Isospectral potentials from modified factorization

Berger, M. S.; Ussembayev, Nail S.

doi:10.1103/physreva.82.022121

Cited by 22 publications

(25 citation statements)

References 16 publications

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“…Yet the limit ε 2 → ε 1 permits to avoid the problem in elegant form and gives rise to the confluent version of Susy QM . Additional results on 2‐step Darboux transformations can be found in, eg, Berger and Ussembayev and Midya Nevertheless, staying in the first‐order approach, the oscillation theorems satisfied by the seed function u with eigenvalue E n ≤ ε ≤ E n +1 prohibit the construction of real‐valued potentials V ( x ) that are free of singularities in Dom V 0 . The situation is different for the complex‐valued potentials V λ ( x ) since the nonlinear superposition of u p and v removes the possibility of zeros in , so the function is regular on

Dom V_{0} \subset double-struckR

.…”

Section: The Quest Of New Modelsmentioning

confidence: 96%

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

Blanco-Garcia

Rosas-Ortiz

Zelaya

2018

Math Methods in App Sciences

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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.

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Dom V_{0} \subset double-struckR

.…”

Section: The Quest Of New Modelsmentioning

confidence: 96%

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

Blanco-Garcia

Rosas-Ortiz

Zelaya

2018

Math Methods in App Sciences

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“…Another method has employed ground-or excited-state wavefunctions of the Morse potential to construct nonsingular isospectral potentials [8,9] by resorting to the well-known nonuniqueness of factorization [10]. The latter indeed allows one to avoid the singularities arising from the use of excited-state wavefunctions in standard SUSYQM [1].…”

Section: Introductionmentioning

confidence: 99%

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Quesne

2012

Int. J. Mod. Phys. A

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The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials V A,B,ext (x), obtained from a conventional Morse potential V A−1,B (x) by the addition of a bound state below the spectrum of the latter, is re-obtained. More importantly, the existence of another family of extended potentials, strictly isospectral to V A+1,B (x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of 'enlarged' shape invariance property, in the sense that their partner, obtained by translating both the parameter A and the degree m of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones. Running head: Rational Extensions of Morse Potential

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“…1 can be generalized by analyzing additional factorization procedures of the SUSY Hamiltonian given by Eq. (11) [75,76]. As such, SUSY fiber transformations can likewise be employed to study new degenerate axially symmetric potentials with a different azimuthal relation from those for the unbroken and broken SUSY cases.…”

Section: Discussionmentioning

confidence: 99%

Supersymmetric Transformations in Optical Fibers

2018

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Supersymmetry (SUSY) has recently emerged as a tool to design unique optical structures with degenerate spectra. Here, we study several fundamental aspects and variants of one-dimensional SUSY in axially symmetric optical media, including their basic spectral features and the conditions for degeneracy breaking. Surprisingly, we find that the SUSY degeneracy theorem is partially (totally) violated in optical systems connected by isospectral (broken) SUSY transformations due to a degradation of the paraxial approximation. In addition, we show that isospectral constructions provide a dimension-independent design control over the group delay in SUSY fibers. Moreover, we find that the studied unbroken and isospectral SUSY transformations allow us to generate refractive-index superpartners with an extremely large phase-matching bandwidth spanning the S þ C þ L optical bands. These singular features define a class of optical fibers with a number of potential applications. To illustrate this, we numerically demonstrate the possibility of building photonic lanterns supporting broadband heterogeneous supermodes with large effective area, a broadband all-fiber true-mode (de)multiplexer requiring no mode conversion, and different mode-filtering, mode-conversion, and pulse-shaping devices. Finally, we discuss the possibility of extrapolating our results to acoustics and quantum mechanics.

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Isospectral potentials from modified factorization

Cited by 22 publications

References 16 publications

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

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

Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Supersymmetric Transformations in Optical Fibers

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