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In these three lectures we will discuss some fundamental aspects of thetheory of self-adjoint extensions of the covariant Laplace-Beltrami and Diracoperators on compact Riemannian manifolds with smooth boundary emphasizing therelation with the theory of global boundary conditions.  Self-adjoint extensions of symmetric operators, specially of theLaplace-Beltrami and Dirac operators, are fundamental in Quantum Physics asthey determine either the energy of quantum systems and/or their unitaryevolution. The well-known von Neumann's theory of self-adjoint extensions ofsymmetric operators is not always easily applicable to differential operators,while the description of extensions in terms of boundary conditions constitutesa more natural approach. Thus an effort is done in offering a description ofself-adjoint extensions in terms of global boundary conditions showing how animportant family of self-adjoint extensions for the Laplace-Beltrami and Diracoperators are easily describable in this way.  Moreover boundary conditions play in most cases an significant physical roleand give rise to important physical phenomena like the Casimir effect. Thegeometrical and topological structure of the space of global boundaryconditions determining regular self-adjoint extensions for these fundamentaldifferential operators is described. It is shown that there is a naturalhomology class dual of the Maslov class of the space.  A new feature of the theory that is succinctly presented here is the relationbetween topology change on the system and the topology of the space ofself-adjoint extensions of its Hamiltonian. Some examples will be commented andthe one-dimensional case will be thoroughly discussed.

aip conference proceedings

Three lectures on global boundary conditions and the theory of  self--adjoint extensions of the covariant Laplace--Beltrami and Dirac  operators on Riemannian manifolds with boundary