Abstract
Solutions of the wave equation on a two-dimensional infinite cone are studied. The Laplacian is determined using the theory of extensions and is given by the boundary condition at a conical point. If the boundary condition corresponds to the Friedrichs extension (i.e., the domain of definition of the Laplacian consists of functions bounded in a neighborhood of a conical point), then the solution of the problem is known and is expressed in terms of Bessel functions. We obtain a solution to the Cauchy problem in the case of general boundary conditions using the spectral expansion of the Laplace operator.
DOI 10.1134/S1061920822040173
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Luc. Hillairet, “Spectral Theory of Translation Surfaces: A Short Introduction”, Actes de Séminaire de Théorie Spectrale et Géometrie, 28 (2009–2010), 51–62.
F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, vol. 2, Plenum Press, New York–London, 1980.
J. Wunsch, “Diffractive Propagation on Conic Manifolds”, Séminaire Laurent Schwartz – ÉDP et Applications, 2015–2016, Exp. No. IX, (2017), 15 pp.
T. Ratiu, T. A. Filatova, A. I. Shafarevich, “Noncompact Lagrangian Manifolds Corresponding to the Spectral Series of the Schrödinger Operator With Delta-Potential on a Surface of Revolution”, Dokl. Math., 86 (2012), 694–696.
T. S. Ratiu, A. A. Suleimanova, A. I. Shafarevich, “Spectral Series of the Schrodinger Operator with Delta-Potential on a Three-Dimensional Spherically Symmetric Manifold”, Russ. J. Math. Phys., 20:3 (2013), 326–335.
A. I. Shafarevich, O. A. Shchegortsova, “Semiclassical Asymptotics of the Solution to the Cauchy Problem for the Schrödinger Equation with a Delta Potential Localized on a Codimension 1 Surface”, Proceedings of the Steklov Institute of Mathematics, 310:1 (2020), 304–313.
A. I. Shafarevich, “Lagrangian tori and quantization conditions corresponding to spectral series of the Laplace operator on a surface of revolution with conical points”, Proc. Steklov Inst. Math., 307 (2019), 294–302.
M. V. Fedoryuk, “Integral Transforms”, In Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya, 13, VINITI, Moscow, 1986, 211–253.
E. C. Titchmarsh, An Introduction to the Theory of Fourier Integrals, Oxford University Press, 1948.
J. Ponce de Leon, “Revisiting the Orthogonality of Bessel Functions of the First Kind on an Infinite Interval”, European Journal of Physics, 36:1 (2015), 015016.
M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1970.
Funding
This work was supported financially by the Russian Science Foundation (grant no. 22-11-00272).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vlasov, A.A., Shafarevich, A.I. Solution of the Cauchy Problem for the Wave Equation on a Cone with a Non-Friedrichs Laplacian. Russ. J. Math. Phys. 29, 588–594 (2022). https://doi.org/10.1134/S1061920822040173
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920822040173