A discrete Hardy--Laptev--Weidl-type inequality and associated Schrödinger-type operators (Q1009453)

While the classical Hardy inequality, \[ \int_{\mathbb{R}^n} \frac{|f(x)|^2}{|x|^2}\, dx \leq \Big|\frac{2}{n-2}\Big|^2\int_{\mathbb{R}^n}|\nabla f(x)|^2\, dx, \] is valid only in the three- and higher-dimensional cases, \textit{A.\,Laptev} and \textit{T.\,Weidl} [Oper.\ Theory, Adv.\ Appl.\ 108, 298--305 (1999; Zbl 0977.26005)] established a two-dimensional Hardy-type inequality, \[ \int_{\mathbb{R}^2} \frac{|f(x)|^2}{|x|^2} \,dx \leq C\int_{\mathbb{R}^2}|\nabla_{\mathbf{A}} f(x)|^2\, dx, \] where \(\nabla_{\mathbf{A}}\) is the magnetic gradient with an Aharonov--Bohm magnetic potential \(\mathbf{A}\). The inequality can be proved by observing the spectral properties of the operator \[ -\frac{\partial^2}{\partial u^2}-\frac{\partial^2}{\partial v^2} \] on the set of functions in \(C_0^{\infty}(\mathbb{R} \times [0, 2\pi])\) satisfying certain boundary conditions. The present paper studies a discrete variant of the Laptev--Weidl inequality and analyses the spectral properties of discrete anologues of selfadjoint operators associated with the operators \[ -\frac{\partial^2}{\partial u^2}-\frac{\partial^2}{\partial v^2} \] and \[ -e^{-2u}\big(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\big). \] In particular, it is shown that, for suitable potentials, the discrete Schrödinger-type operator in the Aharonov--Bohm field has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satisfies a CLR-type inequality.