Hermitian symplectic geometry and extension theory

By M. Harmer. 2000

160 citations recorded by the source corpus.

Here we give brief account of Hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore, we find an explicit parametrization of the Lagrange Grassmannian in terms of the unitary matrices . This allows us to explicitly describe all self-adjoint boundary conditions for the Schrodinger operator on the graph in terms of a unitary matrix. We show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix.

Subfields: Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0b7ece70313781588ce111ed49b8bcb578038811

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Peter Kuchment
2. Pavel Kurasov
3. F. Stenberg

In Computational Theory and Mathematics

1. Peter Kuchment
2. Pavel Kurasov
3. F. Stenberg

In Mathematical Physics

1. Peter Kuchment
2. Pavel Kurasov
3. F. Stenberg
4. Konstantin Pankrashkin
5. S. Naboko
6. Nicolas Popoff

Citations in the corpus

• On the inverse scattering problem on branching graphs (2002, background): “It appears that the language of Lagrangian planes of the corresponding symplectic boundary form makes all calculations explicit [40, 30 ].”
• Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs (2004, background): “We will not discuss any details of these origins of quantum graphs, referring the reader instead to [10, 12, 19, 20, 23, 25, 26, 27, 28, 29, 32, 33, 34, 37] for further information, recent surveys, and literature.”
• Quantum graphs: I. Some basic structures (2004, background): “All such conditions were completely described in [43] (see also the earlier paper [33] for some special cases and [41, 48] for an alternative consideration that represents the boundary conditions in terms of vertex scattering matrices).”
• Graph models for waves in thin structures (2002, methodology): “This is done either by using the standard theory of extensions of symmetric operators, or by a more recent version of it that amounts to finding Lagrangian planes with respect to the symplectic boundary form that corresponds to the maximal operator (see for instance [5, 100, 126, 140, 144] for the a”