Abstract
Boundary value problems for a second-order ordinary differential equation with a nonsmooth potential are considered. An asymptotic representation for an operator semigroup is established based on a previously derived theorem on the eigenvalue asymptotics for the Hill operator. This semigroup is used to describe weak solutions of a mixed problem for a parabolic equation.
Similar content being viewed by others
REFERENCES
Chernyatin, V.A., Obosnovanie metoda Fur’e v smeshannoi zadache dlya uravnenii v chastnykh proizvodnykh (Justification of the Fourier Method in a Mixed Problem for Partial Differential Equations), Moscow: Izd. Mosk. Univ., 1991.
Burlutskaya, M.Sh. and Khromov, A.P., Classical solution of mixed problems under minimum requirements for initial data by the Fourier method, Izv. Sarat. Univ. Nov. Ser. Mat. Mekh., 2014, vol. 14, no. 2, pp. 171–198.
Kornev, V.V. and Khromov, A.P., A mixed problem for an inhomogeneous wave equation with a summable potential, Comput. Math. Math. Phys., 2017, vol. 57, no. 10, pp. 1666–1681.
Baskakov, A.G. and Uskova, N.B., A generalized Fourier method for the system of first-order differential equations with an involution and a group of operators, Differ. Equations, 2018, vol. 54, no. 2, pp. 277–281.
Baskakov, A.G., Krishtal, I.A., and Uskova, N.B., Linear differential operator with an involution as a generator of an operator group, Oper. Matrices, 2018, vol. 12, no. 3, pp. 723–756.
Baskakov, A.G. and Uskova, N.B., Fourier method for first-order differential equations with an involution and groups of operators, Ufa Math. J., 2018, vol. 10, no. 3, pp. 11–34.
Baskakov, A.G. and Polyakov, D.M., The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential, Sb. Math., 2017, vol. 208, no. 1, pp. 1–43.
Polyakov, D.M., A one-dimensional Schrödinger operator with square-integrable potential, Sib. Math. J., 2018, vol. 59, no. 3, pp. 470–485.
Hille, E. and Phillips, R.S., Functional Analysis and Semigroups, Colloq. Publ., vol. 31, Providence: Am. Math. Soc., 1957.
Engel, K.-J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equation, New York: Springer, 2000.
Kirillov, A.A., Elementy teorii predstavlenii (Elements of Representation Theory), Moscow: Nauka, 1978.
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00732 for A.G. Baskakov and project no. 18-31-00205 for D.M. Polyakov.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Baskakov, A.G., Polyakov, D.M. Fourier Method for a Mixed Problem with the Hill Operator. Diff Equat 56, 679–684 (2020). https://doi.org/10.1134/S0012266120060014
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266120060014