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arXiv:1305.5164

Critical parameters for non-hermitian Hamiltonians

Francisco Marcelo Fernández, Javier Garcia

Published Apr 30, 2013• 5 citations
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Abstract

We calculate accurate critical parameters for a class of non-hermitian Hamiltonians by means of the diagonalization method. We study three one-dimensional models and two perturbed rigid rotors with PT symmetry. One of the latter models illustrates the necessity of a more general condition for the appearance of real eigenvalues that we also discuss here.

DOI
10.48550/arxiv.1305.5164

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We calculate accurate critical parameters for a class of non-hermitian Hamiltonians by means of the diagonalization method. We study three one-dimensional models and two perturbed rigid rotors with PT symmetry. One of the latter models illustrates the necessity of a more general condition for the appearance of real eigenvalues that we also discuss here.
In quantum-mechanics, the Hamiltonian potential is said to be non-Hermitian if the parameter is chosen to be imaginary [1]. Several papers have shown a great deal of interest in the study of non-Hermitian complex potentials with real spectrum [2–5]. This new class of non-Hermitian Hamiltonian potentials has been studied heavily [6], and it has been shown that, when the PT symmetry is not broken, such Hamiltonians define unitary theories of quantum mechanics [7]. In 1998 Bender [8] generalized the new kind of Hamiltonian complex potentials to a two-parameter class
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
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.
With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x1+ip2, p=p1+ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT-symmetric condition is satisfied. Thus PT- symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.
We notice that non-linear oscillator Hamiltonian having the character of non-Hermitian (H≠H+)(H \ne H^{+})(H=H+), non-PT Symmetry ([H,PT]≠0)([H,PT] \ne 0)([H,PT]=0) and non-Pseudohermiticity (ηHη−1≠H+)(\eta H \eta^{-1}\ne H^{+})(ηHη−1=H+) can yield real eigenvalues.