New Concept of Solvability in Quantum Mechanics

TL;DR

This paper proposes a generalized concept of solvability in quantum mechanics by allowing non-numerical metrics and generalized Hermitian conjugation, expanding the class of solvable models beyond traditional self-adjoint Hamiltonians.

Contribution

It introduces a new framework for quantum solvability involving nontrivial metrics and generalized Hermitian conjugation, with illustrative models demonstrating the approach.

Findings

Presented solvable quantum models with nontrivial metrics

Argued for a generalized notion of Hermiticity in quantum mechanics

Expanded the class of solvable Hamiltonians beyond traditional self-adjoint forms

Abstract

In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of Hamiltonian) requires a generalization allowing for a non-numerical form of metric. A few illustrative solvable quantum models of this type are presented.