The formula for the regularized trace of the Sturm-Liouville operator with a logarithmic potential
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Abstract
We have obtained a regularized trace formula for the Sturm-Liouville operator on a semi-axis with a logarithmic potential.
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Ishkin, K. K., & Valiullina, L. G. . (2019). The formula for the regularized trace of the Sturm-Liouville operator with a logarithmic potential. Tamkang Journal of Mathematics, 50(3), 269–280. https://doi.org/10.5556/j.tkjm.50.2019.3354
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References
[1] M. A. Naimark, Linear Differential Operators, 2nd ed., Nauka, Moscow (1969); English transl. of 1st ed., Parts I,II, Ungar, New York (1967, 1968).
[2] A. M. Molchanov, On conditions for discreteness of the spectrum of self-adjoint differ- ential equations of the second order, (Russian) Trudy Moskov. Mat. Obsch., 2 (1953), 169–199.
[3] L.G. Valiullina, Kh.K. Ishkin, On the localization conditions for the spectrum of a non-self-adjoint Sturm–Liouville operator with slowly growing potential, Differential equations. Spectral theory, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 141, VINITI, M., (2017), 48–60.
[4] D. Ray, On spectra of second order differential operator, Trans. of Amer. Math. Soc., 77 (1954), 320–321.
[5] M. Rid., B. Simon, Methods of modern mathematical physics, Methods of modern mathematical physics, Academic Press, New York–San Francisco–London, 2 (1975).
[6] M. E. Taylor, Pseudodifferential operators, Prinston, N. Y. (1981).
[7] K.Kh. Boimatov, Sturm–Liouville operators with matrix potentials, Math. Notes, 16, no. 6 (1974), 921–932.
[8] Kh. K. Ishkin, Asymptotic behavior of the spectrum and the regularized trace of higher- order singular differential operators, Diff. Equ., 31, (1995), no. 10, 1622–1632.
[9] R.T. Seeley, The powers As of an elliptic operator A (preprint); Russian transl., Matematika, 12 (1968), no. 1, 96–112.
[10] L. A. Dikii, The zeta function of an ordinary differential equation on a finite interval, (Russian) Izv. Akad. Nauk SSSR. Ser. Mat., 19, (1955), 187–200.
[11] S. Minakshisundaram, A. Pleijel, Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds, Canadian Journal of Mathematics, 1, no. 3 (1949), 242–256.
[12] E. K. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, Oxford (1951).
[13] M.V. Fedoryuk, Asymptotic Analysis. Linear Ordinary Differential Equations, Springer-Verlag, Berlin–Heidelberg, 1993.
[14] M. Rid., B. Simon, Methods of modern mathematical physics, Methods of modern mathematical physics, Academic Press, New York–San Francisco–London, 4 (1979).
[2] A. M. Molchanov, On conditions for discreteness of the spectrum of self-adjoint differ- ential equations of the second order, (Russian) Trudy Moskov. Mat. Obsch., 2 (1953), 169–199.
[3] L.G. Valiullina, Kh.K. Ishkin, On the localization conditions for the spectrum of a non-self-adjoint Sturm–Liouville operator with slowly growing potential, Differential equations. Spectral theory, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 141, VINITI, M., (2017), 48–60.
[4] D. Ray, On spectra of second order differential operator, Trans. of Amer. Math. Soc., 77 (1954), 320–321.
[5] M. Rid., B. Simon, Methods of modern mathematical physics, Methods of modern mathematical physics, Academic Press, New York–San Francisco–London, 2 (1975).
[6] M. E. Taylor, Pseudodifferential operators, Prinston, N. Y. (1981).
[7] K.Kh. Boimatov, Sturm–Liouville operators with matrix potentials, Math. Notes, 16, no. 6 (1974), 921–932.
[8] Kh. K. Ishkin, Asymptotic behavior of the spectrum and the regularized trace of higher- order singular differential operators, Diff. Equ., 31, (1995), no. 10, 1622–1632.
[9] R.T. Seeley, The powers As of an elliptic operator A (preprint); Russian transl., Matematika, 12 (1968), no. 1, 96–112.
[10] L. A. Dikii, The zeta function of an ordinary differential equation on a finite interval, (Russian) Izv. Akad. Nauk SSSR. Ser. Mat., 19, (1955), 187–200.
[11] S. Minakshisundaram, A. Pleijel, Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds, Canadian Journal of Mathematics, 1, no. 3 (1949), 242–256.
[12] E. K. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, Oxford (1951).
[13] M.V. Fedoryuk, Asymptotic Analysis. Linear Ordinary Differential Equations, Springer-Verlag, Berlin–Heidelberg, 1993.
[14] M. Rid., B. Simon, Methods of modern mathematical physics, Methods of modern mathematical physics, Academic Press, New York–San Francisco–London, 4 (1979).