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Extended Krein-Adler theorem for the translationally shape invariant potentials. (English) Zbl 1296.81023
In this paper the authors consider extensions of primary translationally shape invariant potentials built from mixed chains ,i.e., containing state-adding and state -deleting Darboux Backlund transformations.The specific case of the harminic oscillator, where the type 3 symmetry corresponds in fact to a “Wick rotation” of the independent variable,has been the subject of long time investigation [V. G. Bagrov and B. F. Samsonov, Theor. Math. Phys. 104, No. 2, 1051–1060 (1995); translation from Teor. Mat. Fiz. 104, No. 2, 356–367 (1995; Zbl 0857.34070); “Darboux transformation and elementary exact solutions of the Schrödinger equation”, Pramana J. Phys. 49, No. 6, 563–580 (1997; doi:10.1007/BF02848330); D. J. Fernández C. et al., Phys. Lett., A 244, No. 5, 309–316 (1998; Zbl 0941.81022); S. Yu. Dubov et al., Chaos 4, No. 1, 47–53 (1994; Zbl 1055.81529); V. M. Tkachuk, J. Phys. A, Math. Gen. 32, No. 7, 1291–1300 (1999; Zbl 1055.81563)]. Oblomkov has in particular shown that the extensions obtained from general mixed chains coincide with the rational extensions of the harmonic oscillator which possess the trivial monodromy property. He extended the seminal paper of Duidtermaat and Grunbaum and subsequent works [O. A. Chalykh, Russ. Math. Surv. 53, No. 2, 377–379 (1998); translation from Usp. Mat. Nauk 53, No. 2, 167–168 (1998; Zbl 0922.35009); V. M. Goncharenko and A. P. Veselov, J. Phys. A, Math. Gen. 31, No. 23, 5315–5326 (1998; Zbl 0921.34077)] to quadratically increasing rational potentials.Recently the authors have proven that this set of extensions contains those which are associated to Hamiltonians exactly solvable by polynomials and whose spectrum is subtended by systems of exceptional Hermite polynomials providing a precise description of the properties of the these exceptional orthogonal polynomials.The authors show that every primary translationally shape invariant potentials possess a “reverse shape invariance property” which allows one to enlarge the Krein-Adler theorem to some mixed chains.This provides in particular new bilinear determinantal or Wronskians identities for the classical orthogonal polynomials.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35K45 Initial value problems for second-order parabolic systems
33C47 Other special orthogonal polynomials and functions
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