Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.2729

q-Sturm-Liouville theory and the corresponding eigenfunction expansions

Lazhar Dhaouadi
View original source ↗Read source document ↗Export metadata ↓

Abstract

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

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lazhar Dhaouadi. 2008-07-17. q-Sturm-Liouville theory and the corresponding eigenfunction expansions. https://arxiv.org/abs/0707.2729

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sharp $L^p$-estimates for wave equation on $ax+b$ groups

Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} ρ$ which is of $ax+b$ type, and $L$ be the positive definite left-invariant distinguished Laplacian on $G$. Let $u=u(t,\cdot)$ be the solution to $\partial_t^2u+Lu=0$ with initial conditions $u|_{t=0}=f_0$ and $\partial_tu|_{t=0}=f_1$. In this article we show that for a fixed $t\in\mathbb{R}\setminus\{0\}$ and every $1<p<\infty$, \begin{align*} \|u(t,\cdot)\|_{L^p(\mathrm{d}ρ)}\leq C_p\,\Big( (1+|t|)^{2|1/p-1/2|}\|f_0\|_{L^p_{α_0}(\mathrm{d}ρ)}+(1+|t|)\,\|f_1\|_{L^p_{α_1}(\mathrm{d}ρ)}\Big) \end{align*} if and only if \begin{align*} α_0\geq n\left|{1\over p}-{1\over2}\right| \quad \mbox{and} \quad α_1\geq n\left|{1\over p}-{1\over2}\right|-1. \end{align*} The polynomial-in-time growth $1+|t|$ multiplying $\|f_1\|_{L^p_{α_1}(\mathrm{d}ρ)}$ in the above estimate is best possible. This endpoint result settles the conjecture raised by Müller and Thiele [Studia Math. \textbf{179} (2007)].

math.CA↗
arXiv2026

On volumes of simplices in intermediate dimensions

A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1}) : x_{k+1}\in E \rbrace$ has positive Lebesgue measure. Here $\text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\ldots,x_{k},x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1 \leq d \leq 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$. The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of Héra, Keleti and Máthé.

math.CA↗
arXiv2026

Stability of concentration in the Paley-Wiener space

We prove stability of concentration in the Paley-Wiener space for each fixed time-bandwidth product. The concentration deficit controls the squared $L^2(\mathbb R)$-distance from a translated first prolate spheroidal wave function and the squared normalized symmetric difference from an interval at the same center. The obtained exponent for the function distance is optimal. The main idea is the introduction of the set discrepancy weighted by the best optimizers in place of the symmetric difference. The proof then combines elementary Hilbert space arguments with classical properties of the prolate spheroidal wave functions.

math.CA↗