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The Topology and Geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators by Asorey, M.; Ibort, A.; Marmo, G. is a scholarly article available to read on EtoBox.

What is The Topology and Geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators about?

The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space $\mathcal{M}$ of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions $\mathcal{M}_\mathrm{ellip}$ is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian $\mathbf{Gr}$. The topology of $\mathcal{M}_\mathrm{ellip}$ is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral

Author

Asorey, M.; Ibort, A.; Marmo, G.

Published

2015

Language

EN