A family of complex potentials with real spectrum

Fernández, Francisco M.; Guardiola, R.; Ros, J.; Znojil, Miloslav

doi:10.1088/0305-4470/32/17/303

Cited by 140 publications

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“…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].…”

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confidence: 99%

Exact solution for Morse oscillator in -symmetric quantum mechanics

Znojil

1999

Physics Letters A

109

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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: 99%

Exact solution for Morse oscillator in -symmetric quantum mechanics

Znojil

1999

Physics Letters A

109

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“…This is because the use of Lie's method does not guarantee the complete group classification of the class under study since it misses discrete equivalence transformations. Moreover, complex potentials have been shown to be of interest in mathematical-physics, quantum physics, optics, nuclear physics, as can be seen in [2,1,6,16]. Finally, a complete group classification of the class LinSchEq V from our point of view is desirable in order to give a unified presentation of results.…”

Section: Background and Motivationmentioning

confidence: 99%

Group classification of linear Schrödinger equations by the algebraic method

Kurujyibwami

2016

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This thesis is devoted to the group classification of linear Schrödinger equations. The study of Lie symmetries of such equations was initiated more than 40 years ago using the classical Lie infinitesimal method for specific types of real-valued potentials. In first papers on this subject, most attention was paid to dynamical transformations, which necessarily change the time and space variables. This is why phase translations were missed. Later, the study of Lie symmetries was extended to nonlinear Schrödinger equations. At the same time, the group classification problem for the class of linear Schrödinger equations with complex potentials remains unsolved.The aim of the present thesis is to carry out the group classification for the class of linear Schrödinger equations with complex potentials. These potentials are nowadays important in quantum mechanics, scattering theory, condensed matter physics, quantum field theory, optics, electromagnetics and so forth. We exhaustively solve the group classification problem for space dimensions one and two.The thesis comprises two parts. The first part is a brief review of Lie symmetries and group classification of differential equations. Next, we outline the equivalence transformations in a class of differential equations, normalization properties of such class and the algebraic method for group classification of differential equations.The second part consists of two research papers. In the first paper, the algebraic method is applied to solve the group classification problem for (1+1)-dimensional linear Schrödinger equations with complex potentials. With this technique, the problem of the group classification of the class under study is reduced to the classification of certain subalgebras of its equivalence algebra. As a result, we find that the inequivalent cases are exhausted by eight families of potentials and we give the corresponding maximal Lie invariance algebras.In the second paper we carry out the preliminary symmetry analysis of the class of linear Schrödinger equations with complex potentials in the multi-dimensional case. Using the direct method, we find the equivalence groupoid and the equivalence group of this class. Due to the multi-dimensionality, the results of the computations are quite different from the ones presented in Paper I. We obtain the complete group classification of (1+2)-dimensional linear Schrödinger equations with complex potentials.iii Populärvetenskaplig sammanfattningAvhandlingen behandlar gruppklassificeringen av linjära Schrödingerekvationer. Studiet av sådana ekvationers Lie symmetrier påbörjades för mer än 40 år sedan. Man använde då Lies klassiska infinitesimala metod och begränsade sig till speciella typer av reellvärd potential. De första arbetena ägnades huvudsakligen åt dynamiska transformationer. Eftersom dessa nödvändigtvis ändrar både tidsoch rumsvariablerna behandlades inte fastranslationer. Senare utvidgades studierna till icke-linjära Schrödingerekvationer, men gruppklassificeringsproblemet för klassen av linjä...

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“…The final classification result, which is a complete list of inequivalent potentials corresponding to equations with nontrivial Lie symmetries, may be used in quantum theory, quantum field theory, optics and other branches of physics, cf. [17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Equivalence groupoid for (1+2)-dimensional linear Schrödinger equations with complex potentials

Kurujyibwami¹

2015

J. Phys.: Conf. Ser.

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Abstract. We describe admissible point transformations in the class of (1+2)-dimensional linear Schrödinger equations with complex potentials. We prove that any point transformation connecting two equations from this class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of the class. This shows that the class under study is semi-normalized.

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A family of complex potentials with real spectrum

Cited by 140 publications

References 10 publications

Exact solution for Morse oscillator in -symmetric quantum mechanics

Exact solution for Morse oscillator in -symmetric quantum mechanics

Group classification of linear Schrödinger equations by the algebraic method

Equivalence groupoid for (1+2)-dimensional linear Schrödinger equations with complex potentials

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