Bender-Wu branch points in the cubic oscillator

Álvarez, Gabriel

doi:10.1088/0305-4470/28/16/016

Cited by 77 publications

(93 citation statements)

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“…One may even return to the related older literature, say, on the cubic anharmonic oscillator V (x) = ω 2 + igx 3 , all the resonant energies of which remain real and safely bounded below [10,11]. The similar complete suppression of decay has now been observed and/or proved in several other analytic models [12,13].…”

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confidence: 59%

Exact solution for Morse oscillator in -symmetric quantum mechanics

Znojil

1999

Physics Letters A

108

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PT invariance of complex potentials V (x) = [V (−x)]* combines their real symmetry with imaginary antisymmetry. We describe a new exactly solvable model of this type. Its spectrum proves real, discrete and "three-fold", ε = ε n (j) = (2n + α j ) 2 , n = 0, 1, . . . , with α j ≥ 0 and j = 1, 2, 3.PACS 03.65.Ge, 03.65.FdRecently, Daniel Bessis [1] and Bender and Boettcher [2] proposed a certain modification of the interpretation of the one-dimensional Schrödinger-like differential equations. In a way inspired by the methodical relevance of the spatially symmetric harmonic oscillator in field theory [3] they tried to weaken the standard assumption of hermiticity of the Hamiltonian H = H + , i.e., in the strict mathematical language, the requirement of the essentially self-adjoint character of this operator.The new idea has found its first physical applications and immediate tests in field theory (cf., e.g., refs.[4] for explicit illustrations). Its use in the time-independent quantum mechanics has simultaneously been proposed [5]. As long as the timereversal operator T performs the mere Hermitian conjugation in the latter case, a generalization of bound states with definite parity (Pψ = ±ψ ∈ L 2 ) has been found in normalizable states with a parity-plus-time-reversal combined symmetryIn the PT −symmetric quantum mechanics a stability hypothesis Im E = 0 has been tested semi-classically [2,5], numerically [1,6], analytically [7,8] and perturbatively [9,10]. One may even return to the related older literature, say, on the cubic anharmonic oscillator V (x) = ω 2 + igx 3 , all the resonant energies of which remain real and safely bounded below [10,11]. The similar complete suppression of decay has now been observed and/or proved in several other analytic models [12,13].Serious problems arise for the non-analytic PT −symmetric interactions [5]. An indirect clarification of the difficulty is being sought, first of all, in the various analytic PT −analogs of the usual square well [8,9,14]. In the present note, another exactly solvable example of such a type will be proposed and analyzed in detail, therefore.Let us start by characterizing solvable potentials by their so called shape invari-ance. The review [15] lists nine of these models. Once we restrict our attention to the mere confluent-hypergeometric-type equations on full line, we are left with the spatially symmetric harmonic oscillator V (HO) (r) = ω 2 (r − b/ω) 2 − ω and with itsMorse-oscillator partner1The PT −symmetrization of the former model V (HO) has been recalled in the pioneering paper [2]. The latter case (1) is to be considered here.The real and Hermitian version of the model V (M orse) is of a non-ceasing interest in the current literature [16]. Its consequent PT −symmetrization necessitates a few preliminary considerations. In the first step, let us make a detour and return once more to the real and Hermitian V (HO) and to its generalized three-dimensional Schrödinger-equation presentationWe may remind the reader that one can safely return bac...

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confidence: 59%

Exact solution for Morse oscillator in -symmetric quantum mechanics

Znojil

1999

Physics Letters A

108

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“…the general solutions themselves are analytic functions of r (c.f., e.g., [6]). We may construct them in the complex plane which is cut, say, from the origin upwards.…”

Section: Application: Pt −Symmetric Ddt Oscillatorsmentioning

confidence: 99%

PT-symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes

Mustafa

Znojil

2002

J. Phys. A: Math. Gen.

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“…Alvarez [3] discussed the analytical properties of the solutions of the Hamiltonian operator H = 1 2 p 2 + oscillator H = p 2 + ix 3 + λx 2 . The purpose of this paper is the exploration into such relationships as well as into other properties of a class of nonhermitian oscillators.…”

Section: Introductionmentioning

confidence: 99%