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A complete classification of threshold properties for one-dimensional discrete Schr\"{o}dinger operators
We consider the discrete one-dimensional Schr\"{o}dinger operator ,
where and is a self-adjoint operator on
with a decay property given by extending to a compact
operator from to
for some . We give a complete
description of the solutions to , and ,
. Using this description we give
asymptotic expansions of the resolvent of at the two thresholds and
. One of the main results is a precise correspondence between the solutions
to and the leading coefficients in the asymptotic expansion of the
resolvent around . For the resolvent expansion we implement the expansion
scheme of Jensen-Nenciu \cite{JN0, JN1} in the full generality.Comment: 51 page
A complete classification of threshold properties for one-dimensional discrete Schrödinger operators
Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions
We present an explicit formula for the resolvent of the discrete Laplacian on
the square lattice, and compute its asymptotic expansions around thresholds in
low dimensions. As a by-product we obtain a closed formula for the fundamental
solution to the discrete Laplacian. For the proofs we express the resolvent in
a general dimension in terms of the Appell--Lauricella hypergeometric function
of type outside a disk encircling the spectrum. In low dimensions it
reduces to a generalized hypergeometric function, for which certain
transformation formulas are available for the desired expansions
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