Publication: Oscillation Theorems for the Wronskian of an Arbitrary Sequence of Eigenfunctions of Schrodinger's Equation
Loading...
Official URL
Full text at PDC
Publication Date
2015-04
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Kluwer Academic
Citations
Exportar
Abstract
The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schrodinger's equation to have constant sign in its domain of definition. We extend this result by giving explicit formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions. Our results apply in particular to Wronskians of classical orthogonal polynomials, thus generalizing classical results by Karlin and SzegA. Our formulas hold under very mild conditions that are believed to hold for generic values of the parameters. In the Hermite case, our results allow to prove some conjectures recently formulated by Felder et al.
Description
© 2015 Springer-Verlag. The authors would like to thank Robert Milson and Antonio Durán for stimulating discussions. The elegant proof of Lemma 3.1 that uses the irreducibility of Hermite polynomials is in fact entirely due to Robert Milson. MAGF would like to thank the Department of Theoretical Physics II at Universidad Complutense for providing her with office space and all facilities. The research of DGU has been supported in part by the Spanish MINECO-FEDER Grants MTM2012 31714 and FIS2012-38949- C03-01.
UCM subjects
Unesco subjects
Keywords
Citation
[1] N. C. Freeman, and J. J. C. Nimmo, Soliton solutions of the Korteweg de Vries and the Kadomtsev-Petviashvili equations: the Wronskian technique, Proc. Royal Soc. London A (1983) 389 319–329.
[2] L. Infeld and T. E. Hull, The factorization method. Rev. Mod. Phys., 23 (1951) 21.
[3] M.M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955) 121–127.
[4] V.E. Adler, A modification of Crum’s method, ´ Theor. Math. Phys., 101 (1994) 1381–1386.
[5] M.G. Krein, A continual analogue of a Christoffel formula from the theory of orthogonal polynomials, Dokl. Akad. Nauk. SSSR, 113 (1957) 970–973.
[6] D. Gomez-Ullate, Y. Grandati, and R. Milson, Extended Krein-Adler theorem for the translationally shape invariant potentials, J. Math. Phys. 55 (2014) 043510.
[7] S. Karlin and G. Szegő, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960) 1–157.
[8] D. Gomez-Ullate, N. Kamran, and R. Milson, An extended class of orthogonal polynomials defined by a SturmLiouville problem, J. Math. Anal. Appl., 359 (2009) 352–367.
[9] D. Gomez-Ullate, N. Kamran, and R. Milson, An extension of Bochner’s problem: Exceptional invariant subspaces, J. Approx. Theory, 162 (2010) 987–1006.
[10] S. Odake and R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009) 414.
[11] D. Gómez-Ullate, N. Kamran, and R. Milson, Exceptional orthogonal polynomials and th Darboux transformation. J. Phys. A, 43 (2010) 434016–434032.
[12] D. Gómez-Ullate, N. Kamran, and R. Milson, Two-step Darboux transformations and exceptional Laguerre polynomials. J. Math. Anal. Appl., 387 (2012) 410–418.
[13] D. Gomez-Ullate, N. Kamran, and R. Milson, A conjecture on exceptional orthogonal olynomials, Found. Comput. Math. 13 (2013) 615–666.
[14] A. Durán Exceptional Charlier and Hermite orthogonal polynomials, J. Approx Theory, 182 (2014) 29–58.
[15] A. Durán Exceptional Meixner and Laguerre orthogonal polynomials, J. Approx Theory, 184 (2014) 176–208.
[16] D. Gómez-Ullate, Y. Grandati, and R. Milson, Rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials. J. Phys. A 47 (2014) 015203.
[17] D. Gómez-Ullate, F. Marcellan, and R. Milson, Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials, J. Math. Anal. Appl., 399 (2013) 480–495.
[18] P. A. Clarkson, The fourth Painlev´e equation and associated special polynomials. J. Math. Phys. 44 (2003) 5350– 5374.
[19] P. A. Clarkson, On rational solutions of the fourth Painlevé equation and its Hamiltonian, CRM Proc. Lect. Notes 39 (2005) 103–118.
[20] G. V. Filipuk and P. A. Clarkson, The symmetric fourth Painlevé hierarchy and associated special polynomials, Stud. Appl. Math. 121 (2008) 157–188.
[21] G. Felder, A. D. Hemery, and A. P. Veselov, Zeros of Wronskians of Hermite polynomials and Young diagrams, Physica D, 241 (2012) 2131–2137.
[22] L. Zhang and G. Filipuk, On certain Wronskians of multiple orthogonal polynomials, (2014) arXiv:1402.1569 [math.CA]
[23] D. K. Dimitrov, F. R. Rafaeli, Monotonicity of zeros of Laguerre polynomials, J. Comput. Appl. Math. 233 (2009) 699–702.
[24] H. L. Dorwart, Irreducibility of polynomials Amer. Math. Monthly, 42 (1935) 369–381.
[25] I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, II, Sitzungsber. Preuss. Akad. Wiss. Berlin Phys.-Math. Kl., 14 (1929), 370391.