q-Sturm-Liouville theory and the corresponding eigenfunction expansions

Lazhar Dhaouadi (1)

1 Analyse harmonique et fonctions spéciales

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 $$ \Delta_{q}f(x)=\frac{1}{x^{2}}\left[ f(q^{-1}x)-\frac{1+q}{q}f(x)+\frac{1}{q}f(qx)\right]. $$

Domaines: Analyse classique [math.CA]; Mathématiques générales [math.GM]

Lazhar Dhaouadi. q-Sturm-Liouville theory and the corresponding eigenfunction expansions. 2008. ⟨hal-00163435v2⟩