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arXiv:1301.2900 · International Journal of Modern Physics C

Exceptional points, non-normal matrices, hierarchy of spin matrices and an eigenvalue problem

Willi-Hans Steeb, Yorick Hardy

Published Mar 26, 2014• 4 citations
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Abstract

Exceptional points of a class of non-hermitian Hamilton operators Ĥ of the form Ĥ = Ĥ0+ iĤ1are studied, where Ĥ0and Ĥ1are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators Ĥ0and Ĥ1are given by spin matrices for spin s = 1/2, 1, 3/2, …. Since the linear operators studied are non-normal, properties of such operators are described.

DOI
10.1142/s0129183114500594

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Exceptional points of a class of non-hermitian Hamilton operators Ĥ of the form Ĥ = Ĥ0+ iĤ1are studied, where Ĥ0and Ĥ1are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators Ĥ0and Ĥ1are given by spin matrices for spin s = 1/2, 1, 3/2, …. Since the linear operators studied are non-normal, properties of such operators are described.
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The aim of the paper is to study the question whether or not equilibrium states exist in open quantum systems that are embedded in at least two environments and are described by a non-Hermitian Hamilton operator H\cal HH. The eigenfunctions of H\cal HH contain the influence of exceptional points (EPs) as well as that of external mixing (EM) of the states via the environment. As a result, equilibrium states exist (far from EPs). They are different from those of the corresponding closed system. Their wavefunctions are orthogonal although the Hamiltonian is non-Hermitian.
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.
A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum of H0(N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be PT-symmetric]. Despite that, the “physical” parametric domain D[N] is (constructively) shown to exist, guaranteeing that in its interior, the spectrum remains real and non-degenerate, rendering the quantum evolution unitary. Among the non-Hermitian degeneracies occurring at the boundary ∂D[N] of the domain of stability, our main attention is paid to their extreme version corresponding to Kato’s exceptional point of order N (EPN). The localization of the EPNs and, in their vicinity, of the quantum-phase-transition boundaries ∂D[N] is found feasible, at the not too large N, using computer-assisted symbolic manipulations, including, in particular, the Gröbner-basis elimination and the high-precision arithmetics.