Non-Hermitian real operators in quantum mechanics

Natália Bebiano, João da Providência, João P. da Providência

Classes of non-Hermitian operators that have only real eigenvalues are presented.Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra.For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = A * Ĵ, is found.Moreover, we construct a positive definite Hermitian Q such that A is Q-Hermitian, allowing for the standard probabilistic interpretation of quantum mechanics.Finally, it is shown that the considered matrices are similar to Hermitian matrices.

Hynek Bíla

F. Ventriglia

Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

João da Providência, Natália Bebiano, João P. da Providência

Péter R. Śurján, Ágnes Szabados, András Gombás

A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation.

Alexander N. Grigorenko