Dr. Elke Rosenberger

 In the semiclassical limit \( \hbar\to 0\), we analyze a class of self-adjoint Schr\"odinger operators \( H_\hbar = \hbar^2 L + \hbar W + V\cdot \mathrm{id}_\mathscr{E}\) acting on sections of a vector bundle \(\mathscr{E}\) 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 \(m^1,\ldots m^r \in M\), called potential wells.
  Using quasimodes of WKB-type near \(m^j\)
  for eigenfunctions associated with the low lying eigenvalues of \(H_\hbar\), 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 \(\ell + 1\).
  This dimension \(\ell\) determines the polynomial prefactor for exponentially small eigenvalue splitting.

In the limit \(\hbar \to 0\), we analyze a class of Schrödinger operators \( H_\hbar = \hbar^2 L + \hbar W + V\cdot id_{\mathscr{E}}\) acting on sections of a vector bundle \(\mathscr{E}\) over a Riemannian manifold \(M\) where \(L\) is a Laplace type operator, \(W\) is an endomorphism field and the potential energy \(V\) has a non-degenerate minimum at some point \(p\in M\). We construct quasimodes of WKB-type near \(p\) for eigenfunctions associated with the low-lying eigenvalues of \(H_\hbar\). These are obtained from eigenfunctions of the associated harmonic oscillator \(H_{p,\hbar}\) at \(p\), acting on smooth functions on the tangent space.

We analyze a general class of difference operators <tex>H_\varepsilon = T_\varepsilon +V_\varepsilo</tex> on <tex>\ell^2((\varepsilon\mathbb{Z})^d)</tex>, where
<tex>V_\varepsilon</tex> is a multi-well potential and <tex>\varepsilon</tex> is a small parameter. We derive full asymptotic expansions
of the prefactor of the exponentially small eigenvalue splitting due to interactions between two
“wells” (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both
the case where there is a single minimal geodesic (with respect to the natural Finsler metric
induced by the leading symbol<tex> h_0(x, ξ)</tex> of <tex>H_\varepsilon</tex>) connecting the two minima and the case where
the minimal geodesics form an k+1 dimensional manifold, k ≥ 1. These results on the tunneling
problem are as sharp as the classical results for the Schrödinger operator in [Helffer, Sjöstrand,
1984]. Technically, our approach is pseudodifferential and we adapt techniques from [Helffer,
Sjöstrand, 1988] and [Helffer, Parisse, 1994] to our discrete setting.

We analyze a general class of self-adjoint difference operators <tex>H_\varepsilon = T_\varepsilon
</tex>

<tex>+ V_\varepsilon</tex> on <tex>\ell^2(\varepsilon\mathbb{Z}^d)</tex>, where<tex> </tex><tex><tex>V_ε</tex></tex> is a multi-well

potential and ε is a small parameter.

We review some preparatory results on tunneling of the authors, needed for
our presentation of new sharp results on tunneling on the level of complete asymptotic expansions.

The wells are decoupled by introducing certain Dirichlet operators on regions containing only one
potential well. Then the eigenvalue problem for the Hamiltonian <tex>H_\varepsilon</tex> is treated as a small perturbation of these comparison
problems.
After constructing a Finslerian distance d induced by <tex>H_\varepsilon</tex> we show that Dirichlet eigenfunctions decay exponentially with a rate
controlled by this distance to the well. It follows with microlocal techniques that the first
n eigenvalues of <tex>H_\varepsilon</tex> converge to
the first n eigenvalues of the direct sum of harmonic oscillators on <tex>\mathbb{R}^d</tex> located
at the several wells.
In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type
for eigenfunctions associated with the low lying eigenvalues of<tex> H_\varepsilon</tex>. These are obtained
from eigenfunctions or quasimodes for the
operator <tex>H_\varepsilon</tex>, acting on <tex>L^2(\mathbb{R}^d)</tex>, via restriction to the lattice <tex>\varepsilon\mathbb{Z}^d</tex>.

Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator, the remainder is exponentially small and roughly quadratic compared with the
interaction matrix.
We give weighted<tex> \ell^2</tex>-estimates for the difference of eigenfunctions of Dirichlet-operators in
neighbourhoods of the different wells and the associated WKB-expansions at the wells.
In the last step, we
derive full asymptotic expansions for interactions between two ``wells'' (minima)
of the potential energy, in particular for the discrete tunneling
effect. Here we essentially use analysis on phase space, complexified in the momentum variable.
These results are as sharp as the classical results for the Schrödinger operator given by Helffer and Sjöstrand.

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

In the limit <tex>\varepsilon \to 0</tex> we analyze the generators <tex>H_\varepsilon</tex> of families of reversible jump processes in <tex>R^d </tex>associated with a class of symmetric non-local Dirichlet-forms and show exponential decay of the eigenfunctions.

The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being <tex>C^2</tex> or just Lipschitz.
Our estimates are analogous to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice <tex>\varepsilon Z^d</tex>.

Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques.

In the limit ℏ→0, we analyze a class of Schr\"odinger operators H_ℏ = ℏ² L + ℏ W + V id_Eh acting on sections of a vector bundle Eh over a Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has a non-degenerate minimum at some point p ∈ M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low lying eigenvalues of H_ℏ. These are obtained from eigenfunctions of the associated harmonic oscillator H_p,ℏ at p, acting on C^∞(T_pM, Eh_p).

We analyze a general class of difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> acting on the square summable function on <tex>\varepsilon</tex><tex> Z^d</tex>, where <tex>V_\varepsilon</tex> is a multi-well potential and <tex>\varepsilon</tex> is a small parameter.

We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for <tex>H_\varepsilon</tex> as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see Helffer-Sjöstrand (1982)), and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.

We analyze a general class of difference operators <tex>H_\varepsilon = T_\varepsilon+ V_\varepsilon</tex> on <tex>\ell^2(\varepsilon\mathbb{Z}^d</tex>), where <tex>V_\varepsilon</tex> is a one-well potential and <tex>\varepsilon</tex> is a small parameter.

We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of <tex>H_\varepsilon</tex>. These are obtained from eigenfunctions or quasimodes for the operator <tex>H_\varepsilon</tex>, acting on <tex>L^2(R^d)</tex>, via restriction to the lattice <tex>\varepsilon Z^d</tex>.

For a general class of difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> on <tex>\ell^2(\varepsilon\mathbf{Z}^d)</tex>.
where <tex>V_\varepsilon</tex> is a multi-well potential and <tex>\varepsilon</tex> is a small parameter, we analyze the asymptotic behavior as<tex> \varepsilon\to 0</tex> of the (low-lying) eigenvalues and eigenfunctions.

We show that the first n eigenvalues of <tex>H_\varepsilon</tex> converge to the first n eigenvalues of the direct sum of harmonic oscillators on <tex>\mathbf{R}^d</tex> located at the several wells. Our proof is microlocal.

We analyze a general class of self-adjoint difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> on <tex>\ell^2(\varepsilon{\mathbf Z}^d)</tex>, where <tex>V_\varepsilon</tex> is a one-well potential and <tex>\varepsilon</tex> is a small parameter.

We construct a Finslerian distance d induced by<tex> H_\varepsilon</tex> and show that short integral curves are geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by the Finsler distance to the well. This is analog to semiclassical Agmon estimates for Schrödinger operators.

We analyze the asymptotic behavior in the limit <tex>\varepsilon \to 0</tex> for a wide class of difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> with underlying multi-well potential. They act on the square summable functions on the lattice <tex>\varepsilon \mathbb{Z}^d</tex>. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by <tex>H_\varepsilon</tex> and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.