Rational extension of many particle systems


  • Bhabani Prasad Mandal Banaras Hindu University, Institute of Science, Physics Department, Lanka, 221005–Varanasi, Uttar Pradesh India




exceptional orthogonal polynomials, rational extensions, many particle systems, SUSYQM


In this talk, we briefly review the rational extension of many particle systems, and is based on a couple of our recent works. In the first model, the rational extension of the truncated Calogero-Sutherland (TCS) model is discussed analytically. The spectrum is isospectral to the original system and the eigenfunctions are completely expressed in terms of exceptional orthogonal polynomials (EOPs). In the second model, we discuss the rational extension of a quasi exactly solvable (QES) N-particle Calogero model with harmonic confining interaction. New long-range interaction to the rational Calogero model is included to construct this QES many particle system using the technique of supersymmetric quantum mechanics (SUSYQM). Under a specific condition, infinite number of bound states are obtained for this system, and corresponding bound state wave functions are written in terms of EOPs.


Download data is not yet available.


D. Gómez-Ullate, N. Kamran, R. Milson. An extended class of orthogonal polynomials defined by Sturm-Liouville problem. Journal of Mathematical Analysis and Applications 359(1):352–367, 2009. https://doi.org/10.1016/j.jmaa.2009.05.052.

D. Gómez-Ullate, N. Kamran, R. Milson. Exceptional orthogonal polynomials and the Darboux transformation. Journal of Physics A: Mathematical and Theoretical 43(43):434016, 2010. https://doi.org/10.1088/1751-8113/43/43/434016.

D. Gómez-Ullate, N. Kamran, R. Milson. On orthogonal polynomials spanning a non-standard flag. Contemporary Mathematics 563:51, 2012.

D. Gómez-Ullate, Y. Grandati, R. Milson. Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. Journal of Physics A: Mathematical and Theoretical 47(1):015203, 2014. https://doi.org/10.1088/1751-8113/47/1/015203.

C. Quesne. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. Journal of Physics A: Mathematical and Theoretical 41(39):392001, 2008. https://doi.org/10.1088/1751-8113/41/39/392001.

B. Bagchi, C. Quesne, R. Roychoudhary. Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry. Pramana 73:337, 2009. https://doi.org/10.1007/s12043-009-0126-4.

S. Odake, R. Sasaki. Another set of infinitely many exceptional (Xl) Laguerre polynomials. Physics Letters B 684(2-3):173–176, 2010. https://doi.org/10.1016/j.physletb.2009.12.062.

C.-L. Ho, S. Odake, R. Sasaki. Properties of the exceptional (Xl) Laguerre and Jacobi polynomials. SIGMA 7:107, 2011. https://doi.org/10.3842/SIGMA.2011.107.

C.-L. Ho, R. Sasaki. Zeros of the exceptional Laguerre and Jacobi polynomials. International Scholarly Research Notices 2012:920475, 2012. https://doi.org/10.5402/2012/920475.

D. Gómez-Ullate, F. Marcellán, R. Milson. Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials. Journal of Mathematical Analysis and Applications 399(2):480–495, 2013. https://doi.org/10.1016/j.jmaa.2012.10.032.

C. Quesne. Rationally-extended radial oscillators and Laguerre exceptional orthogonal polynomials in kth-order SUSYQM. International Journal of Modern Physics A 26(32):5337–5347, 2011. https://doi.org/10.1142/S0217751X11054942.

Y. Grandati. Solvable rational extensions of the isotonic oscillator. Annals of Physics 326(8):2074–2090, 2011. https://doi.org/10.1016/j.aop.2011.03.001.

B. Midya, B. Roy. Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonian. Physics Letters A 373(45):4117–4122, 2009. https://doi.org/10.1016/j.physleta.2009.09.030.

B. Midya, P. Roy, T. Tanaka. Effect of position-dependent mass on dynamical breaking of type B and type X2 N-fold supersymmetry. Journal of Physics A: Mathematical and Theoretical 45(20):205303, 2012. https://doi.org/10.1088/1751-8113/45/20/205303.

K. Zelaya, V. Hussin. Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method. Journal of Physics A: Mathematical and Theoretical 53(16):165301, 2020. https://doi.org/10.1088/1751-8121/ab78d1.

S. E. Hoffmann, V. Hussin, I. Marquette, Y. Zhang. Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator. Journal of Mathematical Physics 60(5):052105, 2019. https://doi.org/10.1063/1.5091953.

C. Quesne. Quantum oscillator and Kepler–Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions. Journal of Mathematical Physics 57(10):102101, 2016. https://doi.org/10.1063/1.4963726.

S. Sree Ranjani, R. Sandhya, A. K. Kapoo. Shape invariant rational extensions and potentials related to exceptional polynomials. International Journal of Modern Physics A 30(24):1550146, 2015. https://doi.org/10.1142/S0217751X15501468.

B. Bagchi, C. Quesne. An update on the PT -symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally extended supersymmetric partners. Journal of Physics A: Mathematical and Theoretical 43(30):305301, 2010. https://doi.org/10.1088/1751-8113/43/30/305301.

B. Bagchi, Y. Grandati, C. Quesne. Rational extensions of the trigonometric Darboux–Pöschl–Teller potential based on para-Jacobi polynomials. Journal of Mathematical Physics 56(6):062103, 2015. https://doi.org/10.1063/1.4922017.

I. Marquette, C. Quesne. New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials. Journal of Mathematical Physics 54(4):042102, 2013. https://doi.org/10.1063/1.4798807.

D. Dutta, P. Roy. Conditionally exactly solvable potentials and exceptional orthogonal polynomials. Journal of Mathematical Physics 51(4):042101, 2010. https://doi.org/10.1063/1.3339676.

A. Ramos, B. Bagchi, N. Kumari, et al. A short note on “Group theoretic approach to rationally extended shape invariant potentials” [Ann. Phys. 359 (2015) 46-54]. Annals of Physics 382:143–149, 2017. https://doi.org/10.1016/j.aop.2017.05.006.

B. Basu-Mallick, B. P. Mandal, P. Roy. Quasi exactly solvable extension of Calogero model associated with exceptional orthogonal polynomials. Annals of Physics 380:206–212, 2017. https://doi.org/10.1016/j.aop.2017.03.019.

R. K. Yadav, A. Khare, N. Kumari, B. P. Mandal. Rationally extended many-body truncated Calogero–Sutherland model. Annals of Physics 400:189–197, 2019. https://doi.org/10.1016/j.aop.2018.11.009.

R. K. Yadav, S. R. Banerjee, A. Khare, et al. One parameter family of rationally extended isospectral potentials. Annals of Physics 436:168679, 2022. https://doi.org/10.1016/j.aop.2021.168679.

N. Kumari, R. K. Yadav, A. Khare, B. P. Mandal. A class of exactly solvable rationally extended Calogero-Wolfes type 3-body problems. Annals of Physics 385:57–69, 2017. https://doi.org/10.1016/j.aop.2017.07.022.

N. Kumari, R. K. Yadav, A. Khare, B. P. Mandal. A class of exactly solvable rationally extended non-central potentials in two and three dimensions. Journal of Mathematical Physics 59(6):062103, 2018. https://doi.org/10.1063/1.4996282.

B. Midhya, P. Roy. Infinite families of (non)-Hermitian Hamiltonians associated with exceptional Xm Jacobi polynomials. Journal of Physics A: Mathematical and Theoretical 46(17):175201, 2013. https://doi.org/10.1088/1751-8113/46/17/175201.

N. Kumari, R. K. Yadav, A. Khare, et al. Scattering amplitudes for the rationally extended PT symmetric complex potentials. Annals of Physics 373:163–177, 2016. https://doi.org/10.1016/j.aop.2016.07.024.

N. Kumari, R. K. Yadav, A. Khare, B. P. Mandal. Group theoretic approach to rationally extended shape invariant potentials. Annals of Physics 359:46–54, 2015. https://doi.org/10.1016/j.aop.2015.04.002.

R. K. Yadav, A. Khare, B. Bagchi, et al. Parametric symmetries in exactly solvable real and PT symmetric complex potentials. Journal of Mathematical Physics 57(6):062106, 2016. https://doi.org/10.1063/1.4954330.

R. K. Yadav, A. Khare, B. P. Mandal. The scattering amplitude for newly found exactly solvable potential. Annals of Physics 331:313–316, 2013. https://doi.org/10.1016/j.aop.2013.01.006.

R. K. Yadav, A. Khare, B. P. Mandal. The scattering amplitude for one parameter family of shape invariant potentials related to Xm Jacobi polynomials. Physics Letter B 723(4-5):433–435, 2013. https://doi.org/10.1016/j.physletb.2013.05.036.

R. K. Yadav, A. Khare, B. P. Mandal. Rationally extended shape invariant potentials in arbitrary D-dimensions associated with exceptional Xm polynomials. Acta Polytechnica 57(6):477–487, 2017. https://doi.org/10.14311/AP.2017.57.0477.

R. K. Yadav, A. Khare, B. P. Mandal. The scattering amplitude for rationally extended shape invariant Eckart potentials. Physics Letter A 379(3):67–70, 2015. https://doi.org/10.1016/j.physleta.2014.11.009.

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.

W. Witten. Dynamical breaking of supersymmetry. Nuclear Physics B 188(3):513–554, 1981. https://doi.org/10.1016/0550-3213(81)90006-7.

F. Cooper, A. Khare, U. Sukhatme. Supersymmetric Quantum Mechanics. World Scientific, Singapore, 2001. https://doi.org/10.1142/4687.

A. Anderson. Canonical transformations in quantum mechanics. Annals of Physics 232(2):292–331, 1994. https://doi.org/10.1006/aphy.1994.1055.

A. Bhattacharjie, E. C. G. Sudarshan. A class of solvable potentials. Nuovo Cimento 25:864–879, 1962. https://doi.org/10.1007/BF02733153.

G. Darboux. Theorie Generale des Surfaces, Vol. 2. Gauthier-Villars, Paris, 1998.

R. Sasaki, S. Tsujimoto, A. Zhedanov. Exceptional Laguerre and Jacobi polynomials and corresponding potentials through Darboux Crum transformations. Journal of Physics A: Mathematical and Theoretical 43(31):315204, 2010. https://doi.org/10.1088/1751-8113/43/31/315204.

Y. Alhassid, F. Gürsey, F. Iachello. Potential scattering, transfer matrix and group theory. Physical Review Letters 50(12):873–876, 1983. https://doi.org/10.1103/PhysRevLett.50.873.

A. Ushveridze. Quasi-Exactly Solvable Models in Quantum Mechanics. IOP, Bristol, 1994.

A. Khare, B. P. Mandal. A number of quasi-exactly solvable N-body problems. Journal of Mathematical Physics 39(11):5789–5797, 1998. https://doi.org/10.1063/1.532593.

A. Khare, B. P. Mandal. Do quasi-exactly solvable systems always correspond to orthogonal polynomials ? Physics Letter A 239(4-5):197–200, 1998. https://doi.org/10.1016/S0375-9601(97)00897-9.

A. Khare, B. P. Mandal. New QES Hermitian as well as non-Hermitian PT invariant potentials. Pramana: Journal of Physics 73:387–395, 2009.

A. Turbiner. Quasi-exactly-solvable problems andsl(2) algebra. Communications in Mathematical Physics 118:467–474, 1988. https://doi.org/10.1007/BF01466727.

G. Junker, P. Roy. Conditionally exactly solvable problems and non-linear algebras. Physics Letters A 232(3-4):155–161, 1997. https://doi.org/10.1016/S0375-9601(97)00422-2.

G. Levai, P. Roy. Conditionally exactly solvable potentials and supersymmetric transformations. Physics Letters A 264(2-3):117–123, 1999. https://doi.org/10.1016/S0375-9601(99)00778-1.

C. M. Bender, S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters 80(24):5243–5246, 1998. https://doi.org/10.1103/PhysRevLett.80.5243.

C. M. Bender. Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics 70(6):947–1018, 2007. https://doi.org/10.1088/0034-4885/70/6/R03.

A. Mostafazadeh. Pseudo-Hermitian representation of quantum mechanics. International Journal of Geometric Methods in Modern Physics 07(07):1191–1306, 2010. https://doi.org/10.1142/S0219887810004816.

M. Moiseyev. Non-Hermitian quantum mechanics. Cambridge University Press, 2011.

F. Bagarallo, J.-P. Gazeau, F. H. Szafraniec, M. Znojil. Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects. Wiley, Hoboken, 2015.

B. Basu-Mallick, B. P. Mandal. On an exactly solvable BN type Calogero model with non-Hermitian PT invariant interaction. Physics Letters A 284(6):231–237, 2001. https://doi.org/10.1016/S0375-9601(01)00310-3.

A. Khare, B. P. Mandal. A PT-invariant potential with complex QES eigenvalues. Physics Letters A 272(1-2):53–56, 2000. https://doi.org/10.1016/S0375-9601(00)00409-6.

R. Modak, B. P. Mandal. Eigenstate entanglement entropy in a PT -invariant non-Hermitian system. Physical Review A 103(6):062416, 2021. https://doi.org/10.1103/PhysRevA.103.062416.

R. Rawal, B. P. Mandal. Deconfinement to confinement as PT phase transition. Nuclear Physics B 946:114699, 2019. https://doi.org/10.1016/j.nuclphysb.2019.114699.

A. Dwivedi, B. P. Mandal. Higher loop function for non-Hermitian PT symmetric ig3 theory. Annals of Physics 425:168382, 2021. https://doi.org/10.1016/j.aop.2020.168382.

B. Basu-Mallick, T. Bhattacharyya, B. P. Mandal. Phase shift analysis of PT-symmetric nonhermitian extension of AN-1 Calogero model without confining interaction. Modern Physics Letters A 20(7):543–552, 2005. https://doi.org/10.1142/S0217732305015896.

S. R. Jain, A. Khare. An exactly solvable many-body problem in one dimension and the short-range Dyson model. Physics Letters A 262(1):35–39, 1999. https://doi.org/10.1016/S0375-9601(99)00637-4.

S. M. Pittam, M. Beau, M. Olshanii, A. del Campo. Truncated Calogero-Sutherland models. Physical Review B 95(20):205135, 2017. https://doi.org/10.1103/PhysRevB.95.205135.

F. Calogero. Solution of a three-body problem in one dimension. Journal of Mathematical Physics 10(12):2191–2196, 1969. https://doi.org/10.1063/1.1664820.

F. Calogero. Ground state of a one dimensional N-body system. Journal of Mathematical Physics 10(12):2197–2200, 1969. https://doi.org/10.1063/1.1664821.

B. Sutherland. Quantum many-body problem in one dimension: Ground state. Journal of Mathematical Physics 12(2):246–250, 1971. https://doi.org/10.1063/1.1665584.




How to Cite

Mandal, B. P. (2022). Rational extension of many particle systems. Acta Polytechnica, 62(1), 90–99. https://doi.org/10.14311/AP.2022.62.0090



Analytic and Algebraic Methods in Physics