On spectral asymptotic of quasi-exactly solvable quartic potential

Boris Shapiro1, Miloš Tater2
1Department of Mathematics, Stockholm University, Stockholm, Sweden
2Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, Prague, Czech Republic

Tóm tắt

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Từ khóa


Tài liệu tham khảo

Bender, C., Boettcher, S.: Quasi-exactly solvable quartic potential. J. Phys. A 31(14), L273–L277 (1998)

Beraha, S., Kahane, J., Weiss, N.J.: Limits of zeros of recursively defined families of polynomials, in “Studies in Foundations and Combinatorics”, pp. 213–232, Advances in Mathematics Supplementary Studies Vol. 1, ed. G.-C. Rota, Academic Press, New York, (1978)

Bertola, M., Bothner, T.: Zeros of large degree Vorob’ev–Yablonski polynomials via a Hankel determinant identity. Int. Math. Res. Not. IMRN 19, 9330–9399 (2015)

Borcea, J., Bøgvad, R., Shapiro, B.: On rational approximation of algebraic functions. Adv. Math. 204(2), 448–480 (2006)

Borcea, J., Bøgvad, R., Shapiro, B.: Homogenized spectral pencils for exactly solvable operators: asymptotics of polynomial eigenfunctions. Publ. RIMS 45, 525–568 (2009)

Bøgvad, R., Shapiro, B.: On motherbody measures with algebraic Cauchy transform. Enseign. Math. 62(1–2), 117–142 (2016)

Bourget, A., McMillen, T.: On the distribution and interlacing of the zeros of Stieltjes polynomials. Proc. AMS 138, 3267–3275 (2010)

Bourget, A., McMillen, T., Vargas, A.: Interlacing and nonorthogonality of spectral polynomials for the Lamé operator. Proc. AMS 137, 1699–1710 (2009)

Buckingham, R.J., Miller, P.D.: Large-degree asymptotics of rational Painlevé II functions: noncritical behaviour. Nonlinearity 27(10), 2489–2578 (2014)

Buckingham, R.J., Miller, P.D.: Large-degree asymptotics of rational Painlevé II functions: critical behaviour. Nonlinearity 28(6), 1539–1596 (2015)

Clarkson, P.A., Mansfield, E.L.: The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16, R1–R26 (2003)

Coussement, E., Coussement, J., Van Assche, W.: Asymptotic zero distribution for a class of multiple orthogonal polynomials. Trans. Am. Math. Soc. 360, 5571–5588 (2008)

Eremenko, A., Gabrielov, A.: Quasi-exactly solvable quartic: elementary integrals and asymptotics. J. Phys. A Math. Theor. 44, 312001 (2011)

Fisk, S.: Polynomials, roots, and interlacing, arXiv:math/0612833

Kametaka, Y.: On poles of the rational solution of the toda equation of Painlevé-II Type, Proc. Japan Acad. Ser A, 59(A), 358–360 (1983)

Kuijlaars, A.B.J., Van Assche, W.: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99, 167–197 (1999)

Masoero, D.: Poles of integrale tritronquée and anharmonic oscillators. A WKB approach. J. Phys. A: Math. Theor. 43(9), 5201 (2010)

Masoero, D., De Benedetti, V.: Poles of integrale tritronquée and anharmonic oscillators. Asymptotic localization from WKB analysis. Nonlinearity 23, 2501 (2010)

Mukhin, E., Tarasov, V.: On conjectures of A. Eremenko and A. Gabrielov for quasi-exactly solvable quartic. Lett. Math. Phys. 103(6), 653–663 (2013)

Nilsson, N.: Some growth and ramification properties of certain integrals on algebraic manifold. Ark. Mat. 5(1965), 463–476 (1965)

Petkovšek, M., Zakrajšek, H.: Pascal-like determinants are recursive. Adv. Appl. Math. 33, 431–450 (2004)

Shapiro, B., Tater, M.: Asymptotics of spectral polynomials. Acta Polytechnica 47(2–3), 32–35 (2007)

Shapiro, B., Tater, M.: On spectral polynomials of the Heun equation. I, JAT 162, 766–781 (2010)

Shin, K.: Eigenvalues of PT-symmetric oscillators with polynomial potentials. J. Phys. A 38(27), 6147–6166 (2005)

Strebel, K.: Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5, Springer-Verlag, Berlin, (1984), xii+184 pp

Shapiro, B., Takemura, K., Tater, M.: On spectral polynomials of the Heun equation II. Commun. Math. Phys. 311(2), 277–300 (2012)

Taneda, M.: Remarks on the Yablonskii–Vorob’ev polynomials. Nagoya Math. J. 159, 87–111 (2000)

Vorob’ev, A.: On rational solutions of the second Painlevé equations, Diff. Eqns (1) (1965), 58–59 (in Russian)

Yablonskii, A.: On rational solutions of the second Painlevé equation, Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, 30–35 (1959). ((in Russian))