Self-Adjoint Extensions of the Hamiltonian Operator with Symmetric Potentials which are Unbounded from Below

TL;DR

This paper investigates self-adjoint extensions of the Hamiltonian with strongly unbounded symmetric potentials, establishing conditions for hermiticity, analyzing bound states, and exploring potential degeneracies in energy levels.

Contribution

It introduces a novel extension procedure based on Wronskian limits, enabling the study of Hamiltonians with potentials unbounded from below.

Findings

Bound states with even and odd parity are identified.

Degeneracy of energy eigenstates is possible due to relaxed boundary conditions.

Explicit examples demonstrate the applicability of the theoretical framework.

Abstract

We study the self-adjoint extensions of the Hamiltonian operator with symmetric potentials which go to $-\infty$ faster than $-|x|^{2p}$ with $p>1$ as $x\to\pm\infty$. In this extension procedure, one requires the Wronskian between any states in the spectrum to approach to the same limit as $x\to\pm\infty$. Then the boundary terms cancel and the Hamiltonian operator can be shown to be hermitian. Discrete bound states with even and odd parities are obtained. Since the Wronskian is not required to vanish asymptotically, the energy eigenstates could be degenerate. Some explicit examples are given and analyzed.