Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators

Abstract

We analyze a general class of difference operators $H_{ε} = T_{ε} + V_{ε}$ on $ℓ^{2} ({(ε Z)}^{d})$ , where $V_{ε}$ is a multi-well potential and ε is a small parameter. We construct approximate eigenfunctions in neighbourhoods of the different wells and give weighted $ℓ^{2}$ -estimates for the difference of these and the exact eigenfunctions of the associated Dirichlet-operators.

Keywords

semi-classical difference operator tunneling WKB-expansions Dirichlet eigenfunctions asymptotic expansion multi-well potential Agmon estimates

1. Introduction

This paper is motivated by the aim to find complete asymptotic expansions for the tunneling effect between different wells of a difference operator with multi-well potential. More precisely, we investigate a rather general class of families of difference operators ${(H_{ε})}_{ε > 0}$ on the Hilbert space $ℓ^{2} ({(ε Z)}^{d})$ , as the small parameter $ε > 0$ tends to zero. The operator $H_{ε}$ is given by $\begin{array}{l} (1.1) & H_{ε} = (T_{ε} + V_{ε}), where T_{ε} = \sum_{γ \in {(ε Z)}^{d}} a_{γ} τ_{γ}, \\ (1.2) & (τ_{γ} u) (x) = u (x + γ), (a_{γ} u) (x) : = a_{γ} (x; ε) u (x) for x, γ \in {(ε Z)}^{d} \end{array}$ and $V_{ε}$ is a multiplication operator which in leading order is given by a multi-well potential $V_{0} \in C^{\infty} (R^{d})$ , i.e. $V_{0}$ has more than one non-degenerate minima.

By the tunneling effect, the interaction between neighboring potential wells leads to the fact that the eigenvalues and eigenfunctions are different from those of an operator with decoupled wells, which is realized by the direct sum of “Dirichlet-operators” situated at the several wells. Since the interaction is small, it can be treated as a perturbation of the decoupled system.

In [14], we showed that it is possible to approximate the eigenfunctions of the original Hamilton operator $H_{ε}$ with respect to a fixed spectral interval by the eigenfunctions of the several Dirichlet operators situated at the different wells and we gave a representation of $H_{ε}$ with respect to a basis of Dirichlet-eigenfunctions. In [13] we constructed formal asymptotic expansions for the Dirichlet-eigenfunctions at the wells.

In this paper, we show that to these formal expansions we can associate $C_{0}^{\infty}$ -functions defined in a larger neighbourhood of the wells. Moreover, we prove weighted $ℓ^{2}$ -estimates for the difference of these WKB-functions and the Dirichlet eigenfunctions. This will allow us to compute complete asymptotic expansions for the elements of the interaction matrix and to explicitly obtain the leading order term of eigenvalue splitting in a forthcoming paper. This idea is crucial in Helffer–Sjöstrand [8] for their treatment of semiclassical Schrödinger operators (see also Helffer [7] and Dimassi–Sjöstrand [6]). It depends on sharp semiclassical Agmon estimates (see Agmon [1] for the original version).

This paper is based on the thesis Rosenberger [15]. It is the fifth in a series of papers (see Klein–Rosenberger [11–14]); the aim is to develop an analytic approach to the semiclassical eigenvalue problem and tunneling for $H_{ε}$ which in detail and precision is comparable to the well-known analysis for the Schrödinger operator (see Simon [16] and Helffer–Sjöstrand [8]). Our motivation comes from stochastic problems (see Bovier–Eckhoff–Gayrard–Klein [3,4]). A large class of discrete Markov chains, analyzed in [4] with probabilistic techniques, falls into the framework of difference operators treated in this article.

We expect that results similar to this paper hold for WKB expansions associated with eigenfunctions of generators of jump processes in $R^{d}$ , see Klein–Léonard–Rosenberger [10] for first results in this direction.

Hypothesis 1.1.
For some $ε_{0} > 0$ ,
the coefficients $a_{γ} (x; ε)$ in (1.2) are functions $\begin{matrix} (1.3) & a : {(ε Z)}^{d} \times R^{d} \times (0, ε_{0}] \to R, (γ, x, ε) \mapsto a_{γ} (x; ε), \end{matrix}$ satisfying the following conditions:

They have an expansion $\begin{matrix} (1.4) & a_{γ} (x; ε) = \sum_{k = 0}^{N - 1} ε^{k} a_{γ}^{(k)} (x) + R_{γ}^{(N)} (x; ε), N \in N^{}, \end{matrix}$ where* $a_{γ} \in C^{\infty} (R^{d} \times (0, ε_{0}])$ and $a_{γ}^{(k)} \in C^{\infty} (R^{d})$ for all $γ \in {(ε Z)}^{d}$ and $0 ⩽ k ⩽ N - 1$ .

$\sum_{γ \in {(ε Z)}^{d}} a_{γ}^{(0)} = 0$ and $a_{γ}^{(0)} ⩽ 0$ for $γ \neq 0$ .

$a_{γ} (x; ε) = a_{- γ} (x + γ; ε)$ for all $x \in R^{d}$ , $γ \in {(ε Z)}^{d}$ , $ε \in (0, ε_{0}]$ .

For any $c > 0$ and $α \in N^{d}$ there exists $C > 0$ such that for $0 ⩽ k ⩽ N - 1$ , uniformly with respect to $x \in R^{d}$ and $ε \in (0, ε_{0}]$ , $\begin{matrix} (1.5) & {‖ e^{\frac{c | \cdot |}{ε}} \partial_{x}^{α} a_{\cdot}^{(k)} (x) ‖}_{ℓ_{γ}^{2} ({(ε Z)}^{d})} ⩽ C and {‖ e^{\frac{c | \cdot |}{ε}} \partial_{x}^{α} R_{\cdot}^{(N)} (x) ‖}_{ℓ_{γ}^{2} ({(ε Z)}^{d})} ⩽ C ε^{N} . \end{matrix}$

$span {γ \in {(ε Z)}^{d} ∣ a_{γ}^{(0)} (x) < 0} = R^{d}$ for all $x \in R^{d}$ .

For all $ε \in (0, ε_{0}]$ , the potential energy $V_{ε}$ is the restriction to ${(ε Z)}^{d}$ of a function ${\hat{V}}_{ε} \in C^{\infty} (R^{d}, R)$ which has an expansion $\begin{matrix} (1.6) & {\hat{V}}_{ε} (x) = \sum_{ℓ = 0}^{N - 1} ε^{ℓ} V_{ℓ} (x) + R_{N} (x; ε), N \in N^{}, \end{matrix}$ where* $V_{ℓ} \in C^{\infty} (R^{d})$ , $R_{N} \in C^{\infty} (R^{d} \times (0, ε_{0}])$ and for any compact set $K \subset R^{d}$ there exists a constant $C_{K}$ such that ${sup}_{x \in K} | R_{N} (x; ε) | ⩽ C_{K} ε^{N}$ .

$V_{ε}$ is polynomially bounded and there exist constants $R, C > 0$ such that $V_{ε} (x) > C$ for all $| x | ⩾ R$ and $ε \in (0, ε_{0}]$ .

$V_{0} (x) ⩾ 0$ and it takes the value 0 only at a finite number of non-degenerate minima $x_{j}$ , $j \in C = {1, \dots, r}$ , which we call potential wells.

If $T^{d} : = R^{d} / (2 π) Z^{d}$ denotes the d-dimensional torus and $b \in C^{\infty} (R^{d} \times T^{d} \times (0, ε_{0}])$ for some $ε_{0} > 0$ , a family of pseudo-differential operators ${Op}_{ε}^{T} (b) : K ({(ε Z)}^{d}) \to K^{'} ({(ε Z)}^{d})$ is defined by $\begin{matrix} (1.7) & {Op}_{ε}^{T} (b) v (x) : = {(2 π)}^{- d} \sum_{y \in {(ε Z)}^{d}} \int_{{[- π, π]}^{d}} e^{\frac{i}{ε} (y - x) ξ} b (x, ξ; ε) v (y) d ξ, \end{matrix}$ where $\begin{matrix} (1.8) & K ({(ε Z)}^{d}) : = {u : {(ε Z)}^{d} \to C ∣ u has compact support} \end{matrix}$ and $K^{'} ({(ε Z)}^{d}) : = {f : {(ε Z)}^{d} \to C}$ is dual to $K ({(ε Z)}^{d})$ by use of the scalar product ${⟨ u, v ⟩}_{ℓ^{2}} : = \sum_{x} \bar{u} (x) v (x)$ (for details and properties of such pseudo-differential operators see Klein–Rosenberger [12]).

We remark that for $T_{ε}$ defined in (1.1), under the assumptions given in Hypothesis 1.1, one has $T_{ε} = {Op}_{ε}^{T} (t)$ where $t \in C^{\infty} (R^{d} \times T^{d} \times (0, ε_{0}])$ is given by $\begin{matrix} (1.9) & t (x, ξ; ε) = \sum_{γ \in {(ε Z)}^{d}} a_{γ} (x; ε) exp (- \frac{i}{ε} γ \cdot ξ) . \end{matrix}$ Here $t (x, ξ; ε)$ is considered as a function on $R^{2 d} \times (0, ε_{0}]$ , which is $2 π$ -periodic with respect to ξ. By condition (a)(iv) in Hypothesis 1.1, the function $ξ \mapsto t (x, ξ; ε)$ has an analytic continuation to $C^{d}$ . Moreover for all $B > 0$ $\begin{matrix} (1.10) & \sum_{γ} | a_{γ} (x; ε) | e^{\frac{B | γ |}{ε}} ⩽ C and thus sup_{x \in R^{d}} | a_{γ} (x; ε) | ⩽ C e^{- \frac{B | γ |}{ε}} \end{matrix}$ for some $C > 0$ uniformly with respect to x and ε. We further remark that (a)(iv) implies $| a_{γ}^{(k)} (x) - a_{γ}^{(k)} (x + h) | ⩽ C | h |$ for $0 ⩽ k ⩽ N - 1$ uniformly with respect to $γ \in {(ε Z)}^{d}$ and $x, h \in R^{d}$ and (a)(ii), (iii), (iv) imply that $T_{ε}$ is symmetric and bounded and that for some $C_{T} > 0$ $\begin{matrix} (1.11) & {⟨ u, T_{ε} u ⟩}_{ℓ^{2}} ⩾ - C_{T} ε ‖ u ‖_{ℓ^{2}}^{2}, u \in ℓ^{2} ({(ε Z)}^{d}) . \end{matrix}$

Furthermore, we set $\begin{array}{l} t (x, ξ; ε) = \sum_{k = 0}^{N - 1} ε^{k} t_{k} (x, ξ) + {\tilde{t}}_{N} (x, ξ; ε) with \\ (1.12) & t_{k} (x, ξ) : = \sum_{γ \in {(ε Z)}^{d}} a_{γ}^{(k)} (x) e^{- \frac{i}{ε} γ ξ}, 0 ⩽ k ⩽ N - 1, \\ {\hat{t}}_{N} (x, ξ; ε) : = \sum_{γ \in {(ε Z)}^{d}} R_{γ}^{(N)} (x; ε) e^{- \frac{i}{ε} γ ξ} . \end{array}$ Thus, in leading order, the symbol of $H_{ε}$ is $h_{0} : = t_{0} + V_{0}$ . Combining (1.4) and (a)(iii) shows that the $2 π$ -periodic function $R^{d} ∋ ξ \mapsto t_{0} (x, ξ)$ is even with respect to $ξ \mapsto - ξ$ , i.e., $a_{γ}^{(0)} (x) = a_{- γ}^{(0)} (x)$ for all $x \in R^{d}$ , $γ \in {(ε Z)}^{d}$ (see [11], Lemma 1.2) and therefore $\begin{matrix} (1.13) & t_{0} (x, ξ) = \sum_{γ \in {(ε Z)}^{d}} a_{γ}^{(0)} (x) cos (\frac{1}{ε} γ \cdot ξ) . \end{matrix}$ At $ξ = 0$ , for fixed $x \in R^{d}$ the function $t_{0}$ defined in (1.12) has by Hypothesis 1.1(a)(ii) an expansion $\begin{matrix} (1.14) & t_{0} (x, ξ) = ⟨ ξ, B (x) ξ ⟩ + \sum_{\begin{array}{c} | α | = 2 n \\ n ⩾ 2 \end{array}} B_{α} (x) ξ^{α} as | ξ | \to 0, \end{matrix}$ where $α \in N^{d}$ , $B \in C^{\infty} (R^{d}, M (d \times d, R))$ , for any $x \in R^{d}$ the matrix $B (x)$ is positive definite and symmetric and $B_{α}$ are real functions. By straightforward calculations one gets for $1 ⩽ μ, ν ⩽ d$ $\begin{matrix} (1.15) & B_{ν μ} (x) = - \frac{1}{2 ε^{2}} \sum_{γ \in {(ε Z)}^{d}} a_{γ}^{(0)} (x) γ_{ν} γ_{μ} . \end{matrix}$

In order to work in the context of [12], we shall assume:
Hypothesis 1.2.
At the minima $x_{j}$ of $V_{0}$ , $j \in C$ , we assume that $t_{0}$ defined in (1.12) fulfills $\begin{matrix} t_{0} (x_{j}, ξ) > 0 if | ξ | > 0 . \end{matrix}$

We set, using (1.13), $\begin{matrix} (1.16) & \begin{matrix} {\tilde{h}}_{0} : R^{2 d} \to R, {\tilde{h}}_{0} (x, ξ) : = - h_{0} (x, i ξ) = : {\tilde{t}}_{0} (x, ξ) - V_{0} (x), where \\ {\tilde{t}}_{0} (x, ξ) = - \sum_{η \in Z^{d}} a_{ε η}^{(0)} (x) cosh (η \cdot ξ) . \end{matrix} \end{matrix}$ For any set $D \subset R^{d}$ , we denote the restriction to the lattice by $D_{ε} : = D \cap {(ε Z)}^{d}$ .

The first step in this paper is the construction of quasimodes for $H_{ε}$ . For a one well operator, such quasimodes were constructed near the potential well in [13], using the formal asymptotic expansions obtained in that paper. Here we generalize to our multiwell case and extend the quasimodes to larger domains, using the Hamiltonian flow associated to ${\tilde{h}}_{0}$ . In an adapted form, we restate some of the results of [13] in the Appendix.

To this end, we consider $H_{ε}$ as an operator on $L^{2} (R^{d})$ , i.e. we construct quasimodes with respect to the operator on $L^{2} (R^{d})$ $\begin{matrix} (1.17) & {\hat{H}}_{ε} = {\hat{T}}_{ε} + {\hat{V}}_{ε} with {\hat{T}}_{ε} = \sum_{γ \in {(ε Z)}^{d}} a_{γ} τ_{γ}, \end{matrix}$ where $a_{γ} τ_{γ}$ is the translation operator defined in (1.2).

The operator on $L^{2} (R^{d})$ associated to $h_{0}$ is given by $\begin{matrix} (1.18) & {\hat{H}}_{0} : = \sum_{γ} a_{γ}^{(0)} τ_{γ} + V_{0} . \end{matrix}$ The harmonic oscillator at the potential wells $x_{j}$ , $j \in C$ , associated to ${\hat{H}}_{0}$ is given by the operator on $L^{2} (R^{d})$ $\begin{matrix} (1.19) & {\hat{H}}_{0, q}^{j} (x, ε D) = - ε^{2} ⟨ D, B_{j} D ⟩ + ⟨ x, A_{j} x ⟩ + ε (t_{1} (x_{j}, 0) + V_{1} (x_{j})), \end{matrix}$ where $A_{j} = D^{2} V_{0} |_{x_{j}}$ and $B_{j} = B (x_{j})$ for B given in (1.15). We introduce the unitary transformation $\begin{matrix} (1.20) & U_{j} f (x) = | det B_{j} |^{- \frac{1}{4}} f (C_{j} x), C_{j} : = R_{j} B_{j}^{- \frac{1}{2}} \end{matrix}$ on $L^{2} (R^{d})$ , where $R_{j} \in SO (d, R)$ , such that $Λ_{j} : = R_{j} B_{j}^{\frac{1}{2}} A_{j} B_{j}^{\frac{1}{2}} R_{j}^{T}$ is diagonal. Then the operator $\begin{matrix} (1.21) & {\hat{H}}_{0, j}^{'} : = U_{j}^{- 1} τ_{- x_{j}} {\hat{H}}_{0} τ_{x_{j}} U_{j} \end{matrix}$ and is centered at 0 and, for $z = x - x_{j}$ , the associated symbol $h_{0, j} (z, ζ) = h_{0} (C_{j}^{- 1} z, C_{j}^{T} ζ)$ with respect to the ε-quantization given in (1.7) is in leading order diagonal: $\begin{matrix} h_{0, j} (z, ζ) = ζ^{2} + ⟨ z, Λ z ⟩ + O (| z |^{3}) + O (| ζ |^{3}), Λ = diag (λ_{1}, \dots, λ_{n}) \end{matrix}$ (the transformation of $h_{0}$ follows from the usual change of variable formulae for pseudo-differential operators in view of e.g. (A.4) in [12]). Moreover, for ${\hat{H}}_{ε}$ and $U_{j}$ given in (1.17) and (1.20), we set $\begin{matrix} (1.22) & {\hat{H}}_{ε, j}^{'} : = U_{j}^{- 1} τ_{- x_{j}} {\hat{H}}_{ε} τ_{x_{j}} U_{j} . \end{matrix}$

By Hypothesis 1.1, ${\tilde{h}}_{0}$ is hyperregular, and even and strictly convex in each fibre (cf. [11]). We can thus introduce the associated Finsler distance $d = d_{ℓ}$ on $R^{d}$ as in [11], Definition 2.16, where we set $\tilde{M} : = R^{d} ∖ {x_{k}, k \in C}$ . Analog to [11], Theorem 1.6, it can be shown that d is locally Lipschitz and that for any $j \in C$ , the distance $d^{j} (x) : = d (x, x_{j})$ fulfills the generalized eikonal equation and inequality respectively $\begin{array}{l} (1.23) & {\tilde{h}}_{0} (x, \nabla d^{j} (x)) = 0, x \in Ω^{j}, \\ (1.24) & {\tilde{h}}_{0} (x, \nabla d^{j} (x)) ⩽ 0, x \in R^{d}, \end{array}$ where $Ω^{j}$ is some neighborhood of $x_{j}$ .

It follows directly that for each $j \in C$ $\begin{matrix} (1.25) & {\hat{d}}^{j} (z) : = d^{j} (x_{j} + C_{j}^{- 1} z) \end{matrix}$ is the Finsler distance associated to ${\tilde{h}}_{0, j}$ defined in (1.26), satisfies the eikonal equation $\begin{matrix} (1.26) & {\tilde{h}}_{0, j} (z, \nabla {\hat{d}}^{j} (z)) : = - h_{0, j} (z, i \nabla {\hat{d}}^{j} (z)) = 0, z \in C_{j} (Ω^{j} - x_{j}), \end{matrix}$ where $Ω^{j}$ is some neighborhood of $x_{j}$ and fulfills $\begin{matrix} (1.27) & | {\hat{d}}^{j} (z) - {\hat{d}}_{0}^{j} (z) | = O (| z |^{3}) (z \to C_{j} x_{j}) for {\hat{d}}_{0}^{j} (z) = \sum_{ν = 1}^{d} \frac{λ_{ν}^{j}}{2} z_{ν}^{2}, \end{matrix}$ where $λ_{ν}^{j} > 0$ , $ν = 1, \dots, d$ , are the eigenvalues of $A_{j}$ .

We remark that, assuming only Hypothesis 1.1, it is possible that balls $B_{r} (x) : = {y \in R^{d} ∣ d (x, y) ⩽ r}$ , $r < \infty$ , are unbounded in the Euclidean distance (and thus not compact). In this paper, we shall not discuss consequences of this effect.
Remark 1.3.
Since d is locally Lipschitz-continuous, it follows from (1.10) that for any $B > 0$ and any bounded region $Σ \subset R^{d}$ there exists a constant $C > 0$ such that $\begin{matrix} (1.28) & \sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ | γ | < B \end{array}} {‖ a_{γ} (\cdot; ε) e^{\frac{d (\cdot, \cdot + γ)}{ε}} ‖}_{l^{\infty} (Σ)} ⩽ C . \end{matrix}$

For a global estimate on the decay of $a_{γ}$ for large γ, we assume in addition:
Hypothesis 1.4.
There exist constants $η > 0$ and $C > 0$ such that for all $x \in R^{d}$ $\begin{matrix} {‖ a_{(\cdot)} (x; ε) e^{\frac{1}{ε} d (x, x + \cdot)} | \cdot |^{\frac{d + η}{2}} ‖}_{ℓ^{2} ({(ε Z)}^{d})} ⩽ C . \end{matrix}$

For the WKB-expansion of the eigenfunctions constructed in [13] and the weighted norm-estimates for the Dirichlet eigenfunctions shown in [11], it was essential that the potential $V_{0}$ had exactly one minimum. To use these results it is necessary to restrict $H_{ε}$ to regions around the wells, which exclude all other wells.

For $Σ \subset R^{d}$ we define the space $ℓ_{Σ_{ε}}^{2} : = i_{Σ_{ε}} (ℓ^{2} (Σ_{ε})) \subset ℓ^{2} ({(ε Z)}^{d})$ where $i_{Σ_{ε}}$ denotes the embedding via zero extension. Then we define the Dirichlet operator $\begin{matrix} (1.29) & H_{ε}^{Σ} : = 1_{Σ_{ε}} H_{ε} |_{ℓ_{Σ_{ε}}^{2}} : ℓ_{Σ_{ε}}^{2} \to ℓ_{Σ_{ε}}^{2} \end{matrix}$ with domain $D (H_{ε}^{Σ}) = {u \in ℓ_{Σ_{ε}}^{2} ∣ V_{ε} u \in ℓ_{Σ_{ε}}^{2}}$ .
Hypothesis 1.5.

For $j \in C$ , we choose a compact manifold $M_{j} \subset R^{d}$ with $C^{2}$ -boundary such that $x_{j} \in M_{j}$ , $d^{j} \in C^{1} (M_{j})$ and $x_{k} \notin M_{j}$ for $k \in C$ , $k \neq j$ .

Given $M_{j}$ , $j \in C$ , let $I_{ε} = [α (ε), β (ε)]$ be an interval, such that $α (ε), β (ε) \to 0$ for $ε \to 0$ . Furthermore there exists a function $a (ε) > 0$ with the property $| log a (ε) | = o (\frac{1}{ε})$ , $ε \to 0$ , such that none of the operators $H_{ε}, H_{ε}^{M_{1}}, \dots, H_{ε}^{M_{r}}$ has spectrum in $[α (ε) - 2 a (ε), α (ε) [$ or $] β (ε), β (ε) + 2 a (ε)]$ .

The lattice subset associated to $M_{j}$ is denoted by $M_{j, ε} : = M_{j} \cap {(ε Z)}^{d}$ and we denote the eigenvalues of $H_{ε}$ and of the Dirichlet operators $H_{ε}^{M_{j}}$ defined in (1.29) inside the spectral interval $I_{ε}$ and the corresponding real orthonormal systems of eigenfunctions (these exist because all operators commute with complex conjugation) $\begin{matrix} (1.30) & \begin{matrix} spec (H_{ε}) \cap I_{ε} = {λ_{1}, \dots, λ_{N}}, u_{1}, \dots, u_{N} \in ℓ^{2} ({(ε Z)}^{d}), \\ F : = span {u_{1}, \dots, u_{N}}, \\ spec (H_{ε}^{M_{j}}) \cap I_{ε} = {μ_{j, 1}, \dots, μ_{j, n_{j}}}, v_{j, 1}, \dots, v_{j, n_{j}} \in ℓ_{M_{j, ε}}^{2}, j \in C, \\ E_{j} : = span {v_{j, 1}, \dots, v_{j, n_{j}}}, E : = ⨁ E_{j} . \end{matrix} \end{matrix}$ We write $\begin{matrix} (1.31) & v_{α} with α = (α_{1}, α_{2}) \in J : = {(j, k) ∣ j \in C, 1 ⩽ k ⩽ n_{j}} and j (α) : = α_{1} . \end{matrix}$ We remark that the number of eigenvalues N, $n_{j}$ , $j \in C$ with respect to $I_{ε}$ as defined in (1.30) may depend on ε.

For a fixed spectral interval it is shown in [14] that the difference between the exact spectrum and the spectra of Dirichlet realizations of $H_{ε}$ near the different wells is exponentially small and determined by the Finsler distance between the two nearest neighboring wells.

The first result in this paper generalizes the results of [13] where we constructed quasimodes for $H_{ε}$ (with only one potential well $x_{0} = 0$ ) in a small neighborhood of $x_{0}$ . Here we consider the case of several potential wells and construct quasimodes globally on $M_{j}$ under some additional assumptions.
Hypothesis 1.6.
Let $X_{{\tilde{h}}_{0}}$ denote the Hamiltonian vector field of ${\tilde{h}}_{0}$ defined in (1.16) , $F_{t}$ denote the flow of $X_{{\tilde{h}}_{0}}$ and set $\begin{matrix} (1.32) & Λ_{\pm} : = {(x, ξ) \in T^{} R^{d} ∣ F_{t} (x, ξ) \to (x_{j}, 0) for t \to \mp \infty} . \end{matrix}$
For* $j \in C$ , let $Ω^{j} \subset R^{d}$ containing $x_{j}$ , such that the following holds:
For $π : T^{} R^{d} \to R^{d}$ denoting the bundle projection* $π (x, ξ) = x$ , we have $\begin{matrix} Λ_{+} (Ω^{j}) : = π^{- 1} (Ω^{j}) \cap Λ_{+} = {(x, \nabla d^{j} (x)) \in T^{} R^{d} ∣ x \in Ω^{j}} . \end{matrix}$

$d^{j} \in C^{2} (Ω^{j})$ .

$π (F_{t} (x, ξ)) \in Ω^{j}$ for all* $(x, ξ) \in π^{- 1} (Ω^{j}) \cap Λ_{+}$ and all $t ⩽ 0$ .
The restriction of $Ω^{j}$ to the lattice is denoted by $Ω_{ε}^{j}$ .

For $j \in C$ let $M_{j}$ satisfy Hypothesis 1.5 and assume in addition that $M_{j} \subset Ω^{j}$ and that (a) holds for $M_{j}$ replacing $Ω^{j}$ .

By [11], Theorem 1.5, the base integral curves of $X_{{\tilde{h}}_{0}}$ on $R^{d} ∖ {x_{1}, \dots, x_{m}}$ with energy 0 are geodesics with respect to d and vice versa. Thus the above hypothesis implies in particular that there is a unique minimal geodesic between any point in $Ω^{j}$ and $x_{j}$ (see Fig. 1).

Fig. 1.
The regions $M_{j}$ and $Ω^{j}$ and the projection of the Hamilton flow to x-space.

Clearly, $Λ_{+} (Ω^{j})$ is a Lagrange manifold (by (a)(i)) and since the flow $F_{t}$ preserves ${\tilde{h}}_{0}$ , we have $Λ_{+} (Ω^{j}) \subset {\tilde{h}}_{0}^{- 1} (0)$ by (1.32). Thus the eikonal equation ${\tilde{h}}_{0} (x, \nabla d^{j} (x)) = 0$ holds for $x \in Ω^{j}$ . It follows from the construction of the solution of the eikonal equation in [13] that in fact $d^{j} \in C^{\infty} (Ω^{j})$ .

We recall from Remark 1.4 in [13] that, locally near $x_{j}$ , the Lagrange manifolds $Λ_{\pm}$ are parametrized by $\pm d^{j}$ (this follows combining Lemma 1.3 in [13] with the proof of Theorem 1.5 in [11]). Thus, geometrically speaking, Hypothesis 1.6(a) means that $Λ_{\pm} (Ω^{j})$ projects diffeomorphically to $Ω^{j}$ .

If ${\hat{Ω}}^{j}$ denotes an open neighborhood of $Ω^{j}$ given in Hypothesis 1.6, let $χ_{j} \in C_{0}^{\infty} (R^{d})$ denote a cut-off function with $χ_{j} (x) = 1$ for $x \in Ω^{j}$ and $supp χ_{j} \subset {\hat{Ω}}^{j}$ . Then we set $\begin{matrix} (1.33) & {\tilde{d}}^{j} (x) : = χ_{j} (x) d^{j} (x) + (1 - χ_{j} (x)) | x - x_{j} | . \end{matrix}$
Theorem 1.7.
Let $H_{ε}$ be a Hamilton operator satisfying Hypotheses 1.1 , 1.2 and 1.4. For $i, j \in C$ , let $Ω^{j}$ , $M_{j}$ satisfy Hypothesis 1.6 and set $S_{i j} : = d (x_{i}, x_{j})$ . Furthermore we assume that $ε E^{j}$ denotes an eigenvalue of ${\hat{H}}_{0, q}^{j}$ defined in (1.19) with multiplicity $m_{j}$ . Then, for $j \in C$ , there are functions $b_{k}^{j} \in C_{0}^{\infty} (R^{d} \times (0, ε_{0}])$ , $b_{k, ℓ}^{j} \in C_{0}^{\infty} (R^{d})$ , $k = 1, \dots, m_{j}$ , $ℓ \in \frac{Z}{2}$ , $ℓ ⩾ - N_{j}$ for some $N_{j} \in N$ , supported in $Ω_{j}$ , such that for all $M \in \frac{Z}{2}$ there are $C_{M} < \infty$ satisfying $\begin{matrix} (1.34) & | b_{k}^{j} (z; ε) - \sum_{\begin{array}{c} ℓ \in \frac{Z}{2} \\ - N_{j} ⩽ ℓ ⩽ M \end{array}} ε^{ℓ} b_{k, ℓ}^{j} (z) | ⩽ C_{M} ε^{M + \frac{1}{2}} (z \in R^{d}) \end{matrix}$ and real functions $E_{k}^{j} (ε)$ with asymptotic expansion $\begin{matrix} (1.35) & E_{k}^{j} (ε) \sim E^{j} + \sum_{s \in \frac{N^{}}{2}} ε^{s} E_{k, s}^{j}, \end{matrix}$ such that:*
For ${\tilde{d}}^{j}$ defined in (1.33), the functions $\begin{matrix} (1.36) & {\hat{v}}_{j, k} : = ε^{- \frac{d}{4}} e^{- \frac{{\tilde{d}}^{j}}{ε}} b_{k}^{j} \end{matrix}$ are almost orthonormal in the sense that $\begin{matrix} (1.37) & {⟨ {\hat{v}}_{j, k}, {\hat{v}}_{i, ℓ} ⟩}_{L^{2}} = δ_{i j} δ_{k ℓ} + δ_{i j} O (ε^{\infty}) + O (ε^{- (N_{i} + N_{j} + \frac{d}{2})} e^{- \frac{S_{i j}}{ε}}) . \end{matrix}$

For any $x_{0} \in R^{d}$ , the functions $\begin{matrix} (1.38) & {\hat{v}}_{j, k, x_{0}}^{ε} : = ε^{\frac{d}{2}} r_{G_{x_{0}}} {\hat{v}}_{j, k} \end{matrix}$ on the lattice $G_{x_{0}} = {(ε Z)}^{d} + x_{0}$ are approximate eigenfunctions for the operator $H_{ε}$ with respect to the approximate eigenvalues given in (1.35), i.e., $\begin{matrix} (1.39) & e^{\frac{d^{j} (x)}{ε}} (H_{ε} - ε E_{k}^{j} (ε)) {\hat{v}}_{j, k, x_{0}}^{ε} (x) = O (ε^{\infty}) (x \in M_{j} \cap G_{x_{0}}, ε \to 0) . \end{matrix}$

For the restricted approximate eigenfunctions ${\hat{v}}_{j, k, x_{0}}^{ε}$ , we have $\begin{matrix} (1.40) & {⟨ {\hat{v}}_{j, k, x_{0}}^{ε}, {\hat{v}}_{i, ℓ, x_{0}}^{ε} ⟩}_{ℓ^{2} (G_{x_{0}})} = δ_{i j} δ_{k ℓ} + δ_{i j} O (ε^{\infty}) + O (ε^{- (N_{i} + N_{j} + \frac{d}{2})} e^{- \frac{S_{i j}}{ε}}) . \end{matrix}$

The second result of this paper is the following theorem, in which we compare the asymptotic eigenfunctions with respect to $ε E^{j}$ derived in Theorem 1.7 with the exact Dirichlet eigenfunctions associated to a spectral interval around this eigenvalue of diameter $C_{0} ε^{\frac{3}{2}}$ .
Theorem 1.8.
Let $H_{ε}$ satisfy Hypotheses 1.1 , 1.2 and 1.4. For $j \in C$ , let $Ω^{j}$ , $M_{j}$ satisfy Hypothesis 1.6. Furthermore we assume that $ε E^{j}$ denotes an eigenvalue of ${\hat{H}}_{0, q}^{j}$ defined in (1.19) with multiplicity $m_{j}$ and we set $I_{ε} (E^{j}) = [ε E^{j} - C_{0} ε^{\frac{3}{2}}, ε E^{j} + C_{0} ε^{\frac{3}{2}}]$ . Let $v_{j, k}$ , $1 ⩽ k ⩽ m_{j}$ , denote real orthonormal eigenfunctions of the Dirichlet-operator $H_{ε}^{M_{j}}$ defined in (1.29) with respect to the spectral interval $I_{ε} (E^{j})$ . Let ${\hat{v}}_{j, k}^{ε} : = {\hat{v}}_{j, k, 0}^{ε}$ , $1 ⩽ k ⩽ m_{j}$ , be the WKB-functions associated to $ε E_{j}$ , as defined in Theorem 1.7 , (1.38) and set ${\tilde{v}}_{j, k}^{ε} : = \sum_{ℓ = 1}^{m_{j}} c_{k, ℓ} (ε) {\hat{v}}_{j, ℓ}^{ε}$ where the orthogonal matrix $c_{k, ℓ} (ε)$ is given in (3.9) for $ε > 0$ small enough.

Then for every compact set $K \subset M_{j}$ and any $N \in N$ $\begin{matrix} {‖ e^{\frac{d^{j}}{ε}} (v_{j, k} - {\tilde{v}}_{j, k}) ‖}_{ℓ^{2} (K_{ε})} = O (ε^{N}) (ε \to 0), \end{matrix}$ where $K_{ε} : = K \cap {(ε Z)}^{d}$ .

We remark that $c_{k, ℓ} (ε) = 0$ if $E_{k}^{j} (ε)$ is not asymptotically equal to $μ_{j, ℓ}$ and ${(c_{j, ℓ})}_{1 ⩽ j, ℓ ⩽ m_{j}}$ can be chosen to be the identity matrix if all $E_{k}^{j} (ε)$ have different expansions.

The outline of the paper is as follows.

Section 2 consists of the proof of Theorem 1.7, in Section 3 we prove Theorem 1.8. In the Appendix, we restate (adapted versions of) former results used in the proofs.
2. Proof of Theorem 1.7

For $j \in C$ , we use the system of formal power series solutions ${\hat{b}}_{k}^{j} \in A_{j}$ of $H_{ε, j, A}^{'}$ (orthonormal with respect to ${⟨ \cdot, \cdot ⟩}_{A_{j}}$ ) with associated eigenvalues $ε E_{k}^{j} (ε)$ given in Theorem A.4.

By Borel summation (with respect to z), we can find $C^{\infty}$ -functions $g_{k ℓ}^{j}$ , compactly supported in $C_{j} (Ω^{j} - x_{j})$ , possessing ${\hat{b}}_{k}^{j}$ as Taylor series at $C_{j} x_{j}$ . We define a formal asymptotic series in $Ω^{j}$ by $\begin{matrix} g_{k}^{j} (x; ε) : = \sum_{\begin{array}{c} ℓ \in \frac{Z}{2} \\ ℓ ⩾ - N \end{array}} ε^{ℓ} U_{j} g_{k ℓ}^{j} (x - x_{j}), \end{matrix}$ where $U_{j}$ is the unitary transformation given in (1.20). Then by Theorem A.4 and since ${\tilde{d}}^{j} = d^{j}$ on $Ω^{j}$ and ${\tilde{\hat{d}}}^{j} = {\hat{d}}^{j}$ on $C_{j} (Ω^{j} - x_{j})$ , with $x = x_{j} + z$ , $\begin{matrix} (2.1) & \begin{matrix} e^{\frac{{\hat{d}}^{j} (z)}{ε}} ({\hat{H}}_{ε, j}^{'} - ε E_{k}^{j} (ε)) e^{- \frac{{\hat{d}}^{j} (z)}{ε}} \sum_{\begin{array}{c} ℓ \in \frac{Z}{2} \\ ℓ ⩾ - N \end{array}} ε^{ℓ} g_{k ℓ}^{j} (z) & = e^{\frac{d^{j} (x)}{ε}} ({\hat{H}}_{ε} - ε E_{k}^{j} (ε)) e^{- \frac{d^{j} (x)}{ε}} g_{k}^{j} (x, ε) \\ = r_{k}^{j} (x; ε) \end{matrix} \end{matrix}$ in the sense of formal power series where $r_{k}^{j} (x; ε) = \sum_{\begin{array}{c} ℓ \in \frac{Z}{2} \\ ℓ ⩾ - N \end{array}} ε^{ℓ} r_{k ℓ}^{j} (x)$ has the property, that each $r_{k ℓ}^{j}$ is in $C_{0}^{\infty} (Ω^{j})$ and vanishes to infinite order at $x = x_{j}$ . It remains to show that it is possible to modify the functions $U_{j} g_{k, ℓ}^{j} (\cdot - x_{j})$ by (uniquely determined) functions $c_{k ℓ}^{j}$ vanishing at $x_{j}$ to infinite order such that, for the resulting functions ${\tilde{b}}_{k ℓ}^{j} : = U_{j} g_{k ℓ}^{j} (\cdot - x_{j}) - c_{k ℓ}^{j}$ , the formal series $\begin{matrix} (2.2) & {\tilde{b}}_{k}^{j} (x; ε) : = \sum_{\begin{array}{c} ℓ ⩾ - N \\ ℓ \in Z / 2 \end{array}} ε^{ℓ} {\tilde{b}}_{k ℓ}^{j} (x) \end{matrix}$ solves, for $x \in M_{j}$ , the equation $\begin{matrix} (2.3) & e^{\frac{d^{j} (x)}{ε}} ({\hat{H}}_{ε} - ε E_{k}^{j} (ε)) e^{- \frac{d^{j} (x)}{ε}} {\tilde{b}}_{k}^{j} (x, ε) = 0 \end{matrix}$ in the sense of formal power series. To this end, we have to show that the equation $\begin{matrix} (2.4) & e^{\frac{{\tilde{d}}^{j} (x)}{ε}} ({\hat{H}}_{ε} - ε E_{k}^{j} (ε)) e^{- \frac{{\tilde{d}}^{j} (x)}{ε}} c_{k}^{j} (x, ε) = r_{k}^{j} (x; ε) \end{matrix}$ has a unique formal power series solution $c_{k}^{j} (x; ε) = \sum_{ℓ ⩾ - N} ε^{ℓ} c_{k ℓ}^{j} (x)$ with coefficients $c_{k ℓ}^{j} \in C^{\infty} (R^{d})$ vanishing to infinite order at $x = x_{j}$ . By the definition of ${\hat{H}}_{ε}$ in (1.17), the assumptions in Hypothesis 1.1 and (A.10), the left-hand side of (2.4) is given by $\begin{array}{l} e^{\frac{{\tilde{d}}^{j} (x)}{ε}} [{\hat{T}}_{ε} + {\hat{V}}_{ε} - ε (E^{j} + \sum_{m \in N^{*} / 2} ε^{m} E_{k m}^{j})] e^{- \frac{{\tilde{d}}^{j} (x)}{ε}} \sum_{\begin{array}{c} ℓ ⩾ - N \\ ℓ \in Z / 2 \end{array}} ε^{ℓ} c_{k ℓ}^{j} (x) \\ = \sum_{\begin{array}{c} ℓ ⩾ - N \\ ℓ \in Z / 2 \end{array}} ε^{ℓ} {\sum_{γ \in {(ε Z)}^{d}} [\sum_{m \in N} ε^{m} a_{γ}^{(m)} (x) e^{\frac{1}{ε} ({\tilde{d}}^{j} (x) - {\tilde{d}}^{j} (x + γ))} c_{k ℓ}^{j} (x + γ)] \\ (2.5) & + \sum_{m \in N / 2} ε^{m} (V_{m} (x) - ε E_{k m}^{j}) c_{k ℓ}^{j} (x)}, \end{array}$ where we set $E_{k 0}^{j} : = E^{j}$ and $V_{m} = 0$ for $m \notin N$ . To get the different orders in ε of the kinetic term, i.e. the first summand in RHS (2.5), we expand ${\tilde{d}}^{j}$ and $c_{k ℓ}^{j}$ at x and set $η : = \frac{γ}{ε} \in Z^{d}$ . Taylor expansion gives $\begin{matrix} (2.6) & \begin{matrix} \frac{1}{ε} ({\tilde{d}}^{j} (x) - {\tilde{d}}^{j} (x + ε η)) \\ = - \nabla {\tilde{d}}^{j} (x) \cdot η - \frac{ε}{2} D^{2} {\tilde{d}}^{j} |_{x} {[η]}^{2} - \frac{ε^{2}}{2} \int_{0}^{1} {(1 - t)}^{2} D^{3} {\tilde{d}}^{j} |_{x + t ε η} {[η]}^{3} d t \end{matrix} \end{matrix}$ and $\begin{matrix} (2.7) & c_{k, ℓ}^{j} (x + ε η) = c_{k, ℓ}^{j} (x) + ε η \cdot \nabla c_{k, ℓ}^{j} (x) + ε^{2} \int_{0}^{1} (1 - t) D^{2} c_{k, ℓ}^{j} |_{x + t ε η} {[η]}^{2} d t . \end{matrix}$ Combining (2.6) with the expansion of the exponential function at zero gives $\begin{array}{l} e^{\frac{1}{ε} ({\tilde{d}}^{j} (x) - {\tilde{d}}^{j} (x + ε η))} = & e^{- \nabla {\tilde{d}}^{j} (x) \cdot η} (1 - \frac{ε}{2} D^{2} {\tilde{d}}^{j} |_{x} {[η]}^{2} + \frac{ε^{2}}{4} {(D^{2} {\tilde{d}}^{j} |_{x} {[η]}^{2})}^{2} + O (ε^{4})) \\ (2.8) & \times (1 - \frac{ε^{2}}{2} \int_{0}^{1} {(1 - t)}^{2} D^{3} {\tilde{d}}^{j} |_{x + t ε η} {[η]}^{3} d t + O (ε^{4})) (1 + O (ε^{3})) . \end{array}$ To lowest order (i.e. to order $ε^{- N}$ ), Eq. (2.4) is given by $\begin{matrix} (2.9) & (t_{0} (x, \nabla {\tilde{d}}^{j} (x)) + V_{0} (x)) c_{k, - N}^{j} (x) = r_{k, - N}^{j} (x) . \end{matrix}$ By the eikonal Eq. (1.26) (which holds in $Ω^{j}$ by Hypothesis 1.6), the left-hand side of (2.9) vanishes in $Ω^{j}$ . The same argument applies Eq. (2.4) to order $ε^{- N + \frac{1}{2}}$ : $\begin{matrix} (t_{0} (x, \nabla {\tilde{d}}^{j} (x)) + V_{0} (x)) c_{k, - N + \frac{1}{2}}^{j} (x) = r_{k, - N + \frac{1}{2}}^{j} (x) . \end{matrix}$ The first non-vanishing term in the expansion of (2.4) arises from the action of the first-order part of the conjugated operator on $c_{k, - N}^{j} (x)$ , which is given by $\begin{array}{l} {\sum_{γ \in {(ε Z)}^{d}} e^{- \frac{1}{ε} \nabla {\tilde{d}}^{j} (x) \cdot γ} [a_{γ}^{(0)} (x) (\frac{1}{ε} γ \cdot \nabla - \frac{1}{2 ε} ⟨ γ, D^{2} {\tilde{d}}^{j} |_{x} γ ⟩) + a_{γ}^{(1)} (x)] + V_{1} (x) - E} c_{k, - N}^{j} (x) \\ (2.10) & = r_{k, - N + 1}^{j} . \end{array}$ This equation takes the form $\begin{matrix} (2.11) & (P (x, \partial_{x}) + f (x)) u (x) = v (x) \end{matrix}$ for the differential operator $\begin{matrix} (2.12) & P (x, \partial_{x}) = Z (x) \cdot \nabla for Z (x) : = \sum_{η \in Z^{d}} a_{ε η}^{(0)} (x) e^{- η \cdot \nabla {\tilde{d}}^{j} (x)} η, \end{matrix}$ which is well defined by the exponential decay of $a_{ε η}^{(0)}$ (see Hypothesis 1.1(a)(iv)) and since $\begin{matrix} (2.13) & | \nabla_{x} {\tilde{d}}^{j} |_{\sqrt{ε} y} | = O (\sqrt{ε}) and \partial_{x}^{α} {\tilde{d}}^{j} |_{\sqrt{ε} y} = O (1), | α | > 1 (ε \to 0) . \end{matrix}$ It follows from the definition of ${\tilde{h}}_{0}$ in (1.16) that Z is the velocity field $\begin{matrix} (2.14) & Z (x) = \nabla_{ξ} {\tilde{h}}_{0} (x, ξ = \nabla {\tilde{d}}^{j} (x)) . \end{matrix}$ Thus, by Hypothesis 1.6, $Z (x)$ is the projection of the Hamilton field $X_{{\tilde{h}}_{0}}$ of ${\tilde{h}}_{0}$ , evaluated on the Lagrange manifold $Λ_{+} (Ω^{j})$ , onto the configuration space.

We define a cut-off-function $ζ_{j} \in C_{0}^{\infty} (Ω^{j})$ such that $ζ_{j} (x) = 1$ for $x \in M_{j}$ and we set ${\tilde{c}}_{k, - N}^{j} : = ζ_{j} c_{k, - N}^{j}$ . The next and all higher-order equations in (2.4) result from the action of the first-order part of the conjugated operator given in (2.10) on the respective highest-order part of $c_{k}^{j}$ , which for the ℓth order is the term $c_{k, ℓ - 1}^{j}$ . Additionally to the first-order equation, a term is produced by the action of higher-order terms of the conjugated operator on lower-order terms of $c_{k}^{j}$ , which we replace by ${\tilde{c}}_{k}^{j}$ . Since these lower-order terms are already determined by the preceding transport equations, this additional part can be treated as an additional inhomogeneity of (2.11). Thus all transport equations take the form (2.11) with $f \in C^{\infty} (R^{d})$ and $v \in C_{0}^{\infty} (Ω^{j})$ vanishing to infinite-order at $x = x_{j}$ by the construction of the formal series (A.9). In lowest-order (i.e. in order $ε^{- N}$ and $ε^{- N + \frac{1}{2}}$ ) the transport Eq. (2.11) is homogeneous (i.e. $v = 0$ ).

In order to show that the transport equations can be solved in the space of $C^{\infty}$ -functions vanishing to infinite-order at $x_{j}$ , we first remark that $x_{j}$ is a singular point of the vector field Z. In fact, $\nabla_{ξ} t_{0} (x, 0) = 0$ for any $x \in R^{d}$ by (1.14) and $\nabla d^{j} (x_{j}) = 0$ since, by the eikonal equation together with the assumptions on $V_{0}$ in Hypothesis 1.1, the distance $d^{j}$ has a non-degenerate minimum at $x_{j}$ , i.e. $D^{2} d^{j} |_{x_{j}}$ is a positive definite, symmetric matrix.

By (2.14) together with (1.14) and (1.16) we get for some $α > 0$ $\begin{matrix} (2.15) & ⟨ x, D Z |_{x_{j}} x ⟩ = ⟨ x, - 2 B (x_{j}) D^{2} d^{j} |_{x_{j}} x ⟩ ⩽ - α | x |^{2}, x \in R^{d} . \end{matrix}$ If $μ (t)$ denotes for $t \in (- \infty, 0]$ the integral curve of Z with $μ (0) = x_{j}$ , then, by (2.15), $μ (t)$ approaches $x_{j}$ exponentially fast (see e.g. [17]). Moreover, since by Hypothesis 1.6 the integral curves μ joining any point in $Ω^{j}$ with $x_{j}$ lie within $Ω_{j}$ , it is possible to use the method of characteristics and the variation of constants formula described e.g. in the proof of Proposition 3.5 in Dimassi–Sjöstrand [6]. It follows that (2.11) has a unique solution $u \in C^{\infty} (Ω^{j})$ vanishing to infinite-order at $x_{j}$ .

Since in each step the solution was multiplied with the cut-off-function $ζ_{j}$ , this gives the required compactly supported solution of (2.4) and thus defines ${\tilde{b}}_{k}^{j}$ in (2.2) supported in $Ω^{j}$ and solving (2.3) in a neighborhood of $M_{j}$ .

A Borel procedure with respect to ε gives us a function $b_{k}^{j} \in C^{\infty} (R^{d} \times [0, ε_{0}))$ representing the asymptotic sum ${\tilde{b}}_{k}^{j} (x; ε)$ given in (2.2) which is supported in $Ω^{j}$ and solves (2.3) in a neighborhood of $M_{j}$ . Analogously we define a real function $E_{j} (ε)$ as an asymptotic sum $\begin{matrix} E_{k}^{j} (ε) \sim E^{j} + \sum_{k \in \frac{N^{*}}{2}} ε^{s} E_{k s}^{j} . \end{matrix}$ To make the step from ${\hat{H}}_{ε}$ acting on $C_{0}^{\infty} (R^{d})$ to the operator $H_{ε}$ acting on lattice functions in $K ({(ε Z)}^{d})$ , we use that $G_{x_{0}}$ is invariant under the action of ${\hat{H}}_{ε}$ . Thus the restriction to the lattice commutes with ${\hat{H}}_{ε}$ and, using the restriction operator $r_{G_{x_{0}}}$ , yields (1.39). We therefore have proven Theorem 1.7(b).

For $i = j$ , the approximate orthonormality (1.37) follows from the orthonormality with respect to ${⟨ \cdot, \cdot ⟩}_{A_{j}}$ of the expansion ${\hat{b}}_{k}^{j}$ given in (A.9) (Theorem A.4). This has to be combined with a standard estimate of Laplace type ( $\int e^{- \frac{x^{2}}{ε}} O (x^{\infty}) d x = O (x^{\infty})$ ). If $i \neq j$ , (1.37) follows immediately from the estimate $\begin{matrix} (2.16) & \begin{matrix} {⟨ {\hat{v}}_{j, k}, {\hat{v}}_{i, ℓ} ⟩}_{L^{2}} & = ε^{- \frac{d}{2}} \int_{R^{d}} b_{k}^{j} (x; ε) b_{ℓ}^{i} (x; ε) e^{- \frac{1}{ε} (d^{j} (x) + d^{i} (x))} d x \\ ⩽ ε^{- \frac{d}{2}} e^{- \frac{S_{i j}}{ε}} \int_{M_{j} \cap M_{i}} b_{k}^{j} (x; ε) b_{ℓ}^{i} (x; ε) d x, \end{matrix} \end{matrix}$ where we used the triangle inequality for d. Since $M_{k}$ is compact for any $k \in C$ , (1.37) follows.

The estimate (1.40) for the restricted approximate eigenfunctions follows for $i = j$ from (a) combined with the general fact that for $a \in C_{0}^{\infty} (R^{d}, R)$ and $φ \in C^{\infty} (R^{d}, R)$ satisfying $φ (x_{0}) = 0$ and $D^{2} φ |_{x_{0}} > 0$ for some $x_{0} \in R^{d}$ and $φ (x) > 0$ for $x \in supp a ∖ {x_{0}}$ we have $\begin{matrix} (2.17) & ε^{d} \sum_{x \in {(ε Z)}^{d}} a (x) e^{- \frac{φ (x)}{ε}} = \int_{R^{d}} a (x) e^{- \frac{φ (x)}{ε}} d x + O (ε^{\infty}) . \end{matrix}$ This estimate follows from Proposition C.1 combined with the proof of Corollary C.2 in Di Gesù [5]. In fact, since it is shown in the proof of [5], Corollary C.2, that $\int_{R^{d}} | \partial_{y}^{α} a (h y) e^{- \frac{φ (h y)}{h^{2}}} | d y ⩽ C_{α}$ independent of h for any $α \in N^{d}$ , we can use [5], Proposition C.1, to get $\begin{matrix} h^{2 d} \sum_{y \in {(h Z)}^{d}} a (h y) e^{- \frac{φ (h y)}{h^{2}}} = h^{d} \int_{R^{d}} a (h y) e^{- \frac{φ (h y)}{h^{2}}} d y + O (h^{\infty}) \end{matrix}$ and the substitutions $h = \sqrt{ε}$ and $x = \sqrt{ε} y$ give the right-hand side of (2.17).

For $i \neq j$ , the arguments are as in (2.16) with summation replacing integration.

3. Proof of Theorem 1.8

In spirit, Theorem 1.8 and its proof follow Helffer–Sjöstrand [8]. A major technical difference is due to the fact that for second-order differential operators phase functions φ of Lipschitz type work in Agmon estimates. In our context, $φ \in C^{2}$ is essential (see also [11]). To prove Theorem 1.8, it is crucial to modify the phase function used in [11], and here we have to differ from [8] to get $φ \in C^{2}$ .

First, we need the following result on the exponential decay of eigenfunctions $v_{j, k}$ of the Dirichlet operator $H_{ε}^{M_{j}}$ at $x_{j}$ , $j \in C$ , associated to a spectral interval $I_{ε}$ .

Proposition 3.1.
Given Hypotheses 1.1 , 1.2 , 1.4 and 1.5, we assume that $I_{ε} \subset [0, ε R]$ for some $R > 0$ . Then, using the notation in (1.30), there exists a number $N_{0} \in N$ such that for all $j \in C$ and $1 ⩽ k ⩽ n_{j}$ and for all $ε \in (0, ε_{0}]$ $\begin{matrix} {‖ e^{\frac{d^{j}}{ε}} v_{j, k} ‖}_{ℓ^{2} (M_{j, ε})} = O (ε^{- N_{0}}) . \end{matrix}$
Proof.
We apply the decay estimates for Dirichlet eigenfunctions proven in [11], Theorem 1.8, to $H_{ε}^{M_{j}}$ , formally replacing Σ by $M_{j}$ and $d^{0} = d (0, \cdot)$ by $d^{j}$ . To justify this, we remark that, without changing $H_{ε}^{M_{j}}$ and $d^{j}$ on $M_{j}$ , we could modify the potential $V_{0}$ such that the assumptions of [11] are fulfilled up to the fact that now $V_{0}$ has (exactly one) minimum at $x_{j}$ instead of $x_{0} = 0$ . One now observes that [11], Theorem 1.8, and its proof remain valid, if zero is replaced by an arbitrary point $x_{j} \in R^{d}$ . □

In the next proposition, we show that if $ε E^{j}$ denotes an eigenvalue of multiplicity $m_{j}$ of the harmonic oscillator ${\hat{H}}_{0, q}^{j}$ associated to the Dirichlet operator $H_{ε}^{M_{j}}$ , then $H_{ε}^{M_{j}}$ has $m_{j}$ eigenvalues inside the interval $[ε E^{j} - C_{0} ε^{3 / 2}, ε E^{j} + C_{0} ε^{3 / 2}]$ for some $C_{0} > 0$ .
Proposition 3.2.
Assuming Hypotheses 1.1 , 1.2 , 1.4 and 1.5, let ${\hat{H}}_{0, q}^{j}$ be the harmonic oscillator defined in (1.19) associated to the Dirichlet operator $H_{ε}^{M_{j}}$ on $M_{j}$ , $j \in C$ , given in (1.29). Then there exists a bijection $b : spec (H_{ε}^{M_{j}}) \cap I_{ε} \to spec ({\hat{H}}_{0, q}^{j}) \cap I_{ε}$ and a constant $C_{0} > 0$ such that for all $ε \in (0, ε_{0}]$ $\begin{matrix} | b (λ) - λ | ⩽ C_{0} ε^{\frac{3}{2}} . \end{matrix}$
Proof.
In [14], Theorem 1.4, we proved that there is a bijection $b : spec (H_{ε}) \cap I_{ε} \to spec (H_{ε}^{M_{j}}) \cap I_{ε}$ such that $b (λ) - λ = O (e^{- \frac{C}{ε}})$ for some $C > 0$ and moreover that $# spec (H_{ε}) \cap I_{ε} = O (ε^{- N})$ for some $N \in N$ (both in the limit $ε \to 0$ ). We combine this with a result on the low lying spectra of $H_{ε}$ and the associated harmonic oscillator which, for $A_{j}$ , $B_{j}$ in (1.19) and ${\tilde{A}}^{j} : = B_{j} A_{j} B_{j}$ , is given by $\begin{matrix} (3.1) & K = ⨁_{j} K_{j}, where K_{j} (x) : = - Δ + ⟨ x, {\tilde{A}}^{j} x ⟩ + V_{1} (x_{j}) + t_{1} (x_{j}, 0) . \end{matrix}$ Let $e_{k}$ denote the kth eigenvalue of K (counting multiplicity) and let $E_{k} (ε)$ denote the kth eigenvalue of $H_{ε}$ then it is shown in [12], Theorem 1.3, that $E_{k} (ε) = ε e_{k} + O (ε^{\frac{6}{5}})$ as $ε \to 0$ . Together with the results on the approximating eigenvalues given in Theorem 1.7 these results prove Proposition 3.2. □

By use of Theorem 1.7 and Proposition 3.2, the next proposition on the distance of eigenspaces follows from Helffer–Sjöstrand [8], Propositions 1.4 and 2.5, which we recall in the Appendix (Proposition A.6).
Proposition 3.3.
For $j \in C$ , let $E_{j}$ denote the eigenspace of $H_{ε}^{M_{j}}$ for the interval $I_{ε} (E^{j}) = [ε E^{j} - C_{0} ε^{3 / 2}, ε E^{j} + C_{0} ε^{3 / 2}]$ , where $ε E^{j}$ is an eigenvalue of the harmonic oscillator ${\hat{H}}_{0, q}^{j}$ defined in (1.19) and $C_{0} > 0$ is some constant. Let ${\tilde{E}}_{j}$ denote the span of ${{\hat{v}}_{j, 1}^{ε}, \dots, {\hat{v}}_{j, m_{j}}^{ε}}$ , where ${\hat{v}}_{j, k}^{ε} : = {\hat{v}}_{j, k, 0}^{ε}$ , $1 ⩽ k ⩽ m_{j}$ , are the approximate eigenfunctions with respect to $ε E^{j}$ defined in (1.38). Then, setting $\vec{dist} (E, F) : = ‖ Π_{E} - Π_{F} Π_{E} ‖$ , we get $\begin{matrix} (3.2) & \vec{dist} ({\tilde{E}}_{j}, E_{j}) = \vec{dist} (E_{j}, {\tilde{E}}_{j}) = O (ε^{\infty}) . \end{matrix}$ Moreover, the eigenvalues of $H_{ε}^{M_{j}}$ in $I_{ε} (E^{j})$ are given by $μ_{j, k} = ε E_{k}^{j} (ε) + O (ε^{\infty})$ , $1 ⩽ k ⩽ m_{j}$ , where $E_{k}^{j} (ε)$ is given in (1.35) .
Proof.
By Proposition 3.2, it follows that $dim {\tilde{E}}_{j} = m_{j} = dim E_{j}$ and thus by [8], Proposition 1.4, $\vec{dist} ({\tilde{E}}_{j}, E_{j}) = \vec{dist} (E_{j}, {\tilde{E}}_{j})$ .

We estimate $\vec{dist} ({\tilde{E}}_{j}, E_{j})$ using [8], Proposition 2.5 (see Proposition A.6).

Theorem 1.7 and the definition of the Dirichlet operator in (1.29) gives $\begin{matrix} (3.3) & H_{ε}^{M_{j}} {\hat{v}}_{j, k}^{ε} = ε E_{k}^{j} (ε) 1_{M_{j, ε}} {\hat{v}}_{j, k}^{ε} + 1_{M_{j, ε}} [H_{ε}, 1_{M_{j, ε}}] {\hat{v}}_{j, k}^{ε} + O (ε^{\infty}) e^{- \frac{d^{j}}{ε}} . \end{matrix}$ To estimate the norm δ of the remainder $r_{j}$ in Proposition A.6, we thus have to analyze the norm of (cf. the proof of [14], Theorem 1.4) $\begin{matrix} (3.4) & r_{j, k} (x) : = 1_{M_{j, ε}} [H_{ε}, 1_{M_{j, ε}}] {\hat{v}}_{j, k}^{ε} (x) = 1_{M_{j, ε}} (x) \sum_{γ \in {(ε Z)}^{d}} a_{γ} (x; ε) 1_{M_{j, ε}^{c}} (x + γ) {\hat{v}}_{j, k}^{ε} (x + γ) . \end{matrix}$ We have $\begin{matrix} (3.5) & ‖ r_{j, k} ‖_{ℓ^{2} ({(ε Z)}^{d})}^{2} ⩽ \sum_{x \in M_{j, ε}} {(\sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} | a_{γ} (x; ε) {\hat{v}}_{j, k}^{ε} (x + γ) |)}^{2} . \end{matrix}$ To estimate the right-hand side of (3.5), we use that, for some $C > 0$ , we have ${\tilde{d}}^{j} (x) > C$ for all $x \notin M_{j}$ . Thus for $x \in M_{j, ε}$ , we get by (1.36), $\begin{matrix} (3.6) & \sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} | a_{γ} (x; ε) {\hat{v}}_{j, k}^{ε} (x + γ) | = \sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} | a_{γ} (x; ε) ε^{\frac{d}{4}} e^{- \frac{{\tilde{d}}^{j} (x + γ)}{ε}} b_{k}^{j} (x + γ) | ⩽ e^{- \frac{C}{ε}} A (x), \end{matrix}$ where, using the notation ${⟨ γ ⟩}_{ε} : = \sqrt{ε^{2} + | γ |^{2}}$ and the Cauchy–Schwarz inequality, for any $η > 0$ and $x \in M_{j, ε}$ $\begin{array}{l} A (x) & : = \sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} | a_{γ} (x; ε) b_{k}^{j} (x + γ) | \\ ⩽ {(\sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} {| a_{γ} (x; ε) {⟨ γ ⟩}_{ε}^{\frac{d + η}{2}} |}^{2})}^{\frac{1}{2}} {(\sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} {| {⟨ γ ⟩}_{ε}^{- \frac{d + η}{2}} b_{k}^{j} (x + γ) |}^{2})}^{\frac{1}{2}} \\ (3.7) & ⩽ C {(\sum_{\begin{array}{c} γ \in {(ε Z)}^{d} \\ x + γ \notin M_{j, ε} \end{array}} {| {⟨ γ ⟩}_{ε}^{- \frac{d + η}{2}} b_{k}^{j} (x + γ) |}^{2})}^{\frac{1}{2}}, x \in M_{j, ε}, \end{array}$ where in the second step we choose η according to Hypothesis 1.5(a). Inserting (3.6) and (3.7) in (3.5) and changing the order of summation yields $\begin{matrix} (3.8) & \begin{matrix} ‖ r_{j, k} ‖_{ℓ^{2} ({(ε Z)}^{d})}^{2} & ⩽ e^{- \frac{2 C}{ε}} \tilde{C} {‖ b_{k}^{j} ‖}_{ℓ_{M_{j, ε}}^{2}}^{2} \sum_{γ \in {(ε Z)}^{d}} {⟨ γ ⟩}_{ε}^{- (d + η)} \\ ⩽ C e^{- \frac{\tilde{C}}{ε}} . \end{matrix} \end{matrix}$ Inserting (3.8) into (3.3) gives the estimate $δ = O (ε^{\infty})$ for δ as in Proposition A.6 and in addition the statement on the eigenvalues of $H_{ε}^{M_{j}}$ .

By the choice of $I_{ε} (E^{j})$ together with Proposition 3.2 it follows that the constant a in Proposition A.6 can be chosen as $O (ε^{\frac{3}{2}})$ . Thus we get $\vec{dist} (\tilde{E_{j}}, E_{j}) = O (ε^{\infty})$ . □

It follows from Proposition 3.3 that there is an orthogonal matrix ${(c_{ℓ, k} (ε))}_{1 ⩽ ℓ, k ⩽ m_{j}}$ , such that in $ℓ^{2} (M_{j, ε})$ $\begin{matrix} (3.9) & v_{j, ℓ} = \sum_{k = 1}^{m_{j}} c_{ℓ, k} {\hat{v}}_{j, k}^{ε} + O (ε^{\infty}), \end{matrix}$ where $v_{j, ℓ}$ , $ℓ = 1, \dots, m_{j}$ , are real normalized eigenfunctions of $H_{ε}^{M_{j}}$ with respect to $I_{ε} (E^{j})$ as defined in (1.30). The matrix $(c_{ℓ, k})$ can be chosen such that $c_{ℓ, k} = 0$ if $E_{k}^{j}$ is not asymptotically equal to $μ_{j, ℓ}$ . If all $E_{ℓ}^{j}$ have different expansions, then $(c_{ℓ, k})$ may be chosen as identity matrix.
Lemma 3.4.
Under the assumptions of Hypothesis 1.6, for any $x \in Ω^{j}$ , let $γ (t; x)$ denote the base integral curve of $X_{{\tilde{h}}_{0}}$ given by $\begin{matrix} (3.10) & (- \infty, 0] ∋ t \mapsto π \circ F_{t} (x, \nabla d^{j} (x)) = : γ (t; x) . \end{matrix}$ Let $y \in Ω^{j}$ be such that $y \notin {x_{j}} \cup {γ (t; x) ∣ - \infty < t ⩽ 0}$ , then $\begin{matrix} (3.11) & d^{j} (x) < d^{j} (y) + d (y, x) . \end{matrix}$
Proof.
By the triangle inequality the statement is true for ⩽ instead of <. The idea of the proof is to show that equality only may occur, if y lies on the base integral curve of $X_{{\tilde{h}}_{0}}$ with end point x.

Let $η : [0, 1] \to Ω$ be the curve along the segment ${0} \cup {γ (t; x) ∣ - \infty < t ⩽ 0}$ , parameterized such that $η (0) = x$ and $η (1) = x_{j}$ . Thus, by construction and Hypothesis 1.6, η is a minimal geodesic between $x_{j}$ and x. In Bao–Chern–Shen [2], Theorem 6.3.1, it is shown that minimal geodesics are unique up to reparametrization. Equality in (3.11) would contradict this uniqueness, because this would mean that there are two different curves from $x_{j}$ to x, which minimize the curve length and are thus minimizing geodesics. □

By a standard compactness argument, we have the following:
Corollary 3.5.
Let $K_{1}, K_{2} \subset Ω^{j}$ be compact and assume that $K_{2}$ is disjoint from ${\hat{K}}_{1}$ , the compact union of all minimal geodesics from all points of $K_{1}$ to $x_{j}$ .

Then there exists $δ > 0$ such that for all $x \in {\hat{K}}_{1}$ , $y \in K_{2}$ $\begin{matrix} d^{j} (x) ⩽ (1 - δ) (d^{j} (y) + d_{ℓ} (y, x)) . \end{matrix}$
Proof of Theorem 1.8.
Step 1.
In order to simplify the notation, we fix $(j, k) \in J$ (see (1.31)) and set $\begin{matrix} (3.12) & r : = (H_{ε}^{M_{j}} - μ_{j, k}) w, w : = v_{j, k} - v_{j, k}^{'} \end{matrix}$ for $μ_{j, k}$ denoting the eigenvalue associated to $v_{j, k}$ (see (1.30)), i.e., $\begin{matrix} (3.13) & H_{ε}^{M_{j}} v_{j, k} = μ_{j, k} v_{j, k} . \end{matrix}$ We fix some compact set $K \subset M_{j}$ and write $\begin{matrix} (3.14) & \begin{matrix} 1_{K_{ε}} (H_{ε}^{M_{j}} - μ_{j, k}) v_{j, k}^{'} = A_{1} + A_{2} + A_{3}, where \\ A_{1} = \sum_{ℓ = 1}^{m_{j}} c_{k, ℓ} 1_{K_{ε}} (H_{ε} - ε E_{ℓ}^{j} (ε)) {\hat{v}}_{j, ℓ}^{ε}, \\ A_{2} = \sum_{ℓ = 1}^{m_{j}} c_{k, ℓ} 1_{K_{ε}} (ε E_{ℓ}^{j} (ε) - μ_{j, k}) {\hat{v}}_{j, ℓ}^{ε}, \\ A_{3} = \sum_{ℓ = 1}^{m_{j}} c_{k, ℓ} 1_{K_{ε}} [H_{ε}, 1_{M_{j, ε}}] {\hat{v}}_{j, ℓ}^{ε} . \end{matrix} \end{matrix}$ Theorem 1.7 gives $‖ e^{\frac{d^{j}}{ε}} A_{1} ‖_{ℓ^{2} (K_{ε})} = O (ε^{\infty})$ . By Proposition 3.3 together with the fact that $c_{k, ℓ} = 0$ if $E_{ℓ}^{j}$ and $μ_{j, k}$ are not asymptotically equal and that $d^{j} = {\tilde{d}}^{j}$ on $M_{j}$ (cf. (1.38)) we have the analogue result for $A_{2}$ . From [14], Lemma 5.1, it follows that for any $δ > 0$ and $C > 0$ , the commutator $[H_{ε}, 1_{M_{j, ε}}]$ is supported in $δ M_{j} : = {x ∣ d (x, \partial M_{j}) < δ}$ modulo $O (e^{- C / ε})$ . Since $K \cap δ M_{j} = \emptyset$ for some $δ > 0$ , we get $\begin{matrix} (3.15) & \sum_{j = 1}^{3} {‖ e^{\frac{d^{j}}{ε}} A_{j} ‖}_{ℓ^{2} (K_{ε})} = O (ε^{\infty}) . \end{matrix}$ Inserting (3.15) in (3.14) gives for any compact $K \subset M_{j}$ by (3.12) and (3.13) $\begin{matrix} (3.16) & {‖ e^{\frac{d^{j}}{ε}} r ‖}_{ℓ^{2} (K_{ε})} = O (ε^{\infty}) . \end{matrix}$ Furthermore, by the construction of the WKB function (see (A.9)) and Proposition 3.1 $\begin{matrix} (3.17) & {‖ e^{\frac{d^{j}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})} = O (ε^{- N_{0}}) \end{matrix}$ for some $N_{0} \in N$ . By (3.9) we have $\begin{matrix} (3.18) & ‖ w ‖_{ℓ^{2} (M_{j, ε})} = O (ε^{\infty}) . \end{matrix}$ We now claim that for some $N_{1} \in N$ $\begin{matrix} (3.19) & {‖ e^{\frac{d^{j}}{ε}} r ‖}_{ℓ^{2} (M_{j, ε})} = O (ε^{- N_{1}}) . \end{matrix}$ By (3.17), (3.13) and (3.9) it suffices to show $\begin{matrix} (3.20) & {‖ e^{\frac{d^{j}}{ε}} H_{ε}^{M_{j}} {\hat{v}}_{j, k}^{ε} ‖}_{ℓ^{2} (M_{j, ε})} = O (ε^{- N_{1}}) . \end{matrix}$ To prove (3.20), we write (using (1.36)) $\begin{matrix} (3.21) & (e^{\frac{d^{j}}{ε}} H_{ε}^{M_{j}} {\hat{v}}_{j, k}^{ε}) (x) = \sum_{γ \in {(ε Z)}^{d}} 1_{M_{j, ε}} (x) 1_{M_{j, ε}} (x + γ) a_{γ} (x; ε) e^{\frac{1}{ε} (d^{j} (x) - d^{j} (x + γ))} ε^{\frac{d}{4}} b_{k}^{j} (x + γ) . \end{matrix}$ Now $\begin{matrix} (3.22) & \sum_{γ \in {(ε Z)}^{d}} 1_{M_{j, ε}} (x + γ) | a_{γ} (x; ε) e^{\frac{1}{ε} (d^{j} (x) - d^{j} (x + γ))} | ⩽ \sum_{γ \in {(ε Z)}^{d}} e^{\frac{B | γ |}{ε}} | a_{γ} (x; ε) | ⩽ C \end{matrix}$ uniformly for $x \in M_{j, ε}$ , using that $d^{j}$ is Lipschitz ([11], Theorem 1.6) and (1.10). Furthermore, by (1.34) in Theorem 1.7 $\begin{matrix} (3.23) & {‖ b_{k}^{j} ‖}_{ℓ^{2} (M_{j, ε})} ⩽ C ε^{- N_{2}} \end{matrix}$ for some $N_{2} \in N$ (cf. the WKB-construction (A.9)).

Combining (3.22) and (3.23) with (3.21) gives (3.20).
Step 2.
In order to define an appropriate phase function, let $χ \in C^{\infty} (R_{+}, [0, 1])$ such that $χ (r) = 0$ for $r ⩽ \frac{1}{2}$ and $χ (r) = 1$ for $r ⩾ 1$ . In addition we assume that $0 ⩽ χ^{'} (r) ⩽ \frac{2}{log 2}$ . For $B > 0$ we define $g : Ω^{j} \to [0, 1]$ by $\begin{matrix} (3.24) & g (x) : = χ (\frac{d^{j} (x)}{B ε}), x \in Ω^{j} \end{matrix}$ and set, as in [11], $\begin{matrix} (3.25) & Φ (x) : = d^{j} (x) - \frac{B ε}{2} log (\frac{B}{2}) - g (x) \frac{B ε}{2} log (\frac{2 d^{j} (x)}{B ε}), x \in Ω^{j} . \end{matrix}$ To prove Theorem 1.8, this phase function is not good enough (by the proof in [11], it only gives (3.17)). We need a phase function $Ψ_{N}$ , which improves the decay estimate (3.17) with $ℓ^{2} (M_{j})$ replaced by $ℓ^{2} (K_{ε})$ , by a factor $ε^{N}$ for any $N \in N$ . Therefore, we consider for $N \in N$ $\begin{matrix} (3.26) & Ψ_{N} (x) : = Φ (x) + ζ (x) N ε log \frac{1}{ε} . \end{matrix}$ Here ζ is a well-chosen smooth function with $ζ (x) = 1$ for $x \in K$ , but $ζ (x) ⩽ 0$ in some small neighbourhood of $\partial M_{j}$ .

The first property ( $ζ = 1$ on K) gives the required improvement (and can be implemented by using (3.17)), the second ( $ζ ⩽ 0$ near $\partial M_{j}$ ) seems to be unavoidable (since (3.19) cannot be improved near the boundary $\partial M_{j}$ ).

Furthermore, ζ should decrease along the outgoing integral curves of the velocity field $Z (x) = \nabla_{ξ} {\tilde{h}}_{0} (x, \nabla d^{j} (x))$ (see (2.12) and (2.14)) – otherwise it would mess up the positivity properties needed in the following Agmon-type estimates.

We shall define ζ as the solution of a certain Cauchy problem for the velocity field Z.

First notice that, by slightly increasing the compact set $K \subset M_{j}$ , we may without loss of generality assume that K has smooth boundary and that $\begin{matrix} S : = {x \in \partial K ∣ Z (x) is tangential to K} \end{matrix}$ is a hypersurface. This follows from standard transversality arguments (see e.g. Hirsch [9]). For this K we denote by $\hat{K}$ the union of all minimal geodesics from $x_{j}$ to points in K. Let $\begin{matrix} K_{1} = \hat{K} \cup {(d^{j})}^{- 1} ([0, δ]), \end{matrix}$ where $δ > 0$ shall be chosen such that $K_{1} \subset M_{j}$ . In the following, we shall always assume that $B ε < δ$ . For any choice of B (which we shall make in Step 4), this is true for ε sufficiently small.

Now we define ζ as the (unique) solution of the Cauchy problem $\begin{matrix} (3.27) & Z \cdot \nabla ζ = - f in Ω^{j}, ζ |_{\partial K_{1}} = 1, \end{matrix}$ where f is any function in $C^{\infty} (Ω^{j})$ with $f ⩾ 0$ and $f (x) = 0$ for x in some neighbourhood of $K_{1}$ . We remark that in general $\partial K_{1}$ is not globally smooth (at S and at $\partial \hat{K} \cap {(d^{j})}^{- 1} (δ)$ ), and (3.27) is possibly characteristic (at $\partial \hat{K}$ ). To solve (3.27), first observe that by construction, Z on $\partial K_{1}$ is either tangential or outgoing. Thus, since f vanishes in a neighbourhood of $K_{1}$ , it follows that $ζ = 1$ in a neighbourhood of $K_{1}$ and, since the integral curves of Z finally reach $\partial Ω_{j}$ by Hypothesis 1.6, (3.27) can still be solved by the method of characteristics in all of $Ω_{j}$ . Explicitly, $\begin{matrix} (3.28) & ζ (x) = 1 - \int_{- \infty}^{0} f (γ (s; x)) d s, \end{matrix}$ where $γ (s; x)$ denotes the integral curve of $ν (x) = Z (x) \cdot \nabla$ with $γ (0; x) = x$ . It is clear from (3.28) that, if f is sufficiently large near $\partial M_{j}$ , then $ζ (x) ⩽ 0$ in some neighborhood of $\partial M_{j}$ . Summing up, we have $\begin{matrix} (3.29) & ζ \in C^{\infty} (Ω^{j}), ζ = 1 near K_{1} and ζ ⩽ 0 near \partial M_{j} . \end{matrix}$ We remark that for any $B > 0$ there is $C > 0$ such that for all $x \in M_{j}$ $\begin{matrix} (3.30) & e^{\frac{d^{j} (x)}{ε}} \frac{1}{C} {(1 + \frac{d^{j} (x)}{ε})}^{- \frac{B}{2}} ⩽ e^{\frac{Ψ_{N} (x)}{ε}} ⩽ e^{\frac{d^{j} (x)}{ε}} ε^{- N} C {(1 + \frac{d^{j} (x)}{ε})}^{- \frac{B}{2}} \end{matrix}$ which is a direct consequence of Lemma 3.3 in [11]. Furthermore, for $x \in K_{1}$ , the lower bound holds with an additional factor $ε^{- N}$ on LHS (3.30) (by (3.26) and (3.29)).
Step 3.
Now the proof follows the proof of Theorem 1.8 in [11]. We start to give estimates for ${\hat{V}}_{ε} + V^{Ψ_{N}}$ where $\begin{matrix} (3.31) & \begin{matrix} V^{Ψ_{N}} (x) : = \sum_{γ \in M_{j}^{'} (x)} a_{γ} (x; ε) cosh (\frac{1}{ε} (Ψ_{N} (x + γ) - Ψ_{N} (x))), \\ M_{j}^{'} (x) : = {γ \in {(ε Z)}^{d} ∣ x + γ \in M_{j}} . \end{matrix} \end{matrix}$ We claim that $\begin{matrix} (3.32) & {\hat{V}}_{ε} (x) + V^{Ψ_{N}} (x) ⩾ \{\begin{matrix} - C_{2} ε & for x \in M_{j} \cap {(d^{j})}^{- 1} ([0, B ε]), \\ (\frac{B}{C_{0}} - C_{1}) ε & for x \in M_{j} \cap {(d^{j})}^{- 1} ([B ε, \infty)) \end{matrix} \end{matrix}$ for some $C_{0}, C_{1}, C_{2} > 0$ independent of B.

To prove (3.32), we write $\begin{matrix} (3.33) & {\hat{V}}_{ε} (x) + V^{Ψ_{N}} (x) = ({\hat{V}}_{ε} (x) - V_{0} (x)) + (V^{Ψ_{N}} (x) + {\tilde{t}}_{0}^{M_{j}} (x, \nabla Ψ_{N} (x))) + (V_{0} (x) - {\tilde{t}}_{0}^{M_{j}} (x, \nabla Ψ_{N} (x))), \end{matrix}$ where, for $M_{j}^{'} (x)$ defined in (3.31), we set $\begin{matrix} {\tilde{t}}_{0}^{M_{j}} (x, ξ) : = - \sum_{γ \in M_{j}^{'} (x)} a_{γ}^{(0)} (x) cosh (\frac{ξ \cdot γ}{ε}) . \end{matrix}$ By Hypothesis 1.1 $\begin{matrix} (3.34) & {\hat{V}}_{ε} (x) - V_{0} (x) ⩾ - C_{11} ε, x \in M_{j} . \end{matrix}$ Now we claim that, for some $C_{12} < \infty$ (depending on N, but independent of B) $\begin{matrix} (3.35) & V_{0} (x) - {\tilde{t}}_{0}^{M_{j}} (x, \nabla Ψ_{N} (x)) ⩾ \{\begin{matrix} 0 & for x \in M_{j} \cap {(d^{j})}^{- 1} ([0, B ε]), \\ (\frac{B}{C_{0}} - C_{12}) ε & for x \in M_{j} \cap {(d^{j})}^{- 1} ([B ε, \infty)) . \end{matrix} \end{matrix}$ We first remark that exactly along the lines of the proof of [11], Theorem 1.8, (3.22), we have $\begin{matrix} (3.36) & V_{0} (x) - {\tilde{t}}_{0}^{M_{j}} (x, \nabla Φ (x)) ⩾ \{\begin{matrix} 0 & for x \in M_{j} \cap {(d^{j})}^{- 1} ([0, B ε]), \\ \frac{B}{C_{0}} ε & for x \in M_{j} \cap {(d^{j})}^{- 1} ([B ε, \infty)) . \end{matrix} \end{matrix}$ To show (3.35), we consider the following two cases: Case 1 ( $x \in K_{1}$ ).

In this region $ζ = 1$ , thus $\nabla Ψ_{N} = \nabla Φ$ and (3.35) follows from (3.36).

Case 2 ( $x \notin K_{1}$ ).

We remark that ${\tilde{t}}_{0}^{M_{j}} (x, ξ) ⩽ {\tilde{t}}_{0} (x, ξ)$ by Hypothesis 1.1(a)(ii). Since moreover $\begin{matrix} (3.37) & \nabla Ψ_{N} (x) = \nabla Φ (x) + \nabla ζ (x) N ε log (\frac{1}{ε}), \end{matrix}$ Taylor expansion of ${\tilde{t}}_{0}$ at the point $\nabla Φ (x)$ gives by (3.36) $\begin{matrix} (3.38) & LHS (3.35) ⩾ V_{0} (x) - {\tilde{t}}_{0} (x, \nabla Φ (x)) - I (x) ⩾ \frac{B}{C_{0}} ε - I (x), \end{matrix}$ where, for $γ_{x} (t) : = \nabla Φ (x) + t \nabla ζ (x) N ε log (\frac{1}{ε})$ , the remaining term I is given by $\begin{matrix} (3.39) & I (x) = N ε log (\frac{1}{ε}) \int_{0}^{1} D_{ξ} {\tilde{t}}_{0} (x, γ_{x} (t)) \nabla ζ (x) d t . \end{matrix}$ In order to analyze I we first remark that $| γ_{x} (t) - \nabla d^{j} (x) | = O (N ε log (\frac{1}{ε}))$ and by (1.16) $| D_{ξ} {\tilde{t}}_{0} | ⩽ C$ in compact sets and thus $\begin{matrix} (3.40) & | D_{ξ} {\tilde{t}}_{0} (x, γ_{x} (t)) - D_{ξ} {\tilde{t}}_{0} (x, \nabla d^{j} (x)) | = O (N ε log (\frac{1}{ε})), t \in [0, 1] . \end{matrix}$ Inserting (3.40) in (3.39) gives by the definition of $Z (x)$ as in Step 2 $\begin{matrix} (3.41) & I (x) = N ε log (\frac{1}{ε}) Z (x) \cdot \nabla ζ (x) + O ({(N ε log (\frac{1}{ε}))}^{2}) ⩽ C_{12} ε \end{matrix}$ for some $C_{12} > 0$ , where for the second estimate we used that ζ solves (3.27) with $f ⩾ 0$ and ${(N ε log (\frac{1}{ε}))}^{2} = o (ε)$ . Inserting (3.41) into (3.38) gives (3.35).

We now claim that for some $C_{13} > 0$ $\begin{matrix} (3.42) & | V^{Ψ_{N}} (x) + {\tilde{t}}_{0}^{M_{j}} (x, \nabla Ψ_{N} (x)) | ⩽ C_{13} ε . \end{matrix}$ Setting, for $M_{j}^{'} (x)$ defined in (3.31), $\begin{matrix} (3.43) & V_{0}^{Ψ_{N}} (x) : = \sum_{γ \in M_{j}^{'} (x)} a_{γ}^{(0)} (x) cosh (\frac{1}{ε} (Ψ_{N} (x + γ) - Ψ_{N} (x))), \end{matrix}$ we write $\begin{matrix} (3.44) & \begin{matrix} V^{Ψ_{N}} (x) + {\tilde{t}}_{0}^{M_{j}} (x, \nabla Ψ_{N} (x)) & = (V^{Ψ_{N}} (x) - V_{0}^{Ψ_{N}} (x)) + (V_{0}^{Ψ_{N}} (x) + {\tilde{t}}_{0}^{M_{j}} (x, \nabla Ψ_{N} (x))) \\ = : D_{1} (x) + D_{2} (x) \end{matrix} \end{matrix}$ and analyze $D_{1}$ and $D_{2}$ separately. In order to estimate $| D_{1} (x) |$ we first remark that by (3.37) and since $M_{j}$ and $M_{j}^{'} (x)$ are bounded $\begin{matrix} (3.45) & | \frac{1}{ε} (Ψ_{N} (x) - Ψ_{N} (x + γ)) | ⩽ C \frac{| γ |}{ε} \end{matrix}$ for some $C > 0$ . Thus, by Hypothesis 1.1(a), for some $C > 0$ uniformly in $x \in M_{j}$ $\begin{matrix} (3.46) & | D_{1} (x) | ⩽ \sum_{γ \in M_{j}^{'} (x)} | ε a_{γ}^{(1)} (x) + R_{γ}^{(2)} (x; ε) | cosh (\frac{1}{ε} (Ψ_{N} (x) - Ψ_{N} (x + γ))) ⩽ C ε . \end{matrix}$ In order to estimate $| D_{2} (x) |$ we write $\begin{matrix} (3.47) & | D_{2} (x) | ⩽ \sum_{γ \in M_{j}^{'} (x)} | a_{γ}^{(0)} (x) | | cosh {\frac{1}{ε} (Ψ_{N} (x) - Ψ_{N} (x + γ))} - cosh {\frac{1}{ε} γ \nabla Ψ_{N} (x)} |, x \in M_{j} . \end{matrix}$ By the Mean Value theorem for $cosh z$ and since $| sinh x | ⩽ e^{| x |}$ $\begin{array}{l} | cosh {\frac{1}{ε} (Ψ_{N} (x) - Ψ_{N} (x + γ))} - cosh {- \frac{1}{ε} γ \nabla Ψ_{N} (x)} | \\ (3.48) & ⩽ sup_{t \in [0, 1]} e^{\frac{1}{ε} | (Ψ_{N} (x + γ) - Ψ_{N} (x)) t - γ \nabla Ψ_{N} (x) (1 - t) |} \frac{1}{ε} | Ψ_{N} (x) - Ψ_{N} (x + γ) + γ \nabla Ψ_{N} (x) | . \end{array}$ Using (3.45) and $| γ \nabla Ψ_{N} (x) | ⩽ C | γ |$ , we get for some $D > 0$ uniformly in $x \in M_{j}$ and $t \in [0, 1]$ $\begin{matrix} (3.49) & e^{\frac{1}{ε} | (Ψ_{N} (x + γ) - Ψ_{N} (x)) t + γ \nabla Φ (x) (1 - t) |} ⩽ e^{\frac{D}{ε} | γ |} . \end{matrix}$ Since $Ψ_{N} \in C^{2} (M_{j})$ , we can use second-order Taylor expansion to get $\begin{matrix} (3.50) & \frac{1}{ε} | Ψ_{N} (x) - Ψ_{N} (x + γ) + γ \nabla Ψ_{N} (x) | ⩽ \frac{| γ |^{2}}{ε} sup_{t \in [0, 1]} | D^{2} Ψ_{N} (x + t γ) | ⩽ C \frac{| γ |^{2}}{ε} \end{matrix}$ uniformly in $x \in M_{j}$ . Inserting (3.49) and (3.50) into (3.48) and the result into (3.47) gives by Hypothesis 1.1(a) for some $C, \tilde{C} > 0$ $\begin{matrix} (3.51) & | D_{2} (x) | ⩽ C ε \sum_{γ \in M_{j}^{'} (x)} | a_{γ}^{(0)} (x) | {(\frac{| γ |}{ε})}^{2} e^{\frac{D}{ε} | γ |} ⩽ \tilde{C} ε . \end{matrix}$ Equation (3.51) together with (3.46) prove (3.42). Inserting (3.42), (3.35) and (3.34) into (3.33) proves (3.32).

Step 4.

In this step, we finish the proof by using Lemma A.5 which is proven in [11] (Lemma 3.2). First we remark that $| \nabla Φ (x) | ⩽ | \nabla d^{j} (x) |$ for all $x \in M_{j}$ (this can be seen as in Step 1 in the proof of Theorem 1.8 in [11]). Thus by (3.37) and the construction of ζ in Step 2, it follows that ${sup}_{x \in M_{j}} | \nabla Ψ_{N} (x) |$ is independent of B and thus the same is true for the constant $C_{T, Ψ_{N}}$ given in (A.13) (and (A.14)) by the discussion below Lemma A.5. This allows us to choose B in (3.32) such that $\begin{matrix} (3.52) & (\frac{B}{C_{0}} - C_{1}) ε - μ_{j, k} ⩾ ε (C_{T, Ψ_{N}} + 1) . \end{matrix}$ For $\begin{matrix} Ω_{-} : = {x \in M_{j} ∣ {\hat{V}}_{ε} (x) + V^{Ψ_{N}} (x) - μ_{j, k} < 0} and Ω_{+} : = M_{j} ∖ Ω_{-} \end{matrix}$ and for $C_{T, Ψ_{N}}$ as above, we define the functions $F_{\pm} : M_{j} \to [0, \infty)$ by $\begin{array}{l} (3.53) & F_{+} (x) : = \sqrt{(C_{T, Ψ_{N}} + 1) ε 1_{{d^{j} < B ε}} (x) + ({\hat{V}}_{ε} (x) + V^{Ψ_{N}} (x) - μ_{j, k}) 1_{Ω_{+}} (x)}, \\ (3.54) & F_{-} (x) : = \sqrt{(C_{T, Ψ_{N}} + 1) ε 1_{{d^{j} < B ε}} (x) + (μ_{j, k} - {\hat{V}}_{ε} (x) - V^{Ψ_{N}} (x)) 1_{Ω_{-}} (x)} . \end{array}$ Then $F_{\pm}$ are well defined and (using (3.32) and (3.52)) $\begin{array}{l} (3.55) & F : = F_{+} + F_{-} ⩾ \sqrt{(C_{T, Ψ_{N}} + 1) ε} > 0, F_{-} = O (\sqrt{ε}) and F_{+}^{2} - F_{-}^{2} = {\hat{V}}_{ε} + V^{Ψ_{N}} - μ_{j, k} . \end{array}$ Furthermore, again by (3.32) and (3.52), $\begin{matrix} (3.56) & Ω_{-} \subset {x \in M_{j} ∣ d^{j} (x) < B ε} \cap M_{j} and thus supp F_{-} = {d^{j} ⩽ B ε} \cap M_{j} . \end{matrix}$ Now Lemma A.5 yields for r and w defined in (3.12) and $v = e^{\frac{Ψ_{N}}{ε}} w$ the estimate1

Unfortunately, the analog of (3.57) in the proof of Theorem 1.8 in [11] is wrong (the weight $e^{\frac{Φ}{ε}}$ is missing in the analog of the second term on LHS (3.57)). In order to correct this mistake, one should redefine $F_{\pm}$ in [11] as in the present paper (by the analog of (3.53) and (3.54)) and choose B as in (3.52), where again the discussion below Lemma A.5 is crucial.

\begin{matrix} (3.57) & {‖ F e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})}^{2} - C_{T, Ψ_{N}} ε {‖ e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})}^{2} ⩽ 4 {‖ \frac{1}{F} e^{\frac{Ψ_{N}}{ε}} r ‖}_{ℓ^{2} (M_{j, ε})}^{2} + 8 {‖ F_{-} e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})}^{2} . \end{matrix}

Since

e^{\frac{Ψ_{N}}{ε}} = e^{\frac{Φ}{ε}} ε^{- N}

on K by the definition of

Ψ_{N}

, we have for some

N_{0} \in N

by (3.55) and (3.30) (and the discussion below)

\begin{matrix} (3.58) & {‖ F e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})}^{2} - C_{T, Ψ} ε {‖ e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})}^{2} ⩾ ε {‖ e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (K_{ε})}^{2} ⩾ \frac{1}{C} ε^{1 + N_{0} - 2 N} {‖ e^{\frac{d^{j}}{ε}} w ‖}_{ℓ^{2} (K_{ε})} \end{matrix}

and by (3.56), (3.18) and again (3.30)

\begin{matrix} (3.59) & {‖ F_{-} e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} (M_{j, ε})}^{2} = {‖ F_{-} e^{\frac{Ψ_{N}}{ε}} w ‖}_{ℓ^{2} ({d^{j} < B ε} \cap M_{j, ε})}^{2} ⩽ C ε^{1 - 2 N} ‖ w ‖_{ℓ^{2} ({d^{j} < B ε} \cap M_{j, ε})}^{2} = O (1), \end{matrix}

using (3.18) in the last step. In order to analyze the remaining term on the right-hand side of (3.57), we introduce a compact set

G \subset M_{j, ε}

, which is chosen such that

Ψ_{N} (x) ⩽ Φ (x)

for

x \in M_{j, ε} ∖ G

(by the construction of

Ψ_{N}

, this inequality holds on the neighbourhood of

\partial M_{j}

where

ζ ⩽ 0

). Then we write

\begin{array}{l} {‖ \frac{1}{F} e^{\frac{Ψ_{N}}{ε}} r ‖}_{ℓ^{2} (M_{j, ε})}^{2} = A_{1} + A_{2}, where \\ (3.60) & A_{1} = {‖ \frac{1}{F} e^{\frac{Ψ_{N}}{ε}} r ‖}_{ℓ^{2} (G)}^{2} and A_{2} = {‖ \frac{1}{F} e^{\frac{Ψ_{N}}{ε}} r ‖}_{ℓ^{2} (M_{j, ε} ∖ G)}^{2} . \end{array}

We have by (3.55) together with (3.30) for some

N_{0} \in N

\begin{matrix} (3.61) & A_{1} ⩽ C ε^{- 1 + N_{0} - 2 N} {‖ e^{\frac{d^{j}}{ε}} r ‖}_{ℓ^{2} (G)}^{2} = O (1), \end{matrix}

where in the last step we used (3.16) and the fact that G is compact. For analyzing

A_{2}

, we use that

Ψ_{N} ⩽ Φ

M_{j, ε} ∖ G

to get, again by (3.55) together with (3.30), for some

N_{0}, N_{1} \in N

\begin{matrix} (3.62) & A_{2} ⩽ C ε^{- 1} {‖ e^{\frac{Φ}{ε}} r ‖}_{ℓ^{2} (M_{j} ∖ G)}^{2} ⩽ C ε^{- 1 + N_{0}} {‖ e^{\frac{d^{j}}{ε}} r ‖}_{ℓ^{2} (M_{j} ∖ G)}^{2} = O (ε^{- N_{1}}), \end{matrix}

where we used (3.19) in the last step. Inserting (3.61) and (3.62) into (3.60) and combining the result with (3.58) and (3.59) gives by (3.57) that for any

N \in N

, there exists

ε_{N}

such that for all

ε \in (0, ε_{N})

we have

\begin{matrix} {‖ e^{\frac{d^{j}}{ε}} w ‖}_{ℓ^{2} (K_{ε})} ⩽ C ε^{- 1 - N_{0} - N_{1} + 2 N}, \end{matrix}

proving Theorem 1.8. □

Footnotes

Former results

We restate and adapt some results proven in [11] and [13]. For details and proofs, we refer to these papers. We recall that [11] and [13] only treat the one-well situation (with $x_{j} = 0$ ). Since all statements of this appendix are essentially local, the proofs in [11] and [13] hold unchanged in the present multi-well situation (by use of appropriate cut-off functions reducing the multi-well to the one-well situation).

The space $A_{j} (R^{d}) : = C [[z]] [[ε^{1 / 2}]]$ of formal Laurent series in $ε^{1 / 2}$ with final principal part and coefficients in the space $C [[z]]$ of Laurent series in $z \in R^{d}$ with finite principal part is a vector space over the field $K : = C [[ε^{1 / 2}]]$ of Laurent series in $ε^{1 / 2}$ with final principal part. There exists a non-degenerate sesquilinear form ${⟨ \cdot, \cdot ⟩}_{A_{j}} : A_{j} \times A_{j} \to K$ which formally is given by the asymptotic expansion at $z = 0$ of $\begin{matrix} {⟨ p, q ⟩}_{A_{j}} = ε^{- \frac{d}{2}} \int_{R^{d}} \overline{p} (z; ε) q (z; ε) e^{- 2 \frac{d^{j} (z)}{ε}} d z \end{matrix}$ (see [13]) and we define $\begin{matrix} (A.8) & e^{\frac{{\tilde{\hat{d}}}^{j}}{ε}} {\hat{H}}_{ε, j}^{'} e^{- \frac{{\tilde{\hat{d}}}^{j}}{ε}} |_{A_{j}} = : H_{ε, j, A}^{'}, \end{matrix}$ which is symmetric on $A_{j}$ .

In order to prove Proposition 1.8, we need the following (adapted and reformulated version of (a)) lemma proven (in a more general setting) in [11] (Lemma 3.2). For the Schrödinger operator, this type of result is due to Helffer–Sjöstrand [8], see also Dimassi–Sjöstrand [6].

It follows at once from the proof of Lemma A.5 in [11], that the constant $C_{T, Φ}$ has to fulfill the estimate $\begin{matrix} (A.14) & - \frac{1}{2} \sum_{x, x + γ \in Σ} a_{γ} (x, ε) cosh (\frac{1}{ε} (Φ (x) - Φ (x + γ))) {(v (x) - v (x + γ))}^{2} ⩾ - C_{T, Φ} ε ‖ v ‖^{2} . \end{matrix}$ Since Φ is Lipschitz and $M_{j}$ is compact, we have $Φ (x) - Φ (x + γ) ⩽ {sup}_{y \in M_{j}} | \nabla Φ (y) |$ . It therefore follows from Hypothesis 1.1(a)(ii), (iv), that such a constant exists and that it only depends on $T_{ε}$ (with lower bound $C_{T}$ given in (1.11)) and ${sup}_{y \in M_{j}} | D Φ (y) |$ ).

For the sake of the reader, we recall Proposition 2.5 from Helffer–Sjöstrand [8].

References

[1]

Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrödinger Operators, Mathematical Notes, Vol. 29, Princeton Univ. Press, 1982.

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Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators

Abstract

Keywords

1. Introduction

3. Proof of Theorem 1.8

Case 2 ( x ∉ K 1 ).

Footnotes

Former results

References

Case 2 ( $x \notin K_{1}$ ).