Pseudo-Hermitian Approach to Energy-Dependent Klein-Gordon Models

Abstract

The relativistic Klein-Gordon system is studied as an illustration of Quantum Mechanics using non-Hermitian operators as observables. A version of the model is considered containing a generic coordinate-and energy-dependent phenomenological mass-term m 2 (E, x). We show how similar systems may be assigned a pair of the linear, energy-independent left-and right-acting Hamiltonians with quasi-Hermiticity property and, hence, with the standard probabilistic interpretation.

η-Weak-Pseudo-Hermiticity Generators and Radially Symmetric Hamiltonians

“…For explicit illustration the reader may refer to, e.g., Znojil and Lévai in [3]. However, subsequent recent studies emphasized that the PT -symmetric Hamiltonians constitute a subclass of a very broader class of the so-called pseudo-Hermiticity of these non-Hermitian Hamiltonians [26][27][28][29][30][31][32][33][34][35][36][37]. A Hamiltonian H is pseudo-Hermitian if it obeys the similarity transformation:…”

Strengthened PT-symmetry with

“…In these cases one often employs the partitioned and manifestly Hermitian and involutive P, in the latter three contexts at least [18]. In our present letter we complemented these studies by an illustration of several new possibilities emerging within the scenario [b].…”

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

“…They split the Klein-Gordon wave function into two components and for the components vector they arrived at a Schrödinger-like equation with first order in time derivative. Although the Feshbach-Villars formalism appear in some advanced quantum mechanics books [9,10,11,12,13,14,15], and they were utilized in gaining deeper insight into relativistic physics of Klein paradox pair production, [16,17,18,19,20,21,22,23], in exotic atoms [24,25,26], used in theoretical consideraions [27,28,29,31], study relativistic scattering [32,32] and optics [33] or demosntrate PT symmetry [34,35,36,37], they were hardly used as a computational tool. The equations look like ordinary coupled differential equations, but the components are coupled by the kinetic energy operator, which makes them very hard to solve.…”