The spectrum of self-adjoint extensions associated with exceptional Laguerre differential expressions. (English) Zbl 07797571
Summary: Exceptional Laguerre-type differential expressions form an infinite class of Schrödinger operators having rational potentials and one limit-circle endpoint. In this manuscript, the spectrum of all self-adjoint extensions for a general exceptional Laguerre-type differential expression is given in terms of the Darboux transformations which relate the expression to the classical Laguerre differential expression. The spectrum is extracted from an explicit Weyl \(m\)-function, up to a sign.
The construction relies primarily on two tools: boundary triples, which parameterize the self-adjoint extensions and produce the Weyl \(m\)-functions, and manipulations of Maya diagrams and partitions, which classify the seed functions defining the relevant Darboux transforms. Several examples are presented.
The construction relies primarily on two tools: boundary triples, which parameterize the self-adjoint extensions and produce the Weyl \(m\)-functions, and manipulations of Maya diagrams and partitions, which classify the seed functions defining the relevant Darboux transforms. Several examples are presented.
MSC:
| 47-XX | Operator theory |
| 34L05 | General spectral theory of ordinary differential operators |
| 33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |
| 47E05 | General theory of ordinary differential operators |
| 47A10 | Spectrum, resolvent |
Keywords:
exceptional orthogonal polynomials; self-adjoint extensions; Weyl \(m\)-function; spectral theory; boundary triplesReferences:
| [1] | N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equations. Amer. J. Math. 79 (1957), 597-610 Zbl 0079.10802 MR 88623 · Zbl 0079.10802 · doi:10.2307/2372564 |
| [2] | M. J. Atia, L. L. Littlejohn, and J. Stewart, Spectral theory of X 1 -Laguerre polynomials. Adv. Dyn. Syst. Appl. 8 (2013), no. 2, 181-192 MR 3162140 · Zbl 1333.33018 · doi:10.1016/j.jat.2015.11.001 |
| [3] | J. Behrndt, S. Hassi, and H. de Snoo, Boundary value problems, Weyl functions, and dif-ferential operators. Monogr. Math. 108, Birkhäuser/Springer, Cham, 2020 Zbl 1457.47001 MR 3971207 · Zbl 1457.47001 · doi:10.1007/978-3-030-36714-5 |
| [4] | S. Bochner, Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1929), no. 1, 730-736 Zbl 55.0260.01 MR 1545034 · JFM 55.0260.01 · doi:10.1007/BF01180560 |
| [5] | R. F. Boisvert, C. W. Clark, H. S. Cohl, D. W. Lozier, M. A. McClain, B. R. Miller, A. B. Olde Daalhuis, F, W. J. Olver, B. V. Saunders, B. I. Schneider (eds.), NIST digi-tal library of mathematical functions. Release 1.0.22 of 2019-03-15 httpsW//dlmf.nist.gov/ visited on 20 November 2023 |
| [6] | N. Bonneux and A. B. J. Kuijlaars, Exceptional Laguerre polynomials. Stud. Appl. Math. 141 (2018), no. 4, 547-595 Zbl 1408.33019 MR 3879969 · Zbl 1408.33019 · doi:10.1111/sapm.12204 |
| [7] | M. Bush, D. Frymark, and C. Liaw, Singular boundary conditions for Sturm-Liouville operators via perturbation theory. Canad. J. Math. 75 (2023), no. 4, 1110-1146 Zbl 07720708 MR 4620317 · Zbl 07720708 · doi:10.4153/s0008414x22000293 |
| [8] | M. M. Crum, Associated Sturm-Liouville systems. Quart. J. Math. Oxford Ser. (2) 6 (1955), 121-127 Zbl 0065.31901 MR 72332 · Zbl 0065.31901 · doi:10.1093/qmath/6.1.121 |
| [9] | V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value prob-lems for Hermitian operators with gaps. J. Funct. Anal. 95 (1991), no. 1, 1-95 Zbl 0748.47004 MR 1087947 · Zbl 0748.47004 · doi:10.1016/0022-1236(91)90024-Y |
| [10] | A. J. Durán, Exceptional Meixner and Laguerre orthogonal polynomials. J. Approx. The-ory 184 (2014), 176-208 Zbl 1295.33013 MR 3218798 · Zbl 1295.33013 · doi:10.1016/j.jat.2014.05.009 |
| [11] | A. J. Durán and M. Pérez, Admissibility condition for exceptional Laguerre polynomials. J. Math. Anal. Appl. 424 (2015), no. 2, 1042-1053 Zbl 1305.33019 MR 3292716 · Zbl 1305.33019 · doi:10.1016/j.jmaa.2014.11.035 |
| [12] | D. Frymark, Boundary triples and Weyl m-functions for powers of the Jacobi differential operator. J. Differential Equations 269 (2020), no. 10, 7931-7974 Zbl 1493.34080 MR 4113193 · Zbl 1493.34080 · doi:10.1016/j.jde.2020.05.032 |
| [13] | D. Frymark and C. Liaw, Spectral properties of singular Sturm-Liouville operators via boundary triples and perturbation theory. J. Differential Equations 363 (2023), 391-421 Zbl 07684541 MR 4568769 · Zbl 07684541 · doi:10.1016/j.jde.2023.03.022 |
| [14] | M. Á. García-Ferrero, D. Gómez-Ullate, and R. Milson, A Bochner type characterization theorem for exceptional orthogonal polynomials. J. Math. Anal. Appl. 472 (2019), no. 1, 584-626 Zbl 1421.42014 MR 3906391 · Zbl 1421.42014 · doi:10.1016/j.jmaa.2018.11.042 |
| [15] | F. Gesztesy, L. L. Littlejohn, and R. Nichols, On self-adjoint boundary conditions for singular Sturm-Liouville operators bounded from below. J. Differential Equations 269 (2020), no. 9, 6448-6491 Zbl 1458.34143 MR 4107059 · Zbl 1458.34143 · doi:10.1016/j.jde.2020.05.005 |
| [16] | D. Gómez-Ullate, Y. Grandati, and R. Milson, Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials. J. Phys. A 51 (2018), no. 34, article no. 345201 Zbl 1400.33017 MR 3835328 · Zbl 1400.33017 · doi:10.1088/1751-8121/aace4b |
| [17] | D. Gómez-Ullate, Y. Grandati, and R. Milson, Spectral theory of exceptional Hermite polynomials. In From operator theory to orthogonal polynomials, combinatorics, and number theory-a volume in honor of Lance Littlejohn’s 70th birthday, pp. 173-196, Oper. Theory Adv. Appl. 285, Birkhäuser/Springer, Cham, 2021 Zbl 1493.33007 MR 4367467 · Zbl 1493.33007 · doi:10.1007/978-3-030-75425-9_10 |
| [18] | D. Gómez-Ullate, N. Kamran, and R. Milson, An extended class of orthogonal polynomi-als defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359 (2009), no. 1, 352-367 Zbl 1183.34033 MR 2542180 · Zbl 1183.34033 · doi:10.1016/j.jmaa.2009.05.052 |
| [19] | D. Gómez-Ullate, N. Kamran, and R. Milson, Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43 (2010), no. 43, article no. 434016 Zbl 1202.81036 MR 2727790 · Zbl 1202.81036 · doi:10.1088/1751-8113/43/43/434016 |
| [20] | D. Gómez-Ullate, N. Kamran, and R. Milson, An extension of Bochner’s problem: excep-tional invariant subspaces. J. Approx. Theory 162 (2010), no. 5, 987-1006 Zbl 1214.34079 MR 2610341 · Zbl 1214.34079 · doi:10.1016/j.jat.2009.11.002 |
| [21] | D. Gómez-Ullate, N. Kamran, and R. Milson, Two-step Darboux transformations and exceptional Laguerre polynomials. J. Math. Anal. Appl. 387 (2012), no. 1, 410-418 Zbl 1232.33013 MR 2845760 · Zbl 1232.33013 · doi:10.1016/j.jmaa.2011.09.014 |
| [22] | D. Gómez-Ullate, N. Kamran, and R. Milson, A conjecture on exceptional orthogonal polynomials. Found. Comput. Math. 13 (2013), no. 4, 615-666 Zbl 1282.33019 MR 3085680 · Zbl 1282.33019 · doi:10.1007/s10208-012-9128-6 |
| [23] | Y. Grandati, Multistep DBT and regular rational extensions of the isotonic oscillator. Ann. Physics 327 (2012), no. 10, 2411-2431 Zbl 1255.81169 MR 2970643 · Zbl 1255.81169 · doi:10.1016/j.aop.2012.07.004 |
| [24] | J. S. Kelly, C. Liaw, and J. Osborn, Moment representations of exceptional X 1 orthogonal polynomials. J. Math. Anal. Appl. 455 (2017), no. 2, 1848-1869 Zbl 1370.33010 MR 3671258 · Zbl 1370.33010 · doi:10.1016/j.jmaa.2017.05.037 |
| [25] | C. Liaw, L. Littlejohn, and J. S. Kelly, Spectral analysis for the exceptional X m -Jacobi equation. Electron. J. Differential Equations (2015), article no. 194 Zbl 1320.33017 MR 3386555 · Zbl 1320.33017 |
| [26] | C. Liaw, L. L. Littlejohn, R. Milson, and J. Stewart, The spectral analysis of three families of exceptional Laguerre polynomials. J. Approx. Theory 202 (2016), 5-41 Zbl 1333.33018 MR 3440071 · Zbl 1333.33018 · doi:10.1016/j.jat.2015.11.001 |
| [27] | C. Liaw, L. L. Littlejohn, J. Stewart, and Q. Wicks, A spectral study of the second-order exceptional X 1 -Jacobi differential expression and a related non-classical Jacobi differen-tial expression. J. Math. Anal. Appl. 422 (2015), no. 1, 212-239 Zbl 1303.33012 MR 3263455 · Zbl 1303.33012 · doi:10.1016/j.jmaa.2014.08.016 |
| [28] | M. Marletta and A. Zettl, The Friedrichs extension of singular differential operators. J. Differential Equations 160 (2000), no. 2, 404-421 Zbl 0954.47012 MR 1736997 · Zbl 0954.47012 · doi:10.1006/jdeq.1999.3685 |
| [29] | I. Marquette and C. Quesne, New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials. J. Math. Phys. 54 (2013), no. 4, article no. 042102 Zbl 1282.70036 MR 3088222 · Zbl 1282.70036 · doi:10.1063/1.4798807 |
| [30] | B. Midya and B. Roy, Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonians. Phys. Lett. A 373 (2009), no. 45, 4117-4122 Zbl 1234.81072 MR 2552404 · Zbl 1234.81072 · doi:10.1016/j.physleta.2009.09.030 |
| [31] | M. A. Naimark, Linear differential operators. Part II: Linear differential operators in Hilbert space. Frederick Ungar, New York, 1968 Zbl 0227.34020 MR 0262880 · Zbl 0227.34020 |
| [32] | M. Noumi, Painlevé equations through symmetry. Transl. Math. Monogr. 223, American Mathematical Society, Providence, RI, 2004 Zbl 1077.34003 MR 2044201 · doi:10.1090/mmono/223 |
| [33] | S. Odake and R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials. Phys. Lett. B 679 (2009), no. 4, 414-417 MR 2569488 · doi:10.1016/j.physletb.2009.08.004 |
| [34] | S. Odake and R. Sasaki, Another set of infinitely many exceptional .X ‘/ Laguerre poly-nomials. Phys. Lett. B 684 (2010), no. 2-3, 173-176 MR 2588057 · doi:10.1016/j.physletb.2009.12.062 |
| [35] | G. Post and A. Turbiner, Classification of linear differential operators with an invariant subspace of monomials. Russian J. Math. Phys. 3 (1995), no. 1, 113-122 Zbl 0913.34069 MR 1337491 · Zbl 0913.34069 |
| [36] | C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and super-symmetry. J. Phys. A 41 (2008), no. 39, article no. 392001 Zbl 1192.81174 MR 2439200 · Zbl 1192.81174 · doi:10.1088/1751-8113/41/39/392001 |
| [37] | R. Sasaki, S. Tsujimoto, and A. Zhedanov, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations. J. Phys. A 43 (2010), no. 31, article no. 315204 Zbl 1200.33013 MR 2665675 · Zbl 1200.33013 · doi:10.1088/1751-8113/43/31/315204 |
| [38] | B. Simon, Spectral analysis of rank one perturbations and applications. In Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993), pp. 109-149, CRM Proc. Lecture Notes 8, American Mathematical Society, Providence, RI, 1995 Zbl 0824.47019 MR 1332038 · Zbl 0824.47019 · doi:10.1090/crmp/008/04 |
| [39] | B. Simon, Trace ideals and their applications. Second edn., Math. Surveys Monogr. 120, American Mathematical Society, Providence, RI, 2005 Zbl 1074.47001 MR 2154153 · doi:10.1090/surv/120 |
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