On the class of potentials with trivial monodromy
DOI:
https://doi.org/10.26577/JMMCS-2018-3-559Keywords:
spectral instability, spectrum localization, Sturm–Liouville equation, trivial monodromyAbstract
We consider the problem of describing the class TM(Ω;A) potentials meromorphic in a simply
connected domain Ω with a set of poles A satisfying the trivial monodromy condition: any solution
of the corresponding Sturm–Liouville equation for all values of the spectral parameter has no
branch points at any point in A.We have shown that in the case of a finite A the linear (with respect
to the usual addition) space TM(Ω;A) has finite dimension modulo the subspace TM0(Ω;A) of
functions holomorphic in Ω and having at points A, zeros of a given multiplicity (its own for each
point). Thus, for a finite A, a complete description of TM(Ω; A;M) is obtained in terms of any
finite set of functions – solutions of an interpolation problem with multiple nodes at points of the
set A. The result obtained summarizes the well-known results on classes of potentials with trivial
monodromy on the C, decreasing at infinity (J.J. Duistermaat, F.A. Gr¨unbaum) or growing not
faster than the second (A. Oblomkov) or the sixth (J.Gibbons, A.P. Veselov) of degree. In the case
when the set A is countable and has a unique limit point, a sufficiently wide class of functions that
satisfy the condition of trivial monodromy is constructed.
References
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[18] Ishkin, Kh.K. “On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential.”Ufa mathematical journal 5, no. 1 (2013): 36-55. https://doi.org/10.13108/2013-5-1-36
[19] Ishkin, Kh.K., and Rezbayev, A.V. “On the conditions for the existence of triangular transformation operator for a binomial differential equations.” International Conference “Functional Analysis in Interdisciplinary Applications” (FAIA 2017), AIP Conference Proceedings (Astana, Okt. 02-05, 2017), 1880, eds. T. Kal’menov, M. Sadybekov, American Institute of Physics, Melville, NY, 2017, 060006. https://doi.org/10.1134/S1064562418020175
[20] Ishkin, Kh.K. “Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators.” Doklady Mathematics 97, no. 2, (2018): 170-173. https://doi.org/10.1134/S1064562418020175
[21] Ishkin, Kh.K. “On a Trivial Monodromy Criterion for the Sturm–Liouville equation.” Math. Notes 94, no. 4 (2013): 508-523. https://doi.org/10.1134/S0001434613090216
[22] Ishkin, Kh.K. “A localization criterion for the spectrum of the Sturm–Liouville operator on a curve.” St. Petersburg Math. J. 28, no. 1, (2017): 37-63. https://doi.org/10.1090/spmj/1438
[23] Duistermaat, J.J., and Gr¨unbaum, F.A. “Differential equations in the spectral parameter.” Commun. Math. Phys. 103 (1986): 177-240. https://projecteuclid.org/euclid.cmp/1104114705
[24] Darboux,G. “Sur une proposition relative aux ´equations lin´eaires.” C. R. Acad. Sci. Paris 94 (1882):1456-1459.
https://arxiv.org:physics/9908003.
[25] Oblomkov, A.A. “Monodromy-free Schr¨odinger operators with quadratically increasing potentials.”Theor Math Phys 121, no. 3 (1999): 1574-1584. https://doi.org/10.1007/BF02557204
[26] Gibbons,J., Veselov, A.P. “On the rational monodromy-free potentials with sextic growth.” J. Math. Phys. 50, no 1 (2009): 013513. https://doi.org/10.1063/1.3001604
[27] Naimark, M.A. Linear differential operators. 2nd ed., revised and augmented, Nauka, Moscow, 1969; English transl., Frederick Ungar Publ. Co., New York, 1968.
[28] Kato, Тosio. Pertubation theory for linear operators. Berlin-Heidelberg-New York: Springer-Verlag. 1966.
[29] Ishkin, Kh.K. “Necessary Conditions for the Localization of the Spectrum of the Sturm–Liouville Problem on a Curve.”Math. Notes 76, no. 1 (2005): 64-75. https://doi.org/10.1007/s11006-005-0100-5
[2] Lidskii, V.B., and Sadovnichii, V.A. “Asymptotic formulas for the zeros of a class of entire functions.” Mathematics of the USSR-Sbornik 4, no. 4 (1968): 519-527. https://doi.org/10.1070/SM1968v004n04ABEH002812
[3] Davies, E. Brian. “Eigenvalues of an elliptic system.” Math. Zeitschrift 243 (2003): 719-743.
https://doi.org/10.1007/s00209-002-0464-0
[4] Ishkin, Kh.K. “On localization of the spectrum of the problem with complex weight.” J. Math. Sci. 150, no. 6 (2008): 2488-2499. https://doi.org/10.1007/s10958-008-0147-4
[5] Ishkin, Kh.K. “On the Birkhoff–Tamarkin–Langer Conditions and a Conjecture of Davies.” Doklady Mathematics 91, no. 3 (2015): 259–262. https://doi.org/10.1134/S1064562415020040
[6] Birkhoff, George D. “On the asymptotic character of the solutions of certain linear differential operations contain a parameter.” Trans. Аmer. Math. Soc. 9 (1908): 219-231. https://doi.org/10.1090/S0002-9947-1908-1500810-1
[7] Tamarkin, J. “Some general problems of the theory of ordinary linear differential equations and expansions
of arbitrary function in the series of fundamental functions.” Mathematische Zeitschrift 27, no. 1 (1928): 1-54.
https://doi.org/10.1007/BF01171084
[8] Keldysh, M.V. “On the eigenvalues and eigenfunctions of certain classes of non-self-adjoint equations.” Dokl. Akad. Nauk SSSR 77, no. 1 (1951): 11–14.
[9] Aslanyan, Anna, and Davies, E.Brian. “Spectral instability for some Schrodinger operators.” Numer. Math. 72 (2000): 525-552. https://doi.org/10.1007/s002110000149
[10] Davies, E.Brian. “Wild spectral behaviour on anharmonic oscillators.” Bull. London Math. Soc. 32, no. 4 (2000): 432-438. https://doi.org/ 10.1112/S0024609300007050
[11] Davies, E.Brian. “Non-self-adjoint differential operators.” Bull. London Math.Soc. 34, no. 34 (2002): 513-532.
https://doi.org/10.1112/S0024609302001248
[12] Hager, M. “Instabilite spectrale semiclassique d’operateurs.” Annales Henry Poincare 7, no. 6 (2002): 1035-1064. https://doi.org/10.1007/s00023-006-0275-7
[13] Ishkin, Kh.K. “On the spectral instability of the Sturm–Liouville operator with a complex potential.” Differential equations 45, no. 4 (2009): 494-509. https://doi.org/10.1134/S001226610904003X
[14] Ishkin, Kh.K. “A localization criterion for the eigenvalues of a spectrally unstable operator.” Doklady Mathematics 80, no. 3 (2009): 829-832. https://doi.org/10.1134/S106456240906012X
[15] Davies, E.Brian. “Pseudo-spectra, the harmonic oscillator and complex resonances.” Proc. R. Soc. Lond. 455 (1999): 585-599. https://doi.org/10.1098/rspa.1999.0325
[16] Ishkin, Kh.K. “Conditions for localization of the limit spectrum of a model operator associated with the Orr– Sommerfeld equation.” Doklady Mathematics 86, no. 1 (2012): 549-552. https://doi.org/10.1134/S1064562412040357
[17] Nedelec, L. “Pertubations of non-self-adjoint Sturm–Lioville problems with applications to harmonic oscillator.” Methodes and applications of analysis 13, no. 1 (2006): 123-148. https://doi.org/10.4310/MAA.2006.v13.n1.a7
[18] Ishkin, Kh.K. “On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential.”Ufa mathematical journal 5, no. 1 (2013): 36-55. https://doi.org/10.13108/2013-5-1-36
[19] Ishkin, Kh.K., and Rezbayev, A.V. “On the conditions for the existence of triangular transformation operator for a binomial differential equations.” International Conference “Functional Analysis in Interdisciplinary Applications” (FAIA 2017), AIP Conference Proceedings (Astana, Okt. 02-05, 2017), 1880, eds. T. Kal’menov, M. Sadybekov, American Institute of Physics, Melville, NY, 2017, 060006. https://doi.org/10.1134/S1064562418020175
[20] Ishkin, Kh.K. “Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators.” Doklady Mathematics 97, no. 2, (2018): 170-173. https://doi.org/10.1134/S1064562418020175
[21] Ishkin, Kh.K. “On a Trivial Monodromy Criterion for the Sturm–Liouville equation.” Math. Notes 94, no. 4 (2013): 508-523. https://doi.org/10.1134/S0001434613090216
[22] Ishkin, Kh.K. “A localization criterion for the spectrum of the Sturm–Liouville operator on a curve.” St. Petersburg Math. J. 28, no. 1, (2017): 37-63. https://doi.org/10.1090/spmj/1438
[23] Duistermaat, J.J., and Gr¨unbaum, F.A. “Differential equations in the spectral parameter.” Commun. Math. Phys. 103 (1986): 177-240. https://projecteuclid.org/euclid.cmp/1104114705
[24] Darboux,G. “Sur une proposition relative aux ´equations lin´eaires.” C. R. Acad. Sci. Paris 94 (1882):1456-1459.
https://arxiv.org:physics/9908003.
[25] Oblomkov, A.A. “Monodromy-free Schr¨odinger operators with quadratically increasing potentials.”Theor Math Phys 121, no. 3 (1999): 1574-1584. https://doi.org/10.1007/BF02557204
[26] Gibbons,J., Veselov, A.P. “On the rational monodromy-free potentials with sextic growth.” J. Math. Phys. 50, no 1 (2009): 013513. https://doi.org/10.1063/1.3001604
[27] Naimark, M.A. Linear differential operators. 2nd ed., revised and augmented, Nauka, Moscow, 1969; English transl., Frederick Ungar Publ. Co., New York, 1968.
[28] Kato, Тosio. Pertubation theory for linear operators. Berlin-Heidelberg-New York: Springer-Verlag. 1966.
[29] Ishkin, Kh.K. “Necessary Conditions for the Localization of the Spectrum of the Sturm–Liouville Problem on a Curve.”Math. Notes 76, no. 1 (2005): 64-75. https://doi.org/10.1007/s11006-005-0100-5
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How to Cite
Ishkin, K. K., & Akhmetshina, A. D. (2018). On the class of potentials with trivial monodromy. Journal of Mathematics, Mechanics and Computer Science, 99(3), 43–52. https://doi.org/10.26577/JMMCS-2018-3-559
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