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We give a short report on work done in recent years on solvable models for quantum mechanical crystals (crystals with point interactions, thus three dimensional extensions of Kronig Penney's model). We discuss the mathematical definition of the Hamiltonian and its spectral properties in the case of perfect crystals, as well as in the case of crystals with deterministic or randomly distributed point...
We give a survey of some recent mathematical work on resonances, in particular on perturbation series, low energy expansions and on resonances for point interactions. Expansions of the kernels of $$e^{ - it\sqrt {H_ + } } $$ and e−itH in terms of resonances are also given (where H+ is the positive part of the Hamiltonian).
We give a survey of recent results concerning Schrödinger operators describing the motion of a quantum mechanical particle in ℝ3 or ℝ1 under the influence of a potential concentrated at N centers, N≦∞. We dedicate particular attention to the case N=∞, with centers forming a periodic lattice (model of a crystal) or with centers randomly distributed with random strengths (models of disordered...
In several contributions to this conference stochastic processes (especially those of diffusion type) have plaid a role. We would like to give a short exposition of some work done recently on the construction of such processes from a point of view which unifies the finite dimensional case (elliptic operators, heat equation, potential theory, quantum mechanics), the infinite dimensional case (variational...
We discuss diffusion processes on Riemannian manifolds, for which a Newton law holds (in the stochastic sense). We emphasize the existence of a general mechanism for the formation of impenetrable barriers for these processes, corresponding to the nodes of the density of their distribution. We discuss some applications to natural phenomena like the formation of planetary systems, the morphology of...
We give a review of our work concerning the mathematical definition of Feynman path integrals as particular cases of oscillatory integrals on infinite dimensional spaces, to which the finite dimensional theory (in particular the stationary phase method) is extended. Applications are given to quantum mechanics and quantum field theory.
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