infty} {⟨ x ⟩}^{- \infty})$ on the symbolic level. Therefore $f (H) χ_{3}$ , and hence $I_{3}$ , are compact operators.

Combining this with (5.8), (5.9), (5.11) and using the fact that $f (H) χ_{0}^{2} f (H)$ is also a compact operator, we get, with another compact operator $K_{4}$ , $\begin{array}{l} (5.12) & f (H) i [H, A_{F}] f (H) ⩾ \frac{h}{2} min (κ β, ϵ) f {(H)}^{2} + K_{4} . \end{array}$ This ends the proof of the theorem. □

6. Proof of Theorem 2.1

For $h \in] 0, 1]$ and $ℑ z \neq 0$ , we set $\begin{array}{l} H_{j} = h^{2} H_{j} (h), j = 1, 2, H_{0} = - h^{2} Δ, \\ R_{j} (z) = {(z - H_{j})}^{- 1}, j = 0, 1, 2 . \end{array}$

Let $\tilde{f} \in C_{0}^{\infty} (C)$ be an almost analytic extension of f. We recall that $\tilde{f} \in C_{0}^{\infty} (C)$ satisfies $\begin{array}{l} (6.1) & | \overline{\partial} \tilde{f} | ⩽ C_{N} {| ℑ (z) |}^{N} for all N \in N and {\tilde{f}}_{| R} = f . \end{array}$ Since the spectrum of $H_{j}$ is bounded from below, we may choose $z_{0} \in R$ , so that $z_{0}$ is away from $σ (H_{j})$ , $j = 1, 2$ . Set $\begin{array}{l} g (z) = f (z) {(z - z_{0})}^{m} . \end{array}$ By Helffer–Sjöstrand formula (see [5], [7]) we have $\begin{array}{l} g (H_{j}) = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) {(z - z_{0})}^{m} {(z - H_{j})}^{- 1} L (d z), j = 1, 2, \end{array}$ where $L (d z)$ denotes the Lebesgue measure on $C$ . Clearly, $\begin{array}{l} f (H_{j}) = {(H_{j} - z_{0})}^{- m} g (H_{j}) = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) {(z_{0} - z)}^{m} R_{j} {(z_{0})}^{m} R_{j} (z) L (d z) . \end{array}$ From the assumption (A1), the operator ${[R_{\cdot} {(z_{0})}^{m} R_{\cdot} (z)]}_{1}^{2}$ is of trace class for $m > \frac{n}{2}$ , and $\begin{array}{l} (6.2) & tr {[f (H_{\cdot})]}_{1}^{2} = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) {(z_{0} - z)}^{m} tr {[R_{\cdot} {(z_{0})}^{m} R_{\cdot} (z)]}_{1}^{2} L (d z) . \end{array}$

For simplicity, we take $m = 0$ in (6.2). In fact, the main difficulty in the proof comes from the resolvents $R_{j} (z)$ which are not pseudodifferential operators when $ℑ z$ tends to 0. For the general case $m > 0$ , the term $R_{j} (z_{0})$ is a pseudodiferential operator (see Theorem A.4), and the proof remains the same with a minor modification (see Remark 6.3).

From the resolvent equation, we obtain $\begin{array}{l} R_{j} (z) = \sum_{k = 1}^{N} h^{2 k} {(R_{0} (z) V_{j})}^{k} R_{0} (z) + h^{2 (N + 1)} R_{j} (z) V_{j} {(R_{0} (z) V_{j})}^{N} R_{0} (z), j = 1, 2 . \end{array}$ Clearly, $\begin{array}{l} {‖ {[R_{\cdot} (z) V_{\cdot} {(R_{0} (z) V_{\cdot})}^{N}]}_{1}^{2} R_{0} (z) ‖}_{tr} = O (| ℑ z |^{- (N + 2)}), \end{array}$ which together with (6.1) implies $\begin{array}{l} {‖ \int {\bar{\partial}}_{z} \tilde{f} (z) {[R_{\cdot} (z) V_{\cdot} {(R_{0} (z) V_{\cdot})}^{N}]}_{1}^{2} L (d z) ‖}_{tr} = O (1), \end{array}$ and thus (by letting N go to infinity): $\begin{array}{l} (6.3) & tr {[f (H_{\cdot})]}_{1}^{2} \sim \sum_{k = 1}^{\infty} h^{2 k} I_{k}, I_{k} : = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) tr {[{(R_{0} (z) V_{\cdot})}^{k} R_{0} (z)]}_{1}^{2} L (d z) . \end{array}$

Fix $k ⩾ 1$ . We compute the standard symbol of the semiclassical pseudo-differential operator ${(R_{0} (z) V_{\cdot})}^{k} R_{0} (z)$ . Let $m = (m_{1}, \dots, m_{k}) \in N^{k}$ denote multi-indices of dimension k. We also use the notation $\begin{array}{l} {ad}_{H_{0}} P : = [P, H_{0}], {ad}_{H_{0}}^{0} P : = P, {ad}_{H_{0}}^{ν} P : = {ad}_{H_{0}} ({ad}_{H_{0}}^{ν - 1} P), ν = 2, 3, \dots \end{array}$ for a pseudo-differential operator P. When $P = W (x)$ is a multiplication operator in particular, it is a differential operator of the form $\begin{array}{l} {ad}_{H_{0}} W = 2 i h \partial W \cdot (h D) - h^{2} Δ W . \end{array}$

Proposition 6.1.

For $k ⩾ 1$ , we have $\begin{array}{l} {(R_{0} V_{\cdot})}^{k} = \sum_{l = 0}^{\infty} {(- 1)}^{l} P_{k, l} (x, h D; h) R_{0}^{l + k}, \end{array}$ where $P_{k, l} = P_{k, l} [V_{\cdot}]$ is a differential operator of order l defined by $\begin{array}{l} (6.4) & P_{k, l} (x, h D; h) : = \sum_{m \in N^{k}, | m | = l} {ad}_{H_{0}}^{m_{k}} (V_{\cdot} \dots V_{\cdot} {ad}_{H_{0}}^{m_{2}} (V_{\cdot} {ad}_{H_{0}}^{m_{1}} V_{\cdot})) \end{array}$ and its standard symbol $p_{k, l} = p_{k, l} [V_{\cdot}]$ is of the form $\begin{array}{l} (6.5) & p_{k, l} (x, ξ; h) = h^{l} \sum_{ν = 0}^{l} h^{l - ν} p_{k, l, ν} (x, ξ), \end{array}$ where $p_{k, l, ν} (x, ξ)$ is a homogeneous polynomial of degree ν in ξ.

Proof.

Here we omit the suffix and write V for $V_{1}$ and $V_{2}$ . Since $R_{0} V = V R_{0} - [V, R_{0}]$ , $[V, R_{0}] = R_{0} [V, H_{0}] R_{0}$ , it follows by iteration that $\begin{array}{l} R_{0} V = \sum_{m_{1} = 0}^{\infty} {(- 1)}^{m_{1}} ({ad}_{H_{0}}^{m_{1}} V) R_{0}^{m_{1} + 1} . \end{array}$ Notice that ${ad}_{H_{0}}^{m_{1}} V$ is a semiclassical differential operator of order $m_{1}$ with standard symbol $\begin{array}{l} h^{m_{1}} \sum_{q = 0}^{m_{1}} h^{m_{1} - q} \sum_{| α | = q} a_{m_{1}, α} (x) ξ^{α}, a_{m_{1}, α} (x) = {(2 i)}^{| α |} \frac{m_{1}!}{α! (m_{1} - | α |)!} \partial^{α} Δ^{m_{1} - | α |} V (x) . \end{array}$ Next, ${(R_{0} V)}^{2}$ is computed by commutation between $R_{0}$ and $V {ad}_{H_{0}}^{m_{1}} V$ : $\begin{array}{l} {(R_{0} V)}^{2} & = \sum_{m_{1} = 0}^{\infty} {(- 1)}^{m_{1}} R_{0} (V {ad}_{H_{0}}^{m_{1}} V) R_{0}^{m_{1} + 1} \\ = \sum_{m_{1}, m_{2} ⩾ 0} {(- 1)}^{m_{1} + m_{2}} {ad}_{H_{0}}^{m_{2}} (V {ad}_{H_{0}}^{m_{1}} V) R_{0}^{m_{1} + m_{2} + 2} \\ = \sum_{l = 0}^{\infty} {(- 1)}^{l} (\sum_{m_{1} + m_{2} = l} {ad}_{H_{0}}^{m_{2}} (V {ad}_{H_{0}}^{m_{1}} V)) R_{0}^{l + 2} . \end{array}$ Repeating this commutation, we obtain the required formula. □

Remark 6.2.

This proposition gives a development of ${(R_{0} V)}^{k}$ in powers of h since each term of (6.4) is of order l. It has a power of $R_{0}$ on the right. This is convenient for the symbolic calculus for the standard quantization. But thanks to the cyclicity of the trace, it is also useful to send $R_{0}$ to the left. For example one has $\begin{array}{l} {(R_{0} V)}^{2} = & R_{0}^{2} V^{2} + R_{0}^{3} ({ad}_{H_{0}} V) V + R_{0}^{4} ({ad}_{H_{0}}^{2} V) V \\ + R_{0}^{5} ({ad}_{H_{0}}^{3} V) V + R_{0} {(z)}^{6} ({ad}_{H_{0}}^{4} V) V + O (h^{5}), \\ {(R_{0} V)}^{3} = & R_{0}^{3} V^{3} + R_{0}^{4} ({ad}_{H_{0}} V^{2}) V + R_{0}^{4} ({ad}_{H_{0}} V) V^{2} + R_{0}^{5} ({ad}_{H_{0}}^{2} V^{2}) V \\ + R_{0}^{5} ({ad}_{H_{0}}^{2} V) V^{2} + R_{0}^{5} {ad}_{H_{0}} (({ad}_{H_{0}} V) V) V + O (h^{3}), \\ {(R_{0} V)}^{4} = & R_{0}^{4} V^{4} + O (h) . \end{array}$ We use these formulae below for the computation of the first four terms (2.4) and (2.5).

We first check (2.4) and (2.5). Let us begin with $I_{1}$ . We have, 7

⁷

We recall that if A is the t-quantization of a symbol $a (x, ξ)$ with $\partial^{γ} a \in L^{1} (R^{2 n})$ for $| γ | ⩽ 2 n + 1$ , then A is of trace class, and $tr (A) = {(2 π h)}^{- n} \int_{R^{2 n}} a (x, ξ) d x d ξ$ . In particular, it is independent of the type t of the quantization. In this paper, we use the standard quantization (i.e., $t = 1$ ).

\begin{array}{l} I_{1} & = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) tr {[R_{0} (z) V_{\cdot} R_{0} (z)]}_{1}^{2} L (d z) \\ = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) (\int {(z - | ξ |^{2})}^{- 2} d ξ) L (d z) \cdot {(2 π h)}^{- n} \int {[V_{\cdot}]}_{1}^{2} d x \\ = {(2 π h)}^{- n} \int f^{'} (| ξ |^{2}) d ξ \int {[V_{\cdot}]}_{1}^{2} d x . \end{array}

Here we used the identities

\begin{array}{l} tr {[R_{0} (z) V_{\cdot} R_{0} (z)]}_{1}^{2} = & tr {[V_{\cdot} R_{0} {(z)}^{2}]}_{1}^{2} = {(2 π h)}^{- n} \int {[V_{\cdot}]}_{1}^{2} d x \int {(z - | ξ |^{2})}^{- 2} d ξ, \\ (6.6) & - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) {(z - λ)}^{- q - 1} L (d z) = \frac{1}{q!} f^{(q)} (λ) . \end{array}

Similarly, using the formulae in Remark 6.2, (6.6) as well as the fact $\begin{array}{l} (6.7) & \int ξ^{β} f^{(l)} (| ξ |^{2}) d ξ = \{\begin{matrix} 0 & if β \notin {(2 N)}^{n}, \\ \frac{{(- 1)}^{| α |} (2 α)!}{2^{2 | α |} α!} \int f^{(l - | α |)} (| ξ |^{2}) d ξ & if β = 2 α \in {(2 N)}^{n}, \end{matrix} \end{array}$ that we easily see by integration by parts, we get $\begin{array}{l} {(2 π h)}^{n} I_{2} = & \frac{1}{2} \int f^{(2)} (| ξ |^{2}) d ξ \int {[V_{\cdot}^{2}]}_{1}^{2} d x - \frac{h^{2}}{3!} \frac{1}{2} \int f^{(3)} (| ξ |^{2}) d ξ \int {[V_{\cdot} Δ V_{\cdot}]}_{1}^{2} d x \\ + \frac{h^{4}}{4!} \frac{3}{5} \int f^{4)} (| ξ |^{2}) d ξ \int {[V_{\cdot} Δ^{2} V_{\cdot}]}_{1}^{2} d x + O (h^{6}), \\ {(2 π h)}^{n} I_{3} = & \frac{1}{3!} \int f^{(3)} (| ξ |^{2}) d ξ \int {[V_{\cdot}^{3}]}_{1}^{2} d x - \frac{h^{2}}{4!} \frac{2}{5} \int f^{(4)} (| ξ |^{2}) d ξ \int {[V_{\cdot} Δ V_{\cdot}^{2}]}_{1}^{2} d x \\ + \frac{h^{2}}{4!} \frac{4}{5} \int f^{(4)} (| ξ |^{2}) d ξ \int {[V_{\cdot}^{2} Δ V_{\cdot}]}_{1}^{2} d x \\ + \frac{h^{2}}{4!} \frac{2}{5} \int f^{(4)} (| ξ |^{2}) d ξ \int V_{\cdot} {[| \nabla V_{\cdot} |^{2}]}_{1}^{2} d x + O (h^{4}), \\ {(2 π h)}^{n} I_{4} = & \frac{1}{4!} \int f^{(4)} (| ξ |^{2}) d ξ \int {[V_{\cdot}^{4}]}_{1}^{2} d x + O (h^{2}) . \end{array}$ Now (2.4) and (2.5) follow from the fact that $\int_{0}^{\infty} f (| ξ |^{2}) d ξ = n κ_{0} \int_{0}^{\infty} f (r^{2}) r^{n - 1} d r$ .

Next we prove the general assertion of Theorem 2.1. By Proposition 6.1, we have $\begin{array}{l} I_{k} & = - \frac{1}{π} \int {\bar{\partial}}_{z} \tilde{f} (z) tr {[\sum_{l = 0}^{\infty} {(- 1)}^{l} P_{k, l} [V_{\cdot}] R_{0} {(z)}^{l + k + 1}]}_{1}^{2} L (d z) \\ = {(2 π h)}^{- n} \sum_{l = 0}^{\infty} \frac{{(- 1)}^{l}}{(l + k)!} h^{l} \sum_{ν = 0}^{l} h^{l - ν} \iint {[p_{k, l, ν} [V_{\cdot}]]}_{1}^{2} f^{(l + k)} (| ξ |^{2}) d x d ξ, \end{array}$ where we used the identity (6.6). Then by (6.3), we have $\begin{array}{l} {(2 π h)}^{n} tr {[f (H_{\cdot})]}_{1}^{2} \sim \sum_{k = 1}^{\infty} \sum_{l = 0}^{\infty} \sum_{ν = 0}^{l} \frac{{(- 1)}^{l} h^{2 k + 2 l - ν}}{(l + k)!} \iint {[p_{k, l, ν} [V_{\cdot}]]}_{1}^{2} f^{(l + k)} (| ξ |^{2}) d x d ξ . \end{array}$ Recall that $p_{k, l, ν} [V_{\cdot}]$ is a homogeneous polynomial of degree ν in ξ and write $\begin{array}{l} (6.8) & p_{k, l, ν} [V_{\cdot}] = \sum_{| β | = ν} p_{\cdot, k, l, β} (x) ξ^{β}, {[p_{\cdot, k, l, β} (x)]}_{1}^{2} = : q_{k, l, β} (x) . \end{array}$ We then obtain, using again the formula (6.7), $\begin{array}{l} tr {[f (H_{\cdot})]}_{1}^{2} & \sim {(2 π h)}^{- n} \sum_{k = 1}^{\infty} \sum_{l = 0}^{\infty} \sum_{ν = 0}^{l} \sum_{| β | = ν} \frac{{(- 1)}^{l} h^{2 k + 2 l - ν}}{(l + k)!} \int q_{k, l, β} (x) d x \int ξ^{β} f^{(l + k)} (| ξ |^{2}) d ξ \\ \sim {(2 π h)}^{- n} \sum_{k = 1}^{\infty} \sum_{l = 0}^{\infty} \sum_{2 | α | ⩽ l} \frac{{(- 1)}^{l + | α |} (2 α)!}{2^{2 | α |} α! (l + k)!} h^{2 (k + l - | α |)} \int f^{(l + k - | α |)} (| ξ |^{2}) d ξ \\ \sim \sum_{j = 1}^{\infty} c_{j} (f) h^{2 j - n}, \end{array}$ where $\begin{array}{l} c_{j} (f) & = {(2 π)}^{- n} \int f^{(j)} (| ξ |^{2}) d ξ \cdot \sum_{2 | α | ⩽ l ⩽ j + | α | - 1} \frac{{(- 1)}^{l + | α |} (2 α)!}{2^{2 | α |} α! (j + | α |)!} \int q_{j - l + | α |, l, 2 α} (x) d x \\ = \frac{n κ_{0}}{j! {(2 π)}^{n}} \int_{0}^{\infty} f^{(j)} (r^{2}) r^{n - 1} d r \int P_{j} (x) d x, \end{array}$ with $\begin{array}{l} (6.9) & P_{j} (x) = \sum_{2 | α | ⩽ l ⩽ j + | α | - 1} \frac{{(- 1)}^{l + | α |} (2 α)! j!}{2^{2 | α |} α! (j + | α |)!} q_{j - l + | α |, l, 2 α} (x) . \end{array}$ Remark that the sum of the right hand side is finite since $| α | ⩽ j - 1$ and $l ⩽ 2 j - 2$ .

It remains to check that $P_{j} (x)$ given by (6.9) is of the form ${[P_{j} ({D^{α} V_{\cdot}}_{| α | ⩽ 2 j - 4})]}_{1}^{2}$ with a universal polynomial $P_{j}$ of degree j. According to the definition (6.8) of $q_{k, l, β}$ , it is enough to check that $p_{\cdot, j - l + | α |, l, 2 α} (x)$ is a universal polynomial of degree j of $V_{\cdot}$ and its derivatives of order j when $2 | α | ⩽ l ⩽ j + | α | - 1$ . The degree of $p_{k, l, β}$ is not greater than k, and since $2 | α | ⩽ l$ implies $j - l + | α | ⩽ j$ , $p_{\cdot, j - l + | α |, l, 2 α} (x)$ is of degree at most j. The term in $p_{k, l} (x)$ with the highest order derivative of V is $h^{2 (k + l)} Δ^{l} V$ , $k ⩾ 2$ , and that in $P_{j}$ is $h^{2 j} Δ^{j - 2} V$ . This finishes the proof.

Remark 6.3.

For $m \neq 0$ , we decompose ${[R_{\cdot} {(z_{0})}^{m} R_{\cdot} (z)]}_{1}^{2}$ as $\begin{array}{l} {[R_{\cdot} {(z_{0})}^{m} R_{\cdot} (z)]}_{1}^{2} = {[R_{\cdot} {(z_{0})}^{m}]}_{1}^{2} R_{2} (z) + R_{1} {(z_{0})}^{m} {[R_{\cdot} (z)]}_{1}^{2} . \end{array}$ By Theorem A.4, $R_{j} (z_{0})$ is an h-pseudodifferential operator, and we can use the same arguments as above and show that the trace of the operators on the RHS has an asymptotic expansion in powers of $h^{2}$ for $z \in {z \in C, | ℑ z | > h^{δ}}$ with $δ \in] 0, \frac{1}{2} [$ . The remainder of the proof is the same.

Footnotes

h -psedodifferential operator calculus

This section is devoted to some basic results from semiclassical analysis, complex analysis and functional analysis. They are rather standard, but we include them for completeness.

A smooth positive function $m (ρ)$ , $ρ = (x, ξ)$ on the phase space $R^{2 n}$ is called an order function if it satisfies $\begin{array}{l} m (ρ) ⩽ m (ρ^{'}) {⟨ ρ - ρ^{'} ⟩}^{k}, ⟨ ρ ⟩ : = {(1 + | ρ |^{2})}^{1 / 2} \end{array}$ for some $k \in R$ . In particular, $m \equiv 1$ is an order function.

We say that $a (ρ)$ is a symbol in the class $S (m)$ if $\begin{array}{l} (A.1) & | \partial_{ρ}^{α} a (ρ) | ⩽ C_{α} m (ρ) . \end{array}$

If a depends on a semi-classical parameter $h \in] 0, h_{0}]$ and possibly on other parameters as well, we require (A.1) to hold uniformly with respect to these parameters. For h-dependent symbols, we say that $a (x, ξ; h) \in S (m)$ has an asymptotic expansion in powers of h, and we write $\begin{array}{l} a (x, ξ; h) \sim \sum_{j = 0}^{\infty} a_{j} (x, ξ) h^{j}, \end{array}$ if for every $k \in N$ , $a - \sum_{j = 0}^{k} a_{j} h^{j} \in h^{k + 1} S (m)$ .

For $a \in S (m)$ and $t \in [0, 1]$ , we introduce the h-pseudodifferential operator: $\begin{array}{l} {Op}_{h, t} (a) u (x) = \frac{1}{{(2 π h)}^{n}} \iint_{R^{2 n}} e^{i (x - y) \cdot ξ / h} a (t x + (1 - t) y, ξ; h) u (y) d ξ, u \in C_{0}^{\infty} (R^{n}) . \end{array}$ Here the case $t = 1$ corresponds to the classical quantization, which is traditionally used in the theory of partial differential equations, and $t = 1 / 2$ is the Weyl quantization (frequently used in mathematical physics). Even if there are different methods of quantization we get the same classes of pseudodifferential operators. So, each pseudodifferential operator has a classical and Weyl-symbols and the navigation between different symbols is known. Here we announce some results with the classical quantization, and we denote ${Op}_{h, 1} (a) = a (x, h D_{x})$ . and we refer to [9] for other cases.

For the proof we refer to chapters 7-8 in [9].

The following theorem gives an upper bound of the number of zeros of a holomorphic function. It is implicit in [32].

We end this appendix with a theorem on the determinant of a trace class operator (see Lemma 4 and Theorem XIII.105 in [28]).

Acknowledgements

The authors would like to express their sincere gratitude to the referee who carefully read the manuscript and provided valuable comments. The research of the second author has been supported by a JSPS grant-in-aid No. 21K03303.

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Spectral asymptotics for the Schrödinger operator with a non-decaying potential

Abstract

Keywords

1. Introduction

2. Weak asymptotics

2 Theorem 5.35 [18]: Let T be a closed operator on a Hilbert space H and let A be a relatively T-compact operator. Then σ ess ( T ) = σ ess ( T + A ) .

6 We define threshold energies as those energies at which the Mourre estimate (5.3) fails to hold (see Section 3 in [14]).

Footnotes

h -psedodifferential operator calculus

Acknowledgements

References

²
Theorem 5.35 [18]: Let T be a closed operator on a Hilbert space $H$ and let A be a relatively T-compact operator. Then $σ_{ess} (T) = σ_{ess} (T + A)$ .

⁶
We define threshold energies as those energies at which the Mourre estimate (5.3) fails to hold (see Section 3 in [14]).