Abstract
As is known, a nonnegative-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set allows defining forward (and backward) shift operators that can be used to determine the matrix representation of the supersymmetric partner Hamiltonian H(+) in the same basis. We show that if the Hamiltonian is also shape-invariant, then the matrix elements of the Hamiltonian are related such that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of supersymmetric partner Hamiltonians. Moreover, we derive the coherent states associated with this type of Hamiltonian and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also have a tridiagonal matrix representation.
Similar content being viewed by others
References
L. E. Gendenshtein, “Derivation of exact spectra of the Schrödinger equation by means of supersymmetry,” Soviet JETP Lett., 38, 356–359 (1983).
R. Dutt, A. Khare, and U. P. Sukhatme, “Supersymmetry, shape invariance, and exactly solvable potentials,” Amer. J. Phys., 56, 163–168 (1988).
A. A. Andrianov, N. V. Borisov, M. V. Ioffe, and M. I. Eides, “Supersymmetric mechanics: A new look at the equivalence of quantum systems,” Theor. Math. Phys., 61, 965–972 (1984).
A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, “The factorization method and quantum systems with equivalent energy spectra,” Phys. Lett. A, 105, 19–22 (1984).
F. Cooper, A. Khare, and U. P. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep., 251, 267–385 (1995); arXiv:hep-th/9405029v2 (1994).
W. H. Miller Jr., “Lie theory and difference equations: I,” J. Math. Anal. Appl., 28, 383–399 (1969).
V. Spiridonov, L. Vinet, and A. Zhedanov, “Difference Schrödinger operators with linear and exponential discrete spectra,” Lett. Math. Phys., 29, 63–73 (1993).
F. Cooper, A. Khare, and U. P. Sukhtame, Supersymmetry in Quantum Mechanics, World Scientific, Singapore (2001).
M. S. Swanson, A Concise Introduction to Quantum Mechanics, Morgan and Claypool, San Rafael, Calif. (2018).
H. A. Yamani and Z. Mouayn, “Supersymmetry of tridiagonal Hamiltonians,” J. Phys. A: Math. Theor., 47, 265203 (2014).
H. A. Yamani and Z. Mouayn, “Supersymmetry of the Morse oscillator,” Rep. Math. Phys., 78, 281–294 (2016).
E. A. van Dooren, “Spectral properties of birth-death polynomials,” J. Comput. Appl. Math., 284, 251–258 (2015).
J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics, World Scientific, Singapore (1985).
H. Bateman and A. Erdelyi, Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York (1954).
A. D. Alhaidari, “An extended class of L2-series solutions of the wave equation,” Ann. Phys., 317, 152–174 (2005); arXiv:quant-ph/0409002v1 (2004).
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Grundlehren Math. Wiss., Vol. 52), Springer, Berlin (1966).
Acknowledgments
The authors thank a referee for the valuable comments and suggestions that improved the quality of this paper.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Conflicts of interest
The authors declare no conflicts of interest.
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 3, pp. 380–400, June, 2020.
Rights and permissions
About this article
Cite this article
Yamani, H.A., Mouayn, Z. Properties of Shape-Invariant Tridiagonal Hamiltonians. Theor Math Phys 203, 761–779 (2020). https://doi.org/10.1134/S0040577920060057
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577920060057