The Tunneling Effect for a Class of Difference Operators

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.