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SOME CONCLUSIONS ABOUT HERMITIAN OPERATORS EXTENDED TO NON-HERMITIAN CASES

Xinlan Lou, Dazhi XU, Ning WU

Published Oct 1, 2025• 1 citation
Publisher

Abstract

This paper extends some well-known conclusions about Hermitian operators in quantum mechanics to the case of non-Hermitian operators. Specifically, (1) it presents the eigenvalue structure of non-Hermitian operators satisfying algebraic equations; (2) for a class of diagonalizable non-Hermitian operators, it clarifies the relationship between commutativity, degenerate subspaces, and block diagonalization, and provides a visual representation; (3) taking the four-site spin-1/2 Heisenberg XXZ chain as an example, it specifically illustrates the physical implications and applications of these extended results.

DOI
10.26599/phys.2025.9320505

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This paper extends some well-known conclusions about Hermitian operators in quantum mechanics to the case of non-Hermitian operators. Specifically, (1) it presents the eigenvalue structure of non-Hermitian operators satisfying algebraic equations; (2) for a class of diagonalizable non-Hermitian operators, it clarifies the relationship between commutativity, degenerate subspaces, and block diagonalization, and provides a visual representation; (3) taking the four-site spin-1/2 Heisenberg XXZ chain as an example, it specifically illustrates the physical implications and applications of these extended results.
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.
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.
In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.
In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.
In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg’s or the Schrdinger’s picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.
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.