Generating Bounds for the Ground State Energy of the Infinite Quantum Lens Potential

TL;DR

This paper extends the Eigenvalue Moment Method (EMM) to bound the ground state energy of an electron in an infinite quantum lens potential, addressing boundary condition challenges on compact domains.

Contribution

The authors introduce modifications to EMM enabling its application to systems with boundary conditions on compact domains, such as the quantum lens potential.

Findings

Successfully adapted EMM for the quantum lens problem

Generated converging bounds for the ground state energy

Demonstrated effectiveness on a challenging compact domain system

Abstract

Moment based methods have produced efficient multiscale quantization algorithms for solving singular perturbation/strong coupling problems. One of these, the Eigenvalue Moment Method (EMM), developed by Handy et al (Phys. Rev. Lett.{\bf 55}, 931 (1985); ibid, {\bf 60}, 253 (1988b)), generates converging lower and upper bounds to a specific discrete state energy, once the signature property of the associated wavefunction is known. This method is particularly effective for multidimensional, bosonic ground state problems, since the corresponding wavefunction must be of uniform signature, and can be taken to be positive. Despite this, the vast majority of problems studied have been on unbounded domains. The important problem of an electron in an infinite quantum lens potential defines a challenging extension of EMM to systems defined on a compact domain. We investigate this here, and introduce novel modifications to the conventional EMM formalism that facilitate its adaptability to the required boundary conditions.