q-Sturm-Liouville theory and the corresponding eigenfunction expansions

Details q-Sturm-Liouville theory and the corresponding eigenfunction expansions q-Sturm-Liouville theory and the corresponding eigenfunction expansions

2007-07-18

arxiv.org/abs/0707.2729v2

Mathematics

Classical Analysis and ODEs

Scientific paper

The aim of this paper is to study the $q$-Schr\"{o}dinger operator $$ L=
q(x)-\Delta_q, $$ where $q(x)$ is a given function of $x$ defined over
$\mathbb{R}_{q}^{+}=\{q^n,\quad n\in\mathbb Z\}$ and $\Delta_q$ is the
$q$-Laplace operator $$ \Delta_{q}f(x)=\frac{1}{x^{2}}[
f(q^{-1}x)-\frac{1+q}{q}f(x)+\frac{1}{q}f(qx)]. $$

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