Non-Hermitian Hamiltonians with Real Spectrum in Quantum Mechanics

Abstract

Examples are given of non-Hermitian Hamiltonian operators which have a real spectrum. Some of the investigated operators are expressed in terms of the generators of the Weyl-Heisenberg algebra. It is argued that the existence of an involutive operatorĴ which renders the HamiltonianĴ-Hermitian leads to the unambiguous definition of an associated positive definite norm allowing for the standard probabilistic interpretation of quantum mechanics. Non-Hermitian extensions of the PoeschlTeller Hamiltonian are also considered. Hermitian counterparts obtained by similarity transformations are constructed.

Non-Hermitian PT Symmetric Hamiltonian with Position-Dependent Masses: Associated Schrödinger Equation and Finite-Norm Solutions

“…representing the three-dimensional extension of (26). As expected, in the limit γ → 0, one recovers the constantmass free-particle solution with m e ( x) = m and ρ( x) = constant, characterizing a nonintegrability in full space.…”

“…Nowadays, it is known that hermiticity is not a necessary condition for a consistent quantum theory, since it has been demonstrated in the literature that non-Hermitian Hamiltonians may also present real energy eigenvalues, leading to a well-defined quantum theory [14,17,[20][21][22][23][24][25][26][27]. Particularly, among the non-Hermitian Hamiltonians, a great interest has been dedicated to those characterized by a PT symmetry, i.e., symmetric under both P (parity, or space-reflection operator, which reverses position and momentum, x → −x, p → −p) and T (time reversal operator, which reverses time and momentum, t → −t, p → −p, also requiring the reverse of the sign of the complex number, i → −i).…”

“…Hence, the resulting PT operation changes x → −x, t → −t, and i → −i. Recently, an alternative formulation of quantum mechanics has appeared, in which the requirement of Hermiticity of the HamiltonianĤ was replaced by the condition of space-time reflection; consequently, ifĤ presents an unbroken PT symmetry, then its associated energy spectrum should be real [20,[22][23][24][25][26][27].…”

𝒟 $\mathcal {D}$ -Deformed Harmonic Oscillators

“…This model is the two-dimensional version of the one-dimensional Swanson model as discussed in [14], and first introduced in [13], withω 1 = sec(2θ)/2.…”

More mathematics for pseudo-bosons

“…, where θ is a real parameter taking value in − π 4 , π 4 \ {0} =: I, [18]. As before, [x, p] = i1 1.…”