A higher-order deformed Heisenberg spin equation as an exactly solvable dynamical equation

It is shown that first-order Darboux transformations for the two-dimensional massless Dirac equation with scalar potential and for the Schrödinger equation are the same up to a change of coordinates. As a consequence, we obtain a closed-form representation of iterated, higher-order Darboux transformations for our Dirac equation. We use the formalism to generate several new exactly-solvable Dirac systems through higher-order Darboux transformations.

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On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2020.0487 ◽

2020 ◽

Vol 476 (2243) ◽

pp. 20200487

Author(s):

C. Rogers ◽

T. Ruggeri ◽

W. K. Schief

Keyword(s):

Conservation Laws ◽

Classical Limit ◽

Three Dimensional ◽

Classical System ◽

Two Dimensional ◽

Stationary Case ◽

Heisenberg Spin ◽

Lift And Drag ◽

Spin Equation

A classical system of conservation laws descriptive of relativistic gasdynamics is examined. In the two-dimensional stationary case, the system is shown to be invariant under a novel multi-parameter class of reciprocal transformations. The class of invariant transformations originally obtained by Bateman in non-relativistic gasdynamics in connection with lift and drag phenomena is retrieved as a reduction in the classical limit. In the general 3+1-dimensional case, it is demonstrated that Synge’s geometric characterization of the pressure being constant along streamlines encapsulates a three-dimensional extension of an integrable Heisenberg spin equation.

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MATCHING WEAK COUPLING AND QUASICLASSICAL EXPANSIONS FOR DUAL QES PROBLEMS

International Journal of Modern Physics A ◽

10.1142/s0217751x02010777 ◽

2002 ◽

Vol 17 (31) ◽

pp. 4661-4667 ◽

Cited By ~ 1

Author(s):

MICHAEL KAVIC

Keyword(s):

Energy Level ◽

Weak Coupling ◽

Energy Levels ◽

Higher Order ◽

Wkb Approximation ◽

Reflection Property ◽

Exactly Solvable

Certain quasi-exactly solvable systems exhibit an energy reflection property that relates the energy levels of a potential or of a pair of potentials. We investigate two sister potentials and show the existence of this energy reflection relationship between the two potentials. We establish a relationship between the lowest energy edge in the first potential using the weak coupling expansion and the highest energy level in the sister potential using a WKB approximation carried out to higher order.

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HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS

Modern Physics Letters A ◽

10.1142/s0217732311036383 ◽

2011 ◽

Vol 26 (25) ◽

pp. 1843-1852 ◽

Cited By ~ 47

Author(s):

C. QUESNE

Keyword(s):

Quantum Mechanics ◽

Orthogonal Polynomials ◽

Higher Order ◽

Wave Functions ◽

Supersymmetric Quantum Mechanics ◽

Solvable Potentials ◽

Exactly Solvable Potentials ◽

Exactly Solvable ◽

Exceptional Orthogonal Polynomials

Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

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