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Molecular Physics

Real eigenvalues of non-hermitian operators

Péter R. Śurján, Ágnes Szabados, András Gombás

Published Nov 24, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.1080/00268976.2023.2285034

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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.
The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.
In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, the reality of an eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators that may admit real eigenvalues under some symmetry conditions. In general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive-semidefinite part of the non-Hermitian operator. We also prove an uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under a quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula, and measuring the product of noncommuting projectors.
The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.
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.
Aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work starts dealing with: (i) the maximal hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; and with: (ii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-selfadjoint (and even non-hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.
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.