On the Number of Bound States of Point Interactions on Hyperbolic Manifolds

Abstract

We consider the problem of a quantum particle interacting with $N$ attractive point $\delta$-interactions in two and three dimensional Riemannian manifolds and discuss its some spectral properties. The main aim of this paper is to give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds ${\mathbb{H}}^2$ and ${\mathbb{H}}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on ${\mathbb{R}}^2$ and ${\mathbb{H}}^2$.

28 pages, published version, typos and some minor mistakes are corrected

References in corpus (4)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the $\delta$-Function Potential in 1-dimensional Relativistic Quantum Mechanics
• Statistics of wave functions for a point scatterer on the torus
• Existence of Hamiltonians for Some Singular Interactions on Manifolds

Cited by in corpus (3)

• One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
• A Perturbative Approach to the Tunneling Phenomena
• On Schrödinger Operators Modified by $\delta$ Interactions