On quantum integrability and Hamiltonians with pure point spectrum

TL;DR

This paper proves that all finite-dimensional Hamiltonians with pure point spectra are completely integrable and constructs examples with prescribed spectra, advancing understanding of quantum integrability.

Contribution

It establishes that any finite-dimensional pure point spectrum Hamiltonian is completely integrable and constructs Hamiltonians with arbitrary spectra.

Findings

Any n-dimensional pure point spectrum Hamiltonian is completely integrable.

Existence of integrable Hamiltonians with any prescribed closed spectrum.

Applications discussed in quantum integrability context.

Abstract

We prove that any $n$-dimensional Hamiltonian operator with pure point spectrum is completely integrable via self-adjoint first integrals. Furthermore, we establish that given any closed set $\Sigma\subset\mathbb R$ there exists an integrable $n$-dimensional Hamiltonian which realizes it as its spectrum. We develop several applications of these results and discuss their implications in the general framework of quantum integrability.