Two-Particle Bound States at Interfaces and Corners

TL;DR

This paper investigates how two interacting quantum particles form bound states near corners and interfaces in higher dimensions, showing they tend to stick to corners and have finitely many bound states below the continuum.

Contribution

It generalizes previous work to higher dimensions, proving particles prefer corners and the Hamiltonian has finitely many bound states below the essential spectrum.

Findings

Ground state energy decreases when moving from k to k+1 boundary directions.

Particles tend to stick to the intersection of boundary planes.

Hamiltonian has finitely many eigenvalues below the essential spectrum.

Abstract

We study two interacting quantum particles forming a bound state in $d$-dimensional free space, and constrain the particles in $k$ directions to $(0,\infty)^k \times \mathbb{R}^{d-k}$, with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from $k$ to $k+1$. This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all $k$ the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413--1444, 2020) to dimensions $d>1$.