Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity

• Autorzy: Nursultanov M. , Rozenblum G.

Identyfikatory

DOI

10.7494/OpMath.2017.37.1.109

• Języki publikacji: EN

Abstrakty

EN

We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.

Słowa kluczowe

EN

Sturm-Liouville operator; singular potential; asymptotics of eigenvalues

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2017
• Tom: Vol. 37, no. 1
• Strony: 109--139
• Opis fizyczny: Bibliogr. 20 poz.

Twórcy

autor

Nursultanov M.

• Chalmers University of Technology Department of Mathematics Sweden
• The University of Gothenburg Department of Mathematics Sweden

autor

Rozenblum G.

• Chalmers University of Technology Department of Mathematics Sweden The University of Gothenburg Department of Mathematics Sweden
• The University of Gothenburg Department of Mathematics Sweden

Bibliografia

• [1] A. Alenitsyn, Asymptotic properties of the spectrum of a Sturm-Liouville operator in the case of a limit circle, Differential Equations 12 (1977) 2, 298-305; Differentsial’nye Uravneniya 12 (1976) 3, 428-437 [in Russian].
• [2] F. Atkinson, C. Fulton, Some limit circle eigenvalue problems and asymptotic formulas for eigenvalues, [in:] Ordinary and Partial Differential Equations, Lecture Notes in Math., vol. 964, Springer, Berlin, 1982, 28-55.
• [3] F. Atkinson, C. Fulton, Asymptotic formulae for eigenvalues of limit circle problems on a half line, Ann. Mat. Pura Appl. 135 (1983), 363-398.
• [4] F. Atkinson, C. Fulton, Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity. I. Proc. Roy. Soc. Edinburgh Sect. A 99 (1984) 1-2, 51-70.
• [5] F. Atkinson, C. Fulton, Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit circle singularity, II, submitted.
• [6] V. Belogrud, The asymptotic behavior of eigenvalues of non-semi-bounded differential operators [in Russian], Trudy Moskow. Orden. Lenin. Energet. Inst. Vyp. 260 (1975), 11-22.
• [7] V. Belogrud, A. Kostuchenko, The density of the spectrum of a Sturm-Liouville operator [in Russian]. Usp. Mat. Nauk. 28 (1973) 2, 227-228.
• [8] M. Berry, Hermitian boundary conditions at a Dirichlet singularity: the Marletta- -Rozenblum model, J. Phys. A 42 (2009), 165208, 13 pp.
• [9] P. Heywood, On the asymptotic distribution of eigenvalues, Proc. Lond. Math. Soc. 4 (1954), 456-470.
• [10] A.G. Kostuchenko, I.S. Sargsyan, Distribution of Eigenvalues [in Russian], Nauka, Moscow, 1979.
• [11] B.M. Levitan, I.S. Sargsyan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Translated from Russian, Translations of Mathematical Monographs, vol. 39, American Mathematical Society, Providence, R.I., 1975.
• [12] B.M. Levitan, I.S. Sargsyan, Sturm-Liouville and Dirac Operators, Translated from Russian, Mathematics and its Applications (Soviet Series), vol. 59, Kluwer, Dordrecht, 1991.
• [13] K.T. Mynbaev, M.O. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators [in Russian], Nauka, Moscow, 1988.
• [14] M.A. Naimark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert space, English translation, Frederick Ungar Publishing Co., New York, 1968.
• [15] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. 2, Academic Press, New York, London, 1975.
• [16] G. Rozenblum, Asymptotic behavior of the eigenvalues of the Schrodinger operator, Mathematics of the USSR-Sbornik 22 (1974) 3, 349 (Translated from Russian, Mat. Sb. (N.S.) 93 (1974), 347-367). Eigenvalue asymptotics for the Sturm-Liouville operator. . .
• [17] D.B. Sears, E.C. Titchmarsh, Some eigenvalue formulae, Quart. J. Math. 1 (1950), 165-175.
• [18] E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, Oxford, 1946.
• [19] E.C. Titchmarsh, Eigenfunction Expansions Associated With Second-Order Differential Equations. Part I., 2nd ed., Clarendon Press, Oxford, 1962.
• [20] E.C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford, 1939.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).