We show that complex Lie algebras (in particular s1(2, ℂ)) provide us with an elegant method for studying the transition from real to complex eigenvalues of a class of non-Hermitian Hamiltonians: Complexified Scarf II, generalized Pöschl-Teller, and Morse. The characterizations of these Hamiltonians under the so-called pseudo-Hermiticity are also discussed. © 2002 Elsevier Science B.V. All rights reserved.

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University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

Physics Letters, Section A: General, Atomic and Solid State Physics

We demonstrate that the recent paper by Abhinav and Panigrahi entitled 'Supersymmetry, PT-symmetry and spectral bifurcation' [K. Abhinav, P.K. Panigrahi, Ann. Phys. 325 (2010) 1198], which considers two different types of superpotentials for the PT-symmetric complexified Scarf II potential, fails to take into account the invariance under the exchange of its coupling parameters. As a result, they miss the important point that for unbroken PT-symmetry this potential indeed has two series of real energy eigenvalues, to which one can associate two different superpotentials. This fact was first pointed out by the present authors during the study of complex potentials having a complex sl(2) potential algebra. © 2010 Elsevier Inc.

Annals of Physics

Comment on 'Supersymmetry, PT-symmetry and spectral bifurcation'

The PT-symmetric complexi?ed Scarf II potential V(x) =-V 1sech2 x + iV2sechx tanh x, V1 &gt; 0, V2 ≠= 0, is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has recently been identfied as a spectral singularity or zero-width resonance is here demonstrated through the behaviour of the corresponding wavefunctions and some property of the associated pseudo-norms is pointed out. We also construct four different rationally extended supersymmetric partners to V(x), which are PT-symmetric or complex non-PT-symmetric according to the coupling parameters range. A detailed study of one of these partners reveals that SUSY preserves the V(x) spectral singularity existence. © 2010 IOP Publishing Ltd.

Journal of Physics A: Mathematical and Theoretical

An update on the PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally extended supersymmetric partners

The simplest purely imaginary and piecewise constant PT-symmetric potential located inside a larger box is studied. Unless its strength exceeds a certain critical value, all the spectrum of its bound states remains real and discrete. We interpret such a model as an initial element of the generalized non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its first supersymmetric partner in closed form. © World Scientific Publishing Company.

International Journal of Modern Physics A

PT-symmetric supersymmetry in a solvable short-range model

By applying the higher order Darboux algorithm to an exactly solvable non-Hermitian PT symmetric potential, we obtain a hierarchy of new exactly solvable non-Hermitian PT symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is nonlinear pseudosupersymmetry. We also show that this formalism can be used to generate a larger class of new solvable potentials when applied to non-Hermitian systems. © 2005 American Institute of Physics.

Journal of Mathematical Physics

PT symmetric models with nonlinear pseudosupersymmetry

New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials, which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions, which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum. © 2020 Author(s).

New families of non-parity-time-symmetric complex potentials with all-real spectra

The PT-symmetric square well problem is considered in a SUSY framework. When the coupling strength Z lies below the critical value Z 0 (c) where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY partner potentials, depicting an unbroken SUSY situation and reducing to the family of sec 2-like potentials in the Z → 0 limit. For Z above Z 0 (c), there is a rich diversity of SUSY hierarchies, including some with PT-symmetry breaking and some with partial PT-symmetry restoration.

Modern Physics Letters A

PT-symmetric square well and the associated SUSY hierarchies

Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasi-exactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified nonpolynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.

PT-symmetric nonpolynomial oscillators and hyperbolic potential with two known real eigenvalues in a susy framework

The continuity equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT nonsymmetric and its known eigenvalue corresponds to an unbound state.

Generating complex potentials with real eigenvalues in supersymmetric quantum mechanics

We consider a PT-symmetric partner to Khare-Mandal's recently proposed non-Hermitian potential with complex eigen-values. Our potential, which is quasi-exactly solvable, is shown to possess only real eigenvalues. © 2001 Elsevier Science B.V. All rights reserved.

Journal	Data powered by TypesetPhysics Letters, Section A: General, Atomic and Solid State Physics
Publisher	Data powered by TypesetELSEVIER SCIENCE BV
ISSN	0375-9601
Open Access	Yes