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Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions. (English) Zbl 1182.30008
Publ. Res. Inst. Math. Sci. 45, No. 2, 525-568 (2009); corrigendum 48, No. 1, 229-233 (2012).
Authors’ abstract: Consider a homogenized spectral pencil of exactly solvable linear differential operators $$T_{\lambda }=\sum_{i=0}^k Q_{i}(z) \lambda ^{k-i}\frac {d^i}{dz^i}$$, where each $$Q_{i}(z)$$ is a polynomial of degree at most $$i$$ and $$\lambda$$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $$n$$ there exist exactly $$k$$ distinct values $$\lambda _{n,j}, 1\leq j\leq k$$, of the spectral parameter $$\lambda$$ such that the operator $$T_{\lambda }$$ has a polynomial eigenfunction $$p_{n,j}(z)$$ of degree $$n$$. These eigenfunctions split into $$k$$ different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits $$\Psi _{j}(z)=\lim _{n\to \infty } \frac {p^{\prime }_{n,j}(z)}{\lambda _{n,j}p_{n,j}(z)}$$ exist, are analytic and satisfy the algebraic equation $$\sum _{i=0}^k Q_{i}(z) \Psi _{j}^i(z)=0$$ almost everywhere in $$\mathbb {CP}^1$$. As a consequence we obtain a class of algebraic functions possessing a branch near $$\infty \in \mathbb {CP}^1$$ which is representable as the Cauchy transform of a compactly supported probability measure.

##### MSC:
 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 31A35 Connections of harmonic functions with differential equations in two dimensions 34E05 Asymptotic expansions of solutions to ordinary differential equations
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##### References:
 [1] T. Aoki, T. Kawai and Y. Takei, New turning points in the exact WKB analysis for higher-order ordinary differential equations, in Analyse algébrique des perturbations sin- guli‘ eres, I (Marseille-Luminy, 1991), xiii, xv, 69-84, Hermann, Paris. · Zbl 0831.34058 [2] , On the exact WKB analysis for the third order ordinary differential equations with a large parameter, Asian J. Math. 2 (1998), no. 4, 625-640. 567 · Zbl 0963.34045 [3] T. Bergkvist, On asymptotics of polynomial eigenfunctions for exactly solvable differen- tial operators, J. Approx. Theory 149 (2007), no. 2, 151-187. · Zbl 1155.34045 [4] T. Bergkvist and H. Rullg\ring ard, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002), no. 2-3, 153-171. · Zbl 1016.34083 [5] H. L. Berk, W. M. Nevins, K. V. Roberts, New Stokes lines in WKB-theory, J. Math. Phys. 23 (1982), 988-1002. · Zbl 0488.34050 [6] M. V. Berry, Stokes’ phenomenon; smoothing a Victorian discontinuity, Inst. Hautes Études Sci. Publ. Math. No. 68 (1988), 211-221 (1989). · Zbl 0701.58012 [7] M. Berry, K. Mount, Semiclassical approximation in wave mechanics, Rep. Prog. Phys. 35 (1972), 315-397. [8] J. Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry, Translated from the 1987 French original, Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete Vol. 36, Springer, Berlin, x+430 pp., 1998. [9] S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), no. 1, 730-736. · JFM 55.0260.01 [10] J. Borcea and R. Bøgvad, Piecewise harmonic subharmonic functions and positive Cauchy transforms, to appear in Pacific J. Math.; preprint arXiv:math/0506341. · Zbl 1163.31001 [11] P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert prob- lems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295-368. · Zbl 0771.35042 [12] E. Delabaere and D. T. Trinh, Spectral analysis of the complex cubic oscillator, J. Phys. A 33 (2000), no. 48, 8771-8796. · Zbl 1044.81555 [13] J. Ecalle, Les fonctions résurgentes. Publications Mathématiques d’Orsay, Vol. 81-05, 1981. [14] A. Erdélyi, Asymptotic expansions, Dover, New York, vi+108 pp., 1956. · Zbl 0070.29002 [15] M. V. Fedoryuk, Asymptotic analysis, Translated from the Russian by Andrew Rodick, Springer, Berlin, viii+363 pp., 1993. · Zbl 0782.34001 [16] J. Garnett, Analytic capacity and measure, Lecture Notes in Math., 297, Springer, Berlin, iv+138 pp., 1972. · Zbl 0253.30014 [17] L. Hörmander, The analysis of linear partial differential operators. I, Distribution theory and Fourier analysis, Reprint of the second (1990) edition. Classics in Mathematics, Springer, Berlin, 2003. [18] K. Hensel and G. Landsberg, Theorie der algebraischen Funktionen einer Variablen und ihre Anwendung auf algebraische Kurven und Abelsche Integrale, Chelsea, New York, xvi+707 pp, 1965. · Zbl 0199.09902 [19] M. Hukuhara, Sur la théorie des équations différentielles ordinaires, J. Fac. Sci. Univ. Tokyo. Sect. I 7 (1958), 483-510. · Zbl 0081.30203 [20] I. Kaplansky, An introduction to differential algebra, Second edition, Actualités Sci- entifiques et Industrielles, No. 1251, Publications de l’Institut de Mathématique de l’Université de Nancago, Hermann, Paris, 1976. [21] H. L. Krall, Certain differential equations for Tchebycheff polynomials, Duke Math. J. 4 (1938), no. 4, 705-718. · Zbl 0020.02002 [22] B. Malgrange, Équations différentielles ‘ a coefficients polynomiaux, Progr. Math., 96, Birkhäuser Boston, Boston, MA, 1vi+232 pp., 1991. · Zbl 0764.32001 [23] G. Másson and B. Shapiro, On polynomial eigenfunctions of a hypergeometric-type operator, Experiment. Math. 10 (2001), no. 4, 609-618. · Zbl 1008.33008 [24] F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1974. · Zbl 0303.41035 [25] F. Pham, Multiple turning points in exact WKB analysis (variations on a theme of Stokes), in Toward the exact WKB analysis of differential equations, linear or non- linear (Kyoto, 1998 ), 10, 71-85, Kyoto Univ. Press, Kyoto. · Zbl 1017.34091 [26] Y. Sibuya, Global theory of a second order linear differential equation with polynomial coefficients, North Holland Publ., 1975. · Zbl 0322.34006 [27] A. Turbiner, Lie-algebras and linear operators with invariant subspaces, in Lie algebras, cohomology, and new applications to quantum mechanics (Springfield, MO, 1992 ), 263- · Zbl 0809.17023
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