JAMP  Vol.2 No.2 , January 2014
Schrödinger Operators on Graphs and Branched Manifolds
Abstract: We consider the Schrodinger operators on graphs with a finite or countable number of edges and Schr?dinger operators on branched manifolds of variable dimension. In particular, a description of self-adjoint extensions of symmetric Schr?dinger operator, initially defined on a smooth function, whose support does not contain the branch points of the graph and branch points of the manifold. These results are obtained for graphs with a single vertex, graphs with multiple vertices and graphs with a single vertex and countable set of rays.
Cite this paper: Elsheikh, M. (2014) Schrödinger Operators on Graphs and Branched Manifolds. Journal of Applied Mathematics and Physics, 2, 1-9. doi: 10.4236/jamp.2014.22001.

[1]   Y. V. Pokorny, O. M. Penkin, V. L. Pryadier, A. V. Borovskikh, K. P. Lazarev and S. A. Shabrov, “Differential Equations on Geometric Graphs,” M. FIZMATLIT, 2004.

[2]   V. L. Chernyshev and A. I. Shafarevich, “Semiclassical Spectrum of the Schrodinger Operator on a Geometric Graph,” Mathematical Notes, Vol. 82, No. 3-4, 2007, pp. 542-554.

[3]   O. M. Penkin and Y. V. Pokornyi, “On Some Qualitative Properties of Equations for the One-Dimensional CW-complex,” Mathematical Notes, Vol.59, No. 5, 1996, pp. 562-565.

[4]   A. A. Tolchennikov, V. L. Chernyshev and A. I. Shafarevich, “Asymptotic Properties and Classical Dynamical Systems in Quantum Problems on Singular Spaces,” Nelineinaya Dinamika, Vol. 6, No. 3, 2010, pp. 623-638.

[5]   G. G. Amosov and V. Z. Sakbaev, “On Self-Adjoint Extensions of Schrodinger Operators Degenerating on a Pair of Half-Lines and the Corresponding Markovian Cocycles,” Mathematical Notes, Vol. 76, No. 3, 2004, pp. 315-322.

[6]   V. Z. Sakbaev and O. G. Smolyanov, “Dynamics of a Quantum Particle with Discontinuous Position-Dependent Mass,” Doklady Mathematics, Vol. 82, No. 1, 2010, pp. 630-633.

[7]   V. Z. Sakbaev and O. G. Smolyanov, “Diffusion and Quantum Dynamics on Graphs,” Doklady Mathematics, Vol. 88, No. 1, 2013, pp. 404-408.

[8]   M. Gadella, S. Kuru and J. Negro, “Self-Adjoint Hamiltonians with a Mass Jump: General Matching Conditions,” Physics Letters A, 2007, Vol. 362, No. 4, pp. 265-268.

[9]   M. Reed and B. Simon, “Methods of Modern Mathematical Physics ‘1. Functional Analysis’,” Academic Press, New York London, 1972.

[10]   T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin, Heidelberg, New York, 1966.

[11]   N. Dunford and J. T. Schwartz, “Linear Operators ‘Part I: General Theory’,” Interscience Publishers, New York, London, 1958.

[12]   G. N. Jakovlev, “Traces of Functions in the Space on Piecewise Smooth Surfaces,” Mathematics of the USSR-Sbornik, Vol. 3, No. 4, 1967, pp. 481-497.