Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Finite speed of propagation and local boundary conditions for wave equations with point interactions

Authors: Pavel Kurasov and Andrea Posilicano
Journal: Proc. Amer. Math. Soc. 133 (2005), 3071-3078
MSC (2000): Primary 47B25, 81Q10; Secondary 47A55, 47N50, 81Q15
Published electronically: April 25, 2005
MathSciNet review: 2159787
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the boundary conditions entering in the definition of the self-adjoint operator $\Delta^{A,B}$ describing the Laplacian plus a finite number of point interactions are local if and only if the corresponding wave equation $\ddot\phi=\Delta^{A,B}\phi$ has finite speed of propagation.

References [Enhancements On Off] (What's this?)

  • 1. Sergio Albeverio, Friedrich Gesztesy, Raphael Høegh-Krohn, and Helge Holden, Solvable models in quantum mechanics, Texts and Monographs in Physics, Springer-Verlag, New York, 1988. MR 926273
  • 2. S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Mathematical Society Lecture Note Series, vol. 271, Cambridge University Press, Cambridge, 2000. Solvable Schrödinger type operators. MR 1752110
  • 3. Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
  • 4. John W. Hooker and Carl E. Langenhop, On regular systems of linear differential equations with constant coefficients, Rocky Mountain J. Math. 12 (1982), no. 4, 591–614. MR 683854, 10.1216/RMJ-1982-12-4-591
  • 5. Diego Noja and Andrea Posilicano, The wave equation with one point interaction and the (linearized) classical electrodynamics of a point particle, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 3, 351–377 (English, with English and French summaries). MR 1622543
  • 6. Diego Noja and Andrea Posilicano, Delta interactions and electrodynamics of point particles, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999) CMS Conf. Proc., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 505–516. MR 1803443
  • 7. K. V. Pankrashkin, Locality of quadratic forms for point perturbations of Schrödinger operators, Mat. Zametki 70 (2001), no. 3, 425–433 (Russian, with Russian summary); English transl., Math. Notes 70 (2001), no. 3-4, 384–391. MR 1882252, 10.1023/A:1012352029965
  • 8. B. S. Pavlov, Boundary conditions on thin manifolds and the semiboundedness of the three-body Schrödinger operator with point potential, Mat. Sb. (N.S.) 136(178) (1988), no. 2, 163–177, 301 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 161–175. MR 954922
  • 9. Yu. G. Shondin, Semibounded local Hamiltonians for perturbations of the Laplacian on curves with angular points in 𝐑⁴, Teoret. Mat. Fiz. 106 (1996), no. 2, 179–199 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 106 (1996), no. 2, 151–166. MR 1402004, 10.1007/BF02071070

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B25, 81Q10, 47A55, 47N50, 81Q15

Retrieve articles in all journals with MSC (2000): 47B25, 81Q10, 47A55, 47N50, 81Q15

Additional Information

Pavel Kurasov
Affiliation: Department of Mathematics, Lund Institute of Technology, P.O. Box 118, 22100 Lund, Sweden

Andrea Posilicano
Affiliation: Dipartimento di Scienze, Università dell’Insubria, I-22100 Como, Italy

Keywords: Point interactions, singular perturbations, locality, wave equation
Received by editor(s): June 4, 2004
Published electronically: April 25, 2005
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2005 American Mathematical Society