Extensions of solvable potentials with finitely many discrete eigenstates

Abstract

We address the problem of rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates. The overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions, which are an essential ingredient for multi-indexed and iso-spectral extensions of these potentials. They have exactly the same form as the eigenfunctions but their degrees are much higher than n_{max} so that their energies are lower than the groundstate energy.

22 pages, 3 figures. Typo corrected, comments and references added. To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1212.6595

References in corpus (7)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Solvable rational extensions of the isotonic oscillator
• Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Symmetries and the compatibility condition for the new translational shape invariant potentials

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• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
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• Extending Romanovski polynomials in quantum mechanics
• Inter-relations between additive shape invariant superpotentials
• The Scattering amplitude for Rationally extended shape invariant Eckart potentials
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials
• Generation of a Novel Exactly Solvable Potential
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• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• Exactness of SWKB for Shape Invariant Potentials
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• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
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• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Exactness of Semiclassical Quantization Rule for Broken Supersymmetry
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• Generalized Langer Correction and the Exactness of WKB for all Conventional Potentials
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts