On quantum integrability and Hamiltonians with pure point spectrum

A. Enciso, D. Peralta-Salas

Published 2004-06-11, updated 2004-07-06Version 3

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.