turkmath.org

According to the postulates of Quantum Mechanics, the dynamics of quantum systems are generated by a self-adjoint operator, namely Hamiltonian operator associated with the energy of the system. Dirac delta potentials are known as one class of singular interactions, which have many applications in various areas of physics. There are different mathematically rigorous approaches for the description of such systems by some self-adjoint Hamiltonian operator in $L^2(\mathbb{R}^n)$. In this talk, I would like to introduce the subject in a rather elementary way and briefly discuss such interactions in one dimension heuristically and from the Von Neumann's self-adjoint extension point of view. Then, I shall extend the same model onto the two and three dimensional Cartan-Hadamard manifolds with Ricci curvature bounded below by describing the system in terms of "limit" of resolvent of the regularized version of the initial singular Hamiltonian. This will be accomplished by the heat kernel defined on manifolds and its Li-Yau type of estimates.