Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Abstract

Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

“…The novelty is that the eigenvalues of the new refractive indices will be all-real. Moreover, the additional eigenvalue can be incorporated at any position of the spectrum [41]. The formulae ( 7)-( 11) still apply for complex-valued refractive indices, and are particularly important to generate balanced gain-and-loss.…”

“…In contraposition to conventional supersymmetric approaches, the factorization energy can be positioned at any place in the point spectrum of n 1 [41]. Moreover, the missing state (11), now written as…”

“…Moreover, as the reality of the spectrum is not granted a priori for non-Hermitian models, the demonstration that PT symmetry implies real spectrum [35] stimulated the systematic search of PT-symmetric systems in quantum mechanics [36]. Nevertheless, further improvements shown that PT-symmetry is not a necessary condition for the spectrum reality [37,38], a fact confirmed in diverse supersymmetric models of non-Hermiticity [39][40][41][42][43][44], where the imaginary part of a wide class of complex-valued potentials lead to balanced gain and loss probability without the necessity of PT symmetry. The latter property opens new possibilities in optical design, where the gain-loss manipulation includes but is not limited to PT symmetry [45].…”

Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

“…Then, following [63], the solution of (C-1) is of the form σ(t) = [aq 2 1 (t) + bq 1 (t)q 2 (t) + cq 2 2 (t)] 1/2 , (C-3) where {a, b, c} is a set of real constants. To get a function σ > 0, it is necessary to impose the condition b 2 − 4ac = −4 w 2 W 2 0 , with nonnegative constants {a, b, c} [64,65].…”

Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method

“…In turn, N pmq 0 has to be computed explicitly. From the definition of inner product (14) and the reparametrization zpx, tq it is easy to show that N pmq 0 is determined from the same relation of the stationary case. Therefore, the normalization constant of the rational extension of the harmonic oscillator, reported in [39], was used.…”