Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators (Q2816226)

The article continues a series of the authors' articles about the asymptotic expansions of exact and approximate Dirichlet eigenfunctions for the operator \(H_\varepsilon = T_\varepsilon + V_\varepsilon\) in \(\ell^2((\varepsilon{\mathbb Z}^d))\), where NEWLINE\[NEWLINET_\varepsilon = \sum\limits_{\gamma \in (\varepsilon {\mathbb Z})^d} a_\gamma \tau_\gamma, \quad (\tau_\gamma u)(x) = u(x + y), \;\;(a_\gamma u)(x) = a_\gamma;\varepsilon)u(x), \qquad x, y \in (\varepsilon{\mathbb Z}^d),NEWLINE\]NEWLINE \(V_\varepsilon\) is a multiplication operator which in leading order is given by a multi-well potential \(V_0 \in {\mathcal C}^\infty({\mathbb R}^d)\) (in other words, the potential \(V_0\) has more than one non-degenerate minima), \(\varepsilon\) is a small parameter. More precisely, in this article, for formal asymptotic expansions of Dirichlet-eigenfunctions at the wells, associate \({\mathcal C}^\infty_0\)-functions are defined and weighted \(\ell^2\)-estimates for the difference of these \({\mathcal C}^\infty_0\)-functions and the Dirichlet eigenfunctions are obtained. These arguments allow the authors to compute complete asymptotic expansions for the elements of the interaction matrix and to explicitly obtain the leading order term of eigenvalues.