Quantization of rationally deformed Morse potentials by Wronskian transforms of Romanovski-Bessel polynomials


  • Gregory Natanson AI-Solutions Silver Spring MD 20904, U.S.A.




translationally form-invariant Sturm-Liouville equation, generalized Bessel polynomials, Romanovski-Bessel polynomials, rational Darboux-Crum transformations, polynomial Wronskians


The paper advances Odake and Sasaki’s idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral family of solvable rational potentials by including ‘juxtaposed’ pairs of Romanovski-Bessel polynomials into the aforementioned polynomial Wronskians which results in deleting the corresponding pairs of bound energy states.


Download data is not yet available.


A. D. Alhaidari. Exponentially confining potential well. Theoretical and Mathematical Physics volume 206:84–96, 2021. https://doi.org/10.1134/S0040577921010050.

J. Gibbons, A. P. Veselov. On the rational monodromy-free potentials with sextic growth. Journal of Mathematical Physics 50(1):013513, 2009. https://doi.org/10.1063/1.3001604.

H. L. Krall, O. Frink. A new class of orthogonal polynomials: the Bessel polynomials. Transactions of the American Mathematical Society 65:100–105, 1949. https://doi.org/10.1090 S0002-9947-1949-0028473-1.

W. A. Al-Salam. The Bessel polynomials. Duke Mathematical Journal 24(4):529–545, 1957. https://doi.org/10.1215/S0012-7094-57-02460-2.

T. S. Chihara. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.

R. Koekoek, P. A. Lesky, R. F. Swarttouw. Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, Heidelberg, 2010.

D. Gomez-Ullate, N. Kamran, R. Milson. The Darboux transformation and algebraic deformations of shape-invariant potentials. Journal of Physics A: Mathematical and General 37(5):1789–1804, 2004. https://doi.org/10.1088/0305-4470/37/5/022.

Y. Grandati. Solvable rational extensions of the Morse and Kepler-Coulomb potentials. Journal of Mathematical Physics 52(10):103505, 2011. https://doi.org/10.1063/1.3651222.

C.-L. Ho. Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials. Progress of Theoretical Physics 126(2):185–201, 2011. https://doi.org/10.1143/PTP.126.185.

C. Quesne. Revisiting (quasi-)exactly solvable rational extensions of the Morse potential. International Journal of Modern Physics A 27(13):1250073, 2012. https://doi.org/10.1142/S0217751X1250073X.

D. Gomez-Ullate, Y. Grandati, R. Milson. Extended Krein-Adler theorem for the translationally shape invariant potentials. Journal of Mathematical Physics 55(4):043510, 2014. https://doi.org/10.1063/1.4871443.

V. I. Romanovski. Sur quelques classes nouvelles de polynomes orthogonaux. Comptes Rendus de l’Académie des Sciences 188:1023–1025, 1929.

P. A. Lesky. Einordnung der Polynome von Romanovski-Bessel in das Askey-Tableau. Zeitschrift für Angewandte Mathematik und Mechanik 78(9):646–648, 1998. https://doi.org/b8b8sp.

C. Quesne. Extending Romanovski polynomials in quantum mechanics. Journal of Mathematical Physics 54(12):122103, 2013. https://doi.org/10.1063/1.4835555.

N. Cotfas. Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics. Central European Journal of Physics 2(3):456–466, 2004. https://doi.org/10.2478/BF02476425.

N. Cotfas. Shape-invariant hypergeometric type operators with application to quantum mechanics. Central European Journal of Physics 4(3):318–330, 2006. https://doi.org/10.2478/s11534-006-0023-0.

M. A. Jafarizadeh, H. Fakhri. Parasupersymmetry and shape invariance in differential equations of mathematical physics and quantum mechanics. Annals of Physics 262(2):260–276, 1998. https://doi.org/10.1006/aphy.1997.5745.

S. Odake, R. Sasaki. Extensions of solvable potentials with finitely many discrete eigenstates. Journal of Physics A: Mathematical and Theoretical 46(23):235205, 2013. https://doi.org/10.1088/1751-8113/46/23/235205.

S. Odake. Krein–Adler transformations for shape-invariant potentials and pseudo virtual states. Journal of Physics A: Mathematical and Theoretical 46(24):245201, 2013. https://doi.org/10.1088/1751-8113/46/24/245201.

G. Darboux. Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. Gauthier-Villars, Paris, 1915.

M. M. Crum. Associated Sturm-Liouville systems. The Quarterly Journal of Mathematics 6(1):121–127, 1955. https://doi.org/10.1093/qmath/6.1.121.

L. D. Landau, E. M. Lifshity. Quantum Mechanics (Non-relativistic Theory). 3rd ed. Butterworth-Heinemann, 1977.

G. Natanson. Equivalence relations for Darboux-Crum transforms of translationally form-invariant Sturm-Liouville equations, 2021. https://www.researchgate.net/publication/353131294.

D. Gómez-Ullate, Y. Grandati, R. Milson. Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials. Studies in Applied Mathematics 141(4):596–625, 2018. https://doi.org/10.1111/sapm.12225.

W. N. Everitt, L. L. Littlejohn. Orthogonal polynomials and spectral theory: a survey. In C. Brezinski, L. Gori, A. Ronveaux (eds.), Orthogonal Polynomials and their Applications, vol. 9, pp. 21–55. 1991. IMACS Annals on Computing and Applied Mathematics.

W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman. Orthogonal polynomial solutions of linear ordinary differential equations. Journal of Computational and Applied Mathematics 133(1-2):85–109, 2001. https://doi.org/10.1016/S0377-0427(00)00636-1.

A. K. Bose. A class of solvable potentials. Nuovo Cimento 32:679–688, 1964. https://doi.org/10.1007/BF02735890.

G. A. Natanzon. Study of the one-dimensional Schrödinger equation generated from the hypergeometric equation. Vestnik Leningradskogo Universiteta 10:22–28, 1971. English translation https://arxiv.org/PS_cache/physics/pdf/9907/9907032v1.pdf.

B. V. Rudyak, B. N. Zakhariev. New exactly solvable models for Schrödinger equation. Inverse Problems 3(1):125–133, 1987. https://doi.org/10.1088/0266-5611/3/1/014.

L. D. Fadeev, B. Seckler. The inverse problem in the quantum theory of scattering. Journal of Mathematical Physics 4(1):72–104, 1963. https://doi.org/10.1063/1.1703891.

W. A. Schnizer, H. Leeb. Exactly solvable models for the Schrödinger equation from generalized Darboux transformations. Journal of Physics A: Mathematical and General 26(19):5145–5156, 1993. https://doi.org/10.1088/0305-4470/26/19/041.

W. A. Schnizer, H. Leeb. Generalized Darboux transformations: classification of inverse scattering methods for the radial Schrödinger equation. Journal of Physics A: Mathematical and General 27(7):2605–2614, 1994. https://doi.org/10.1088/0305-4470/27/7/035.

D. Gómez-Ullate, Y. Grandati, R. Milson. Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials. Journal of Physics A: Mathematical and Theoretical 51(34):345201, 2018. https://doi.org/10.1088/1751-8121/aace4b.

L. E. Gendenshtein. Derivation of exact spectra of the Schrödinger equation by means of supersymmetry. Journal of Experimental and Theoretical Physics Letters 38:356–359, 1983.

L. E. Gendenshtein, I. V. Krive. Supersymmetry in quantum mechanics. Soviet Physics Uspekhi 28(8):645–666, 1985. https://doi.org/10.1070/PU1985v028n08ABEH003882.

B. F. Samsonov. On the equivalence of the integral and the differential exact solution generation methods for the one-dimensional Schrodinger equation. Journal of Physics A: Mathematical and General 28(23):6989–6998, 1995. https://doi.org/10.1088/0305-4470/28/23/036.

B. F. Samsonov. New features in supersymmetry breakdown in quantum mechanics. Modern Physics Letters A 11(19):1563–1567, 1996. https://doi.org/10.1142/S0217732396001557.

V. G. Bagrov, B. F. Samsonov. Darboux transformation and elementary exact solutions of the Schrödinger equation. Pramana 49:563–580, 1997. https://doi.org/10.1007/BF02848330.

F. Brafman. A set of generating functions for Bessel polynomials. Proceedings of the American Mathematical Society 4:275–277, 1953. https://doi.org/10.1090/S0002-9939-1953-0054100-X.

K. H. Kwon, L. L. Littlejohn. Classification of classical orthogonal polynomials. Journal of the Korean Mathematical Society 34(4):973–1008, 1997.

A. F. Nikiforov, V. B. Uvarov. Special Functions of Mathematical Physics. Birkhauser, Basel, 1988. https://doi.org/10.1007/978-1-4757-1595-8.

G. Natanson. Darboux-Crum nets of Sturm-Liouville problems solvable by quasi-rational functions I. General theory, 2018. https://doi.org/10.13140/RG.2.2.31016.06405/1.

F. Gesztesy, B. Simon, G. Teschl. Zeros of the Wronskian and renormalized oscillation theory. American Journal of Mathematics 118(3):571–594, 1996. https://doi.org/10.1353/ajm.1996.0024.

G. Natanson. Exact quantization of the Milson potential via Romanovski-Routh polynomials, 2015. https://doi.org/10.13140/RG.2.2.24354.09928.

A. Schulze-Halberg. Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations. Applied Mathematics Letters 25(10):1520–1527, 2012. https://doi.org/10.1016/j.aml.2012.01.008.

D. Gomez-Ullate, N. Kamran, R. Milson. An extension of Bochner’s problem: exceptional invariant subspaces. Journal of Approximation Theory 162(5):987–1006, 2010. https://doi.org/10.1016/j.jat.2009.11.002.

S. Bochner. Über Sturm-Liouvillesche Polynomsysteme. Mathematische Zeitschrift 29:730–736, 1929. https://doi.org/10.1007/BF01180560.

V. G. Bagrov, B. F. Samsonov. Darboux transformation of the Schrödinger equation. Physics of Particles and Nuclei 28(4):374–397, 1997. https://doi.org/10.1134/1.953045.

M. G. Krein. On a continuous analogue of the Christoffel formula from the theory of orthogonal polynomials. Doklady Akademii Nauk SSSR 113(5):970–973, 1957.

V. E. Adler. A modification of Crum’s method. Theoretical and Mathematical Physics 101:1381–1386, 1994. https://doi.org/10.1007/BF01035458.

A. J. Durán, M. Pérez, J. L. Varona. Some conjecture on Wronskian and Casorati determinants of orthogonal polynomials. Experimental Mathematics 24(1):123–132, 2015. https://doi.org/10.1080/10586458.2014.958786.

A. Kienast. Untersuchungen über die Lösungen der Differentialgleichung xy ′′ + (γ − x)y ′ − βy. Denkschriften der Schweizerischen Naturforschenden Gesellschaft 57:247, 79 pages, 1921.

W. Lawton. On the zeros of certain polynomials related to Jacobi and Laguerre polynomials. Bulletin of the American Mathematical Society 38:442–448, 1932. https://doi.org/10.1090/S0002-9904-1932-05418-0.

W. Hahn. Bericht über die Nullstellen der Laguerreschen und der Hermiteschen Polynome. Jahresbericht der Deutschen Mathematiker-Vereinigung 1:215–236, 1933.

R. Courant, D. Hilbert. Methods of Mathematical Physics, Vol. 1,. Interscience, New York, 1953.






Analytic and Algebraic Methods in Physics