The tunneling effect for Schrödinger operators on a vector bundle

In the semiclassical limit ħ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \rightarrow 0$$\end{document}, we analyze a class of self-adjoint Schrödinger operators Hħ=ħ2L+ħW+V·idE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\hbar = \hbar ^2 L + \hbar W + V\cdot {\mathrm {id}}_{\mathscr {E}}$$\end{document} acting on sections of a vector bundle E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {E}}$$\end{document} over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m1,…mr∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^1,\ldots m^r \in M$$\end{document}, called potential wells. Using quasimodes of WKB-type near mj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^j$$\end{document} for eigenfunctions associated with the low lying eigenvalues of Hħ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\hbar $$\end{document}, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension ℓ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell + 1$$\end{document}. This dimension ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} determines the polynomial prefactor for exponentially small eigenvalue splitting.


Introduction
In this paper, we study the low lying spectrum of a Schrödinger operator H on a vector bundle E over a smooth oriented Riemannian manifold M . More precisely, in the limit → 0, we analyze the tunneling effect for operators of the form acting on the space Γ ∞ (M, E ) of smooth sections of E , where L is a symmetric Laplace type operator (i.e. a second order differential operator on E with principal symbol σ L (x, ξ) = |ξ| 2 ), W ∈ Γ ∞ (M, End (E )) is a smooth symmetric endomorphism field over M and the potential energy V ∈ C ∞ (M, R) has a finite number of non-degenerate minima m 1 , . . . m r .
Operators of this type arise e.g. in Witten's perturbation of the de Rham complex, where H is the square of the Dirac type operator where φ is a Morse function. This operator acts on (the sections of) E = Λ p M taking values in Λ p M , while its square P = Q 2 φ maps the sections of E into itself. More explicitly, it is given by P = 2 (dd * + d * d) + (L grad φ + L * grad φ ) + dφ 2 id E , where L X denotes the Lie derivative in the direction of the vector field X, and grad φ is the gradient of φ with respect to the Riemannian metric g. Then W := L grad φ +L * grad φ actually is C ∞ (M ) linear and thus an endomorphism field as described above. In this particular case, the endomorphism W is non-vanishing, which is the reason for us to include this (somewhat unusual) term in our considerations. This operator has been considered in detail in [HS4], giving much mathematical detail to the original paper [W]; see also [HKN], which derives full asymptotic expansions of all low-lying eigenvalues using an inductive approach which builds on the results of [BEGK], [BGK], [Eck] for generators of reversible diffusion operators, using a potential theoretic approach based on estimating capacities. It seems to be open if the latter approach could also be applied to operators as considered in this paper, and it is also open if all of the low-lying spectrum of operators as considered in this paper could be analysed by methods close to [HKN]. Here the Witten-Laplacian seems to be special.
We use semi-classical quasimodes of WKB-type constructed in [LR], which are an important step in discussing tunneling problems, i.e. exponentially small splitting of eigenvalues for a selfadjoint realization of H . In the scalar case, for dim M > 1, rigorous results in this field start with the seminal paper [HS1] (for M = R n or M compact).
In everything what follows, let (M, g) be a (smooth) oriented n-dimensional Riemannian manifold and let π : E → M be a complex vector bundle over M equipped with an inner product γ (i.e. a positive definite Hermitian form). Let rkE denote the dimension of a fibre of E . Denoting by d vol the Radon measure on M induced from the Riemannian metric g, this allows to introduce the Hilbert space L 2 (M, E ) of (equivalence classes of) sections in E as the completion of Γ ∞ c (M, E ), the space of compactly supported smooth sections of E 1 , with inner product [u, v] d vol(m) and associated norm u 2 E = u, u E . (1.1) Recall that a differential operator L acting on sections of E is said to be of Laplace type if, in local coordinates x, it has the form where g ij (x) is the inverse matrix of the metric g ij (x) and b j , c ∈ Γ ∞ (M, End (E )) are endomorphism fields. Examples for Laplace-type operators are the Hodge-Laplacian dd * + d * d on p-forms (in particular, for p = 0, this is the Laplace-Beltrami operator) and the square of a generalized Dirac operator acting on spinors. Also, second order elliptic operators L in divergence form, i.e, L = i,j ∂ i a ij ∂ j , on open subsets of R n belong to this class. To the best of our knowledge, even for scalar operators of this form the tunneling effect has not been treated explicitly in the literature. The geometrically formulated theorems of our paper cover in particular this special case, closing an obvious gap in the literature. Moreover, for any Laplace type operator L on E which is symmetric on Γ ∞ c (M, E ) with respect to the inner product . , . E , there exists a unique metric connection ∇ E : Γ ∞ (M, E ) → Γ ∞ (M, T * M ⊗ E ) on E and a symmetric endomorphism field U ∈ Γ ∞ (M, End sym (E )) such that (see [LR], Remark 2.1). In the following we always assume ∇ E to be the metric connection such that (1.3) holds. We denote by . , . m and | . | m the inner product and norm on T * m M induced by g (we feel free to sometimes suppress the index m ∈ M ) and we use the same symbols for the extension of the inner product and the norm to the complexified cotangent bundle T * C M = T * M ⊗ C. Then T * C M ⊗ E is well defined as a bundle (since fibrewise both factors are complex vector spaces), and we denote by . , . ⊗ and . ⊗ the inner product and norm on L 2 (M, T * C M ⊗ E ); both are induced from the inner product on the fibres of T * C M ⊗ E which for complex one-forms α, β and sections u, v of E is given by α ⊗ u, β ⊗ v ⊗ = α, β 1 γ [u, v] and then extends to the full tensor product by linearity (see also the beginning of Section 4 for some more standard details on the inner product ·, · p in the fibres of the complexified exterior bundle Λ p C (M )). We feel free to sometimes suppress the subscript C. As a general rule, all of our inner products are antilinear in the first and linear in the second factor.
Our setup is the following.
Hypothesis 1.1 For > 0, we consider Schrödinger operators H acting on L 2 (M, E ) of the form where L is a symmetric Laplace type operator as above, W ∈ Γ ∞ (M, End sym (E )) is a symmetric endomorphism field and V ∈ C ∞ (M, R). Furthermore, we shall assume: (a) The potential V is non-negative and there is a compact subset K ⊂ M and δ > 0 such that V (m) ≥ δ for all m ∈ M \ K. (b) V has non-degenerate minima at a finite number of points m j ∈ M, j ∈ {1, . . . , r} := C, i.e., V (m j ) = 0 and ∇ 2 V | m j > 0.
(c) If U is the endomorphism field given in equation (1.3) with respect to L in (1.4), the symmetric endomorphism field U + W is bounded from below, i.e. there is a positive constant C < ∞ such that uniformly for ∈ (0, 1]. It is then clear that (for ∈ (0, 1]) the operator H with domain Γ ∞ c (M, E ) is a semi-bounded, symmetric and densely defined operator in the Hilbert space L 2 (M, E ). Thus, by a well known result of abstract spectral theory, its associated semi-bounded quadratic form is closable. Passing to the closure of q and using the representation theorem for symmetric, semi-bounded, closed forms yields a distinguished self-adjoint operator, the Friedrich's extension of H : Γ ∞ c (M, E ) → L 2 (M, E ) (which by usual abuse of notation we shall also denote by H ). We recall that, if M is assumed to be complete, is actually essentially self-adjoint 2 . Furthermore, if M in addition is assumed to be of bounded geometry, various different natural approaches to the definition of Sobolev spaces for E all lead to identical results (see [A] and [E]). For the purpose of this paper, none of this seems to be relevant. We shall stick to the Friedrich's extension of H . Similarly, for any open Ω ⊂ M with compact closure in M we shall define the Dirichlet realization . We remark that the operator H given in (1.4) is not necessarily real, i.e. it does not commute with complex conjugation, in contradistinction to the more special case of the Witten Laplacian P introduced above. Thus, in the general case, the well known Beurling-Deny criteria do not apply, and even the groundstate of our operator H may well be degenerate. It is thus natural to treat tunneling in this degenerate setting and we shall do so in due course.
However, under semi-classical quantization (ξ → −i d in some reasonable sense) its principal -symbol is both real and scalar. This is crucial for our construction. Thus our assumptions exclude Schrödinger operators with magnetic field (the operator (i d + α) * (i d + α), with a 1-form α describing the magnetic potential, has non-real principal -symbol, see e.g. [HK]) or with endomorphism valued potential V as needed e.g. for molecular Hamiltonians in the Born-Oppenheimer approximation (see e.g. [KMSW]). Defining the hyperregular Hamiltoniañ one has σ H (m, ξ) = −h 0 (m, iξ) id E , and one can use the theory developed in [KR1] (or results given in [HS1]) to introduce an adapted geodesic distance on M . There it is shown that the (Agmon)-distance d on M given by the Agmon-metric ds 2 = V g (which is the Jacobi metric of classical mechanics on the Riemannian manifold (M, g) for the Hamiltonian function |ξ| 2 + V at energy zero) is Lipschitz everywhere and smooth near the potential wells m j , j ∈ C. Moreover, defining d j (m) := d(m, m j ) for j ∈ C, by [KR1], Theorem 1.6 (or see [HS1], Proposition 3.1) there are neighborhoods Ω j of m j such that (1.10) These are the eikonal equation and the eikonal inequality. The central result of this paper is a very precise asymptotic formula for the splitting of certain low-lying eigenvalues of the operator H h .
As we shall recall below, power series expansions of low lying eigenvalues are in leading order given by the harmonic approximation and can be derived by a certain perturbation theory.
2 See [C] and [St] for proofs using finite propagation speed and nice partitions of unity, respectively. Both papers do not formally cover precisely the class of operators of Laplace type considered in the present paper, but both methods generalize to our class of operators on bundles (e.g., the propagation speed in [C] only depends on the principal symbol of H for fixed , thus being independent of U and W ).
Furthermore, in situations of symmetry or near symmetry certain eigenvalues, including but not restricted to the groundstate, are almost degenerate in the sense of being exponentially close. On the other hand, such almost degeneracy of eigenvalues can be considered as a spectral picture of (possibly almost) underlying symmetry in a geometrical sense. In such a situation, approximating the spectrum of H h by the spectrum of appropriately chosen operators with Dirichlet boundary conditions on certain subsets Ω j of M might give a truly degenerate spectrum even for the groundstate, and the true spectrum of H h can then perturbatively be recovered through an (exponentially small) so called interaction matrix.
Crucial ingredients are the minimal geodesics between the potential wells and the distance between the wells in the Agmon metric ds 2 = V g.
While even in the scalar case such an analysis of eigenvalue splitting is often restricted to the case of one unique minimal geodesic (and possibly the groundstate), we here analyse the more complex situation where these minimal geodesics might form submanifolds of dimension ℓ + 1 (the above mentioned case of a unique geodesic is then a special case for ℓ = 0). See [KR3,KR4] for somewhat similar results for a class of difference operators on a scaled lattice in R n . Intuitively (in both cases), larger eigenvalue splitting is connected to more tunneling (or larger conductance) between the wells and thus to the dimension of connecting geodesics as these provide (in some sense which we shall not even try to make precise) optimal tunneling paths for a quantum particle. For the operators considered in this paper, this is made precise in our Theorem 6.7 below which is our central result.
The outline of the paper is as follows. In Section 2 we justify the harmonic approximation in our setting and sketch the proof of the basic spectral stability result. In Section 3 we prove Agmon-type estimates in the semiclassical limit for the decay of eigenfunctions for the Dirichlet operators on certain sets Ω j in M containing only one potential well. As usual, these are a crucial ingredient for subsequent WKB expansions. We emphasize that these estimates only involve certain structural identities for our operator; in particular, computations with a full expansion of everything in local coordinates are not required. Section 4 introduces the crucial above mentioned interaction matrix, and Section 5 analyzes it in more detail in important special cases. Section 6 is the heart of the paper, culminating in the above mentioned Theorem 6.7 which contains a full asymptotic expansion of the interaction matrix (and thus of the eigenvalue splitting, including but not restricted to the groundstate). As usual for such a precise result this requires additional geometric assumptions. Most importantly, the outgoing manifolds (with starting point m j ) for the Hamilton field ofh 0 have to be parametrized as Lagrange manifolds by the geodesic distance d j (m j , ·). And it is here that we prove (under appropriate assumptions) that the constructed WKB-expansions are actually very close to the true eigenfunctions of Dirichlet realizations of H h in Ω j which justifies replacing the Dirichlet eigenfunctions in the interaction matrix by these WKB functions. The proof of our main theorem then requires combining all this preparatory work with certain explicit calculations in local coordinates involving a form of the Morse Lemma with parameters and stationary phase.

Harmonic Approximation
In this section we shall show that the lowest N eigenvalues of H are given by the lowest N eigenvalues of the direct sum of associated harmonic oscillators at the potential wells, up to an error O( 6/5 ) as → 0.
Denoting the fibre over m ∈ M by E m := π −1 (m), we define the harmonic oscillator at m j , j = 1, . . . r, associated to H as the operator on C ∞ (T m j M, E m j ) given by where ∆ T m j M denotes the Laplacian on T m j M induced by the metric g m j and ∇ 2 V | m j denotes the Hessian of V at m j .
As a preparation, assume χ k are non-negative smooth functions with k χ 2 k = 1 (i.e. a quadratic partition of unity). Then a short calculation with double commutators gives (called the IMS-Localization formula in [CFKS] Now choose a smooth quadratic partition of unity χ 2 0 + χ 2 1 = 1, subordinate to the open cover Ω 0 = M \K and Ω 1 ⊃ K, and a positive function (2.5) Clearly, V K is relatively compact with respect to H . In fact, let us define the Sobolev space H 1 (M, E ) as the set of those u ∈ L 2 (M, E ) such that the distributional derivative ∇ E u is in L 2 (M, T * M ⊗ E ), with norm ||u|| H 1 = ||u|| E + ||∇ E u|| ⊗ . Since the domain D(H ) of the Friedrichs extension is contained in the form domain, one easily checks that D(H ) consists of functions in H 1 loc (M, E ). Furthermore, Rellich's compactness theorem holds in the following form: If Ω is an open set in M with compact closure, the embedding H 1 (Ω, E ) → L 2 (Ω, E ) is compact 3 . This gives, for λ < 0 in the resolvent set, compactness of V K (H − λ) −1 . Thus Weyl's essential spectrum theorem gives in view of (2.5) (2.6) proving Lemma 2.2. ✷ Theorem 2.3 Assume that H satisfies Hypothesis 1.1. Denote by e ℓ the ℓ-th eigenvalue of r j=1 H m j , , counting multiplicity. Then, for fixed m ∈ N and sufficiently small, H has at least m eigenvalues E ℓ ( ), ℓ = 1, . . . m, below inf σ ess (H ) and Proof. The proof follows closely the arguments of [S1], [S1e], which are also used in [CFKS]. As already noted in [CFKS], only minor changes are required to adapt the proof for a scalar operator on M = R n to a more complicated operator on a bundle E . We shall sketch the main idea for our operator H which is slightly different from the operator in [CFKS].
Let χ ∈ C ∞ 0 (R n ) be a cut-off function with 0 ≤ χ ≤ 1 (and such that 1 − χ 2 is smooth), let (Ω j , φ j ) be centered local charts based at m j ∈ M (i.e. satisfying φ j (m j ) = 0) and consider the pull-back χ j = φ * j (χ( −2/5 ·)) of the scaled cut-off function. For sufficiently small, we have 3 We remark that, since M is neither complete nor of bounded geometry, our definition of the Sobolev space need not necessarily coincide with the usual other natural definitions, see e.g. [E]. For H 1 (Ω, E ), however, all these ambiguities disappear by compactness of Ω.
χ j ∈ C ∞ 0 (M ) (for all 1 ≤ j ≤ r), and χ 0 := 1 − r j=1 χ 2 j is smooth. Thus, the localization formula (2.3) holds. Then, on the support of χ j , one uses Taylor expansion at m j of V up to third order terms and of W up to linear terms. In operator norm, all remainder terms of the Taylor expansion in χ j H χ j for 1 ≤ j ≤ r are of order O( 6/5 ), and so is the localization error 2 j |dχ j | 2 . One now fixes n such that e n < e n+1 (for the eigenvalues of the harmonic oscillators in H m j , ) and denotes by g k , for 1 ≤ k ≤ n, the corresponding normalized eigenfunctions of the appropriate harmonic oscillators H m j , , pulled back from T m j M to M and cut-off by multiplication with χ j . A straightforward computation then gives , which in view of the mini-max formula establishes the upper bound on E k ( ) in (2.7). To establish the lower bound, one uses the mini-max formula again and derives a lower bound in terms of a suitable symmetric finite rank operator (constructed from the restrictions of all localized operators H m j , to the spectral subspace of enery below e ∈ (e n , e n+1 ), which then implies (2.7). These arguments belong to abstract spectral theory (provided the error bounds for localization and Taylor expansion have been established) and do not depend on the geometry encoded in H and E . ✷

Agmon-estimates
In this section we prove exponential decay of eigensections of Dirichlet realizations of H , in the limit → 0. These estimates allow to decouple the wells and are crucial for establishing good error estimates. Technically, the main point of our subsequent discussion is to verify that abstract properties of the metric connection ∇ E are always sufficient. Computations in local coordinates are not required.
and, for some C 1 > 0, Proof. In order to prove (3.1), we first assume φ ∈ C 2 (M, R) and use (1.3) and (1.4) to write The case where φ is only Lipschitz can be deduced from the above as in the scalar case (see [HS1], Prop 1.1) using convolution with a standard mollifier and the dominated convergence theorem.
In order to prove (3.2), we use the definition of F + and F − to write Inserting (3.9) and (3.10) into (3.8) proves (3.2). In order to prove (3.3), we use (3.10) to write where in the second step we used (3.2). For some C 1 > 0 (independent of u and φ) one has, using (1.5), Inserting (3.12) into (3.11) proves the estimate (3.3). ✷ We set S j,k := d(m j , m k ) for j, k ∈ C, j = k and S 0 := min j,k∈C,j =k S j,k . (3.13) Hypothesis 3.2 For S 0 given in (3.13), there exists S ∈ (0, S 0 ) such that for all j ∈ C, the ball In the following, we fix such an S with the additional property that S + ε for ε > 0 sufficiently small still satisfies this condition. For each j ∈ C, we choose a compact manifold M j ⊂ M (with smooth boundary) such that B S (m j ) ⊂M j and m k / ∈ M j for k = j. Let H Mj denote the operator restricted to M j with Dirichlet boundary conditions.
By standard arguments (using compact embedding theorems for Sobolev spaces), H Mj has compact resolvent and thus purely discrete spectrum. (3.14) 4 Thus, in particular, any (in the topology of M ) closed subset of B S (m j ) is compact. Global compactness of M , however, is irrelevant. This is close (but not equivalent) to one of the equivalent statements in the Hopf-Rhinow Theorem, namely: All closed and (with respect to the g-distance) bounded subsets of the Riemannian manifold B S (m j ) are compact. Here, of course, closed is taken in the relative topology of B S (m j ) and B S (m j ) itself is bounded but not compact. Correspondingly, geodesics (for g) may reach the boundary of B S (m j ) in finite time, violating geodesic completeness of B S (m j ). Similarly, taking the closure of Γ ∞ Proof. We fix j ∈ C and set for any Then for sufficiently small it follows from the eikonal equation (1.9) that where for the last step we used that 1 Then by (3.16) and (3.17) Moreover e Φ is of the same order of magnitude as 1

Interaction matrix
In this section we introduce the interaction matrix, which under appropriate spectral conditions allows to compute full asymptotics of eigenvalue splitting on an exponentially small scale. It is our main technical tool.
We start with notations and recall some standard facts. By Λ p C (M ) = Λ p (M ) ⊗ C we denote the complexified exterior bundle; its smooth sections are the complex differential p-forms in Ω p C (M ). We extend the hermitian form γ to a sesquilinear fibrewise pairing where u, v are in E m and α ∈ T * C,m M which in the standard way extends to the full tensor product by linearity. Similarly, we define an extension in the first factor, giving We feel free to (often) suppress the subscript m. In particular, for and similarly for γ[u, which is just an equivalent (and shorter) expression for ∇ E being metric. As in the real case, the Hodge star operator * : where v 1 , . . . v n are oriented orthonormal vectors in T m M . In particular, if i : Σ → M denotes an embedded regular hypersurface in M , oriented by its outer normal field N we have  where δ := (−1) n(k+1)+1 * d * : denotes the codifferential operator.
Theorem 4.3 Under the assumptions given in Hypotheses 1.1, 3.2 and 4.2 and with the notation given above, there exists S 2 ∈ (S, S 0 ) such that for sufficiently small and for all s < S 2 and for all α ∈ J Proof. In order to show (a), we write H ψ α = µ α ψ α + H , χ j(α) v α (4.14) and observe that H , χ j(α) = 2 ∇ E * ∇ E , χ j(α) . (4.15) We claim that, for any χ ∈ C ∞ 0 (M, R), one has as operator on Γ ∞ (M, E ) where ∆ = d * d = δd (see (4.8)). In fact, dropping momentarily for reasons of brevity the superscript E in ∇ E , one readily computes where T = [∇ * , χ]∇ and T * = −∇ * (dχ⊗ ). (4.18) We need a few identities on linear algebra in the fibres of E and T * C M ⊗ E which we include for the convenience of the reader. In view of the usual musical isomorphism dχ, ω 1 = ω(grad χ), which is standard at least for real χ, one gets the first equality in and combining equation (4.23) with (4.20) gives (4.16).
Using the commutator formula (4.16) we can now estimate the commutator term on the rhs of (4.14). The assumption on the cut-off function χ j gives that d j ≥ S 1 on the support of dχ j for some S 1 ∈ (S, S 0 ). Thus one finds for some C > 0, since the estimate on the summand involving ∆χ j , where j = j(α), is trivial and the summand involving ∇ grad χj satisfies, using (4.19), for some C > 0. Taking the supremum over u = 1 gives (4.24). Since equation (4.24) together with Corollary 3.4 shows (a) for any S 2 < S 1 . In order to prove (b), we recall that at each well m j , j ∈ C, the Dirichlet eigenfunctions v j,ℓ are orthonormal and thus by Corollary 3.4 and since d . (4.26) for some N 0 ∈ N. Since moreover d j ≥ S 1 on the support of 1 − χ j for some S 1 ∈ (S, S 0 ) it follows from (4.26), using again Corollary 3.4, that for some N 0 ∈ N and for all α, β ∈ J . (4.27) Then the proofs of (b) and (c) proceed exactly along the lines of [HS1], Theorem 2.4. (d) can be seen as in [HS1], Lemma 2.8. ✷ In the following theorem, we introduce the notion of interaction matrix, refining the analysis of the error term above.
Theorem 4.4 In the setting of Theorem 4.3, for all s < S 2 and sufficiently small, the matrix of Π 0 H | E in the basis ψ α , α ∈ J , is given by where the interaction matrix is given by (4.30) In particular, we have for some . (4.31) Proof. Writing the statement on the matrix representation means that we have to determine w αβ such that (4.33) For any ordered basis of a finite dimensional Hilbert space, we use the notation x := (x 1 , . . . , x n ) and G (4.34) Then, setting ψ := (ψ 1,1 , ψ 1,2 . . . , ψ r,nr−1 , ψ r,nr ) for the basis of the Hilbert space E = j∈C E j , estimate (4.27) yields for any s < S 1 . We define Since G ψ is self-adjoint and positive, an orthonormal system of E is given by and (4.35) yields and therefore by (4.32) and (4.33) we get Π 0 H ψ = ψM + O e − 2s for M = δ αβ µ α + w αβ α,β∈J (4.40) for w αβ given in (4.29). This proves (4.28) and (4.29). To see (4.30), we write Using product rule, some of the terms cancel and we get Using again that d j ≥ S 1 on the support of dχ j , it follows from Corollary 3.4 that the first term on the right hand side is O e − 2s . This proves (4.30). Equation (4.31) follows from (4.30), using (4.26) together with Corollary 3.4 and the fact d j > S 2 on the support of dχ j . ✷ Since H is self-adjoint, it should have a symmetric matrix representation. Moreover, we want to give a matrix representation for H |F . for w αβ given in (4.29).
Proof. First we compute where in the last step we used (4.35). By (4.40) and (4.34) we can write ψ * Π 0 H ψ ≡ G ψ M where here and in the following ≡ is equality modulo O e − 2s . Since φ is orthonormal, by (4.37) and (4.35) the matrix of Π 0 H | E with respect to φ is given by where we used Taylor expansion and that both T and (w αβ ) are of order O e − s . By (4.43) we can write Since Π 0 is the projection on E along F ⊥ , we have ker Π 0 = ker Π F and Π F Π 0 = Π F and the eigenspaces E and F are in bijection via Π 0 | F and Π F | E . Moreover F and F ⊥ are invariant under the action of H and therefore Π F H = H Π F , thus where we used that by (4.38) Π 0 = Π E Π 0 = φ φ * Π 0 and the definition of M . Writing φ α = f α + h α for f α ∈ F and h α ∈ F ⊥ , we get by Theorem 4.3 and therefore (4.48) Thus, analog to (4.44), using (4.46), (4.48) and that g is orthonormal, the matrix of H | F in the basis g is given byM

Interaction matrix in special cases
In this section we give an explicit formula for the interaction matrix element w αβ in the case that the two wells m j(α) , m j(β) are near and the Dirichlet operators have very close eigenvalues inside the chosen spectral interval I . We start with some properties of the one-form γ[∇ E u, v] introduced in (4.1).
Lemma 5.1 For u, v ∈ Γ ∞ (M, E ) and δ the codifferential operator defined in (4.8) (5.1) Proof. First we recall that δ = d * (since M has no boundary). For reasons of brevity we drop the superscript (and later the subscript) E . Thus Lemma 4.1 yields for any φ ∈ C ∞ 0 (M, C) φ, δγ [∇u, v] We now give assumptions leading to a more explicit form for the interaction matrix.
Hypothesis 5.2 Under the assumptions given in Hypotheses 1.1, 3.2 and 4.2 and with the notation given at the beginning of Section 4 we assume that α, β ∈ J are pairs such that for some constant (5.6) Setting j = j(α), k = j(β) to shorten the notation, we define the closed "ellipse" We remark that G j,k is contained in the union B S (m j ) ∪ B S (m k ) which is compact by assumption.
Thus, in particular, G j,k (and Σ j,k to be defined below) are compact in and set Σ j,k := ∂Ω j,k ∩ G j,k .
The following proposition gives an explicit formula for the interaction term by means of a surface integral.
Proposition 5.3 Under the assumptions on the pairs α, β ∈ J given in Hypothesis 5.2 the elements w αβ of the interaction matrix, modulo O −N0 e − 1 (S0+a) for some N 0 ∈ N, are given by (5.11) and N is the outward unit normal on ∂Ω j,k , i.e. the unit normal on Σ j,k pointing from m j to m k .
Proof. We fix a pair α, β ∈ J satisfying (5.6) and write G = G j,k , Ω = Ω j,k and Σ = Σ j,k . Let χ G ∈ C ∞ 0 (M ) be a cut-off function such that χ G = 1 on G and with support close to G, then supp We choose the cut-off functions χ j and χ k in the definition of ψ α and ψ β (see (4.13)) such that χ j = 1 on supp χ G ∩ Ω and χ k = 1 on supp χ G ∩ Ω c .

By product rule
where the last estimate follows from the fact that d j + d k > S 0 + a on the support of dχ G together with the exponential decay properties of v α and v β (Corollary 3.4). To analyze A 1 , we remark that by Hypothesis 1.1 the endomorphism fields U and W are symmetric on E and 2 U + W + V id E commutes with χ G id E , thus The last two estimates follow from assumption (5.6) together with Corollary 3.4. Inserting (5.17) and (5.18) into (5.16) proves (5.15) and thus (5.9).
Applying (4.4) to the hypersurface Σ proves (5.10). ✷ Remark 5.4 (a) If S jk > S 0 + a and |µ α − µ β | = O e − a , then it follows at once from (4.31) that w αβ = O −N0 e − S 0 +a . Thus the formula (5.9) is relevant only if the Agmon distance S jk between the wells and the difference of the Dirichlet eigenvalues in the assumptions of Proposition 5.3 are related by (5.6).
If a is large, then S jk is nearly 2S 0 , but |µ α − µ β | must be very small. If on the contrary a is small, then S jk must be near to S 0 , but |µ α − µ β | is comparatively large (though still exponentially small of course). (b) It is possible to treat the limiting case d(m j(α) , m j(β) ) = S 0 and along the lines of the above proof, choosing a in the construction of Σ jk arbitrarily small, yielding where N is the outward unit normal on ∂Ω j(α),j(β) .

Asymptotic expansion
Using the quasimodes for the Dirichlet operators constructed in [LR], we will give asymptotic expansions for the interaction term w αβ in the case considered in Section 5.
We start with some additional hypotheses: Hypothesis 6.1 Let Xh 0 denote the Hamiltonian vector field on T * M with respect toh 0 defined in (1.8), F t denote its flow and for j ∈ C set Let M j satisfy Hypothesis 3.2. We assume that there is Ω j ⊂⊂ M j , open and containing m j , such that the following holds.
(a) For τ : T * M → M denoting the bundle projection τ (m, ξ) = m, we have ∈ Ω j for all (m, ξ) ∈ τ −1 (Ω j ) ∩ Λ j + and all t ≤ 0. By [KR1], Theorem 1.5, the base integral curves of Xh 0 on M \ {m 1 , . . . m r } 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 m j .
Clearly, Λ + (Ω j ) is a Lagrange manifold (by (a)) and since the flow F t preservesh 0 , we have Λ + (Ω j ) ⊂h −1 0 (0) by (6.1). Thus the eikonal equationh 0 (m, dd j (m)) = 0 holds for all m ∈ Ω j . Since in our setting the (in Ω j ) unique solution d j (·) of the eikonal equation is defined by following the flow of the Hamiltonian field and projecting to the base Ω j , it follows that in fact d j ∈ C ∞ (Ω j ).
The projection of Xh 0 , evaluated on Λ + (Ω j ), onto the configuration space Ω j is given by ∂ ξh0 (m, ξ = dd j (m)) = 2 grad d j (m). Thus the pair (d j , Ω j ) is, for each j ∈ C, an admissible pair in the sense of [LR], Def. 2.6, i.e. d j is the unique non-negative solution of the eikonal equation |dd j (m)| 2 = V (m) for m ∈ Ω j and Φ t (Ω j ) ⊂ Ω j for all t ≤ 0, where Φ t denotes the flow of the vector field 2 grad d j . In particular, Ω j is star-shaped with respect to the vector field 2 grad d j .
By straightforward calculations (compare [LR]) we have on Ω j where ∇ E is the unique metric connection determined by L given in (1.3) and ∆ denotes the Laplace-Beltrami operator acting on functions. The next theorem is a version of the results given in [LR], Theorem 2.7 and Corollary 2.10, adapted to the case of more than one potential well.
Theorem 6.2 Let H be as described in Hypothesis 1.1. For j ∈ C let Ω j and M j satisfy Hypothesis 6.1 and fix K compact in Ω j . Furthermore, we assume that E j denotes an eigenvalue of multiplicity ℓ j of the local harmonic oscillator H m j , at m j as given in (2.1). Then, for 0 sufficiently small and for α = (j, k) , k = 1, . . . , ℓ j , ℓ ∈ Z 2 , ℓ ≥ −N j for some N j ∈ N 2 , there are functions a α ∈ C ∞ ((0, 0 ), Γ ∞ c (M, E )) and sections a α,ℓ ∈ Γ ∞ c (M, E ), compactly supported in Ω j , such that for all N ∈ Z 2 there are C N < ∞ satisfying and real functions E α ( ) with asymptotic expansion such that the following holds: (a) the sections v α ( ) := − n 4 e − d j a α ( ) (6.5) are approximate eigensections for H with respect to the approximate eigenvalues E α ( ) given in (6.4), i.e.
uniformly on K (6.6) (b) for α = (j, k), β = (i, ℓ) as above, the approximate eigensections given in (6.5) are almost orthonormal in the sense that Remark 6.3 With the notation given in (2.2), the lowest order in in the expansion of a α is given by N j(α) = max γ |γ|/2 where γ runs over all multi-indices such that e j γ,ℓ = E j for some ℓ = 1, . . . rk E .
Proof. Combine Theorem 2.3 on the harmonic approximation and Theorem 4.3 on the existence of a bijection between the spectrum of H and the union of the spectra of H Mj , both restricted to a spectral interval I (giving the existence of b and a rough bound O( 6/5 )) with Theorem 6.2 on the existence of asymptotic expansions (which improves the rough bound to O( 3/2 )). ✷ Now we will prove that the difference between the quasimodes of Theorem 6.2 and the Dirichlet eigensection is exponentially small.
Theorem 6.5 Let H be given in Hypothesis 1.1 and for any j ∈ C, let Ω j , M j satisfy Hypothesis 6.1. Furthermore, we assume that E j denotes an eigenvalue of H m j , defined in (2.1) with multiplicity ℓ j and we set let v α denote orthonormal eigensections of the Dirichlet operator H Mj with eigenvalue belonging to the spectral interval I E j . Let K be any compact set in Ω j and let v α (resp. E α ) be the quasimodes (resp. the approximate eigenvalues) associated to E j , as defined in Theorem 6.2, and denote by J j the set (of cardinality ℓ j ) of all such α.
Then there is a unitary ℓ j × ℓ j matrix C j ( ) = c j α,β ( ) α,β∈Ij -possesing a full asymptotic expansion in half-integer powers of -such that for sufficiently small and α ∈ J j We remark that we can choose c j α,β ( ) = 0 if E β ( ) is not asymptotically equal to µ α ( ) (the Dirichlet eigenvalue associated to v α ), thus c j α,β∈Ij can be chosen to be the identity matrix if all E α ( ), α ∈ J j , have different expansions. Note, furthermore, that in view of standard elliptic estimates the basic estimate (6.9) establishes similar bounds on all higher derivatives: Second order derivatives of e d j (v α −ṽ α ) can be bounded from the elliptic equation and (6.9), and mathematical induction then implies bounds on all derivatives. In particular, the Sobolev embedding theorem on the compact subset Ω j of M (where all the standard definitions of Sobolev spaces actually coincide) give the following result: If H denotes an oriented hypersurface in Ω j and dσ the induced surface measure, then (6.9) implies where | · | ⊗ denotes the norm in the fibres of T * M ⊗ E induced from ·, · ⊗ . We also remark that similar considerations apply to the Agmon estimates in Section 3, yielding in particular (6.11) for some N 0 ∈ N, using Corollary 3.4.
Proof. Here one may follow the arguments in [HS1], Theorem 5.8 (the scalar case). Denoting bỹ E j and E j the space spanned by v j,k , 1 ≤ k ≤ ℓ j and v j,k , 1 ≤ k ≤ ℓ j respectively, it follows from Theorem 6.2 and Proposition 6.4 (as in [HS1], using Proposition 2.5 of that paper) that where µ j,k denotes the eigenvalues of H Mj associated to v j,k . This proves (6.8).
From Corollary 3.4 and (6.5) it is clear that the left hand side of (6.9) is of order O( −N0 ) for some N 0 ∈ N. In order to simplify the notation, we fix α = (j, k) ∈ J and set r := H − µ α w , w := v α −ṽ α (6.13) Then Theorem 6.2 shows for any compactK ⊂Ω j (fixed in advance as amplified in Theorem 6.5) (6.14) Furthermore, by (6.8) we have Let χ ∈ C ∞ 0 (Ω j ) be a cut-off function, which is equal to one in a neighborhood of the union K of all minimal geodesics from points in K to m j . For Φ defined in (3.15) we set for N ∈ N and ε > 0 Then a compactness argument (see [HS1], Lemma 5.7) shows that if U is a small neighborhood of K and ε is sufficiently small, then for each N there exists N > 0 such that Φ N (m) = Φ(m) + N ln 1 for all < N and m ∈ U . On the other hand, for any m, m ′ ∈ Ω j we have where d g denotes the distance with respect to the Riemannian metric. Thus for some constant C > 0 and for m in a region bounded away from m j . For ε sufficiently small (independent of N ), we therefore get for some C 0 > 0 We choose B such that B C0 − µα ≥ 1 and define F + and F − as in (3.18), replacing dΦ by dΦ N and E by µ j,k . We remark that e d j / = O −N0 e Φ/ for some N 0 ∈ N and Φ N = Φ + N ln −1 in K. Using also that 1 K [∇ E , χ] = 0, we have for some C > 0 where in the second step we used (the analog of ) (3.20). From (3.3) it follows that where for the last step we used that e Φ N ≤ e Φ −N for the first and third term and the fact that Φ N ≤ Φ on the support of dχ (by the definition of Ψ) together with Corollary 3.4 (4.16) for the second term. ChoosingK = supp χ, the last term on the right hand side of (6.19) is O( ∞ ) by (6.15), the first term is O( ∞ ) by (6.14). Since e Φ/ [H, χ]w 2 = O( −N1 ) for some N 1 ∈ N by the definition of Φ and Corollary 3.4, this proves (6.9). ✷ We shall now combine the approximate Dirichlet eigensections with the formula (5.9) (or (5.20)). Under more special conditions, we shall refine the construction at the beginning of Section 4. We start by giving appropriate assumptions for the index-set J of the relevant set of Dirichlet eigenfunctions and derive an associated spectral interval.
Hypothesis 6.6 Let H be given in Hypothesis 1.1 and for any j ∈ C, let M j , H Mj and S satisfy Hypothesis 3.2.
1) Let E 0 be in the spectrum of the direct sum of the localized harmonic oscillators H m j , , j ∈ C, given in (2.1). Let J be a maximal set of pairs α = (j, k) such that for α ∈ J all asymptotic eigenvalues E α ( ) of H given in Theorem 6.2 with leading order E 0 are equal. Let µ α be the associated eigenvalues and v α be the eigensections of the Dirichlet operators H M j(α) .
(c) there is a constant C > 0 such that for all m ∈ Σ j(α),j(β) and with the notation consists of a unique minimal geodesic (Case I) (in which case we set H j(α),j(β) =: m 0 ) or it is a manifold (possibly singular at the wells) of dimension ℓ + 1 with 1 ≤ ℓ ≤ n − 1 (Case II).
To unify our notation, we set ℓ = 0 in Case I. Estimate (6.20) implies that the transverse Hessian of d j(α) + d j(β) (transverse with respect to G j(α),j(β) ) is non-degenerate at all points of H j(α),j(β) (the intersection of the geodesics with the hypersurface Σ j(α),j(β) ). More precisely, choose near H j(α),j(β) a tubular neighborhood τ of Σ j(α),j(β) and commuting unit vector fields N 1 , . . . N n such that N = N n is normal to Σ j(α),j(β) , N 1 , . . . N n−1 are an orthonormal base in T Σ j(α),j(β) and N 1 , . . . N n−ℓ−1 are transversal to G j(α),j(β) . We remark that N n is not necessarily tangent to the geodesics in G j(α),j(β) and that the vector fields N n−ℓ , . . . N n−1 are possibly only locally defined on H j(α),j(β) (while N 1 , . . . N n−ℓ−1 exist globally on H j(α),j(β) ). Then is called the transverse Hessian of d j(α) + d j(β) at H j(α),j(β) and (6.20) implies that it is positive (in particular non-degenerate) for all points in H j(α),j(β) . Then, using the Morse-Lemma with parameters, the integral in (5.9) (or (5.20)) for w αβ has a complete asymptotic expansion. More precisely, Theorem 6.7 Under the assumptions given in Hypothesis 6.6, for a fixed pair α, β ∈ J , let w αβ be the interaction matrix element with respect to the spectral interval I as given in (4.29). Then there is a sequence (I p ) p∈N/2 in R such that w αβ ∼ −(Nα+N β ) (1−ℓ)/2 e −S j(α),j(β) / p∈N/2 p I p . (6.22) Explicit formulae for the leading order term are slightly different in Case I and Case II (see Hypothesis 6.6, 3d). Partition J into maximal subsets J j associated to one potential minimum m j . For δ ∈ J j and a δ ∈ C ∞ ((0, 0 ), Γ ∞ c (M, E )) given in Theorem 6.2, let v δ = − n 4 e − d j ã δ withã δ := β∈Jj c j δ,β ( )a β be the approximate eigenfunctions and C j ( ) = c j α,β ( ) α,β∈Jj be the unitary matrix as given in Theorem 6.5. We denote by dσ the Riemannian surface measure on H j(α),j(β ) induced by the Riemannian metric g, by N = N n the unit normal vector field on Σ j(α),j(β) pointing from m j(α) to m j(β) and define the transverse Hessian by (6.21). Moreover, df (N ) = N (f ) denotes the normal derivative of f ∈ C ∞ (M, R).
Then the leading order in the expansion in (6.22) is given by Case 1: Case 2: We remark that all I p = I p;α,β depend on α, β. Moreover, the leading order term satisfies I 0;α,β = I 0;β,α (since γ m is Hermitian and switching α and β implies switching the orientation of N ).
Recall that by construction, the eigenvalues of H ε exponentially close to µ α for α ∈ J (given in Hypothesis 6.6) also lie in the spectral interval I. Thus, by Corollary 4.6 specialized to the case of exactly two elements in J , the operator H ε has precisely two eigenvalues λ ± inside I. Up to errors O e − 2σ (for any σ < S 2 ), these are given by the eigenvalues of the 2 × 2-matrix µ αwαβ w αβ µ β , namely Thus in this case the eigenvalue splitting is explicitly given by In the symmetric case with µ α = µ β , the splitting is, modulo O e − 2σ , given by the symmetric interaction termw αβ = 1 2 (w αβ + w βα ). The complete asymptotic expansion ofw αβ (via expansion of both w αβ , w βα )) given in Theorem 6.7 also gives an asymptotic expansion of the eigenvalue Proof. We only prove Case 2. Theorem 6.5 allows to replace modulo terms of order O( ∞ ) the Dirichlet eigenfunctions v α and v β in (5.10) or (5.20) respectively by the associated approximate eigenfunctionsṽ α andṽ β .
In fact, for the first term in the integrand on the rhs of equation (5.10) one obtains where N is the unit normal vector field on Σ j(α),j(β) pointing from m j(α) to m j(β) . Writing α = (j, ℓ) and β = (k, u), and using (for w equal to e d j / v α or e d j / (v α −ṽ α )) the identity straightforward calculation of Σ jk rhs(6.25) dσ gives, by use of the estimate (6.10), Schwarz inequality and Corollary 3.4, an error of order O( ∞ e −S j,k / ). Treating the second term in the integrand on the right hand side of equation (5.10) in the same way proves our claim.