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The trace formula for Schrödinger operators from infinite dimensional oscillatory integrals. (English) Zbl 0866.58020

The authors initiate a new detailed study of the trace formula for Schrödinger operators, exploiting the theory of infinite dimensional oscillatory integrals by finite dimensional approximations. In particular, the explicit computation of contributions given by constant and non constant periodic orbits, for potentials which are quadratic plus a bounded nonlinear part, is provided. The heat semigroup as well as the Schrödinger group are discussed and it is shown in particular that their singular supports are contained in an explicit countable set independent of the bounded part of the potential.

MSC:

58D30 Applications of manifolds of mappings to the sciences
35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
46N50 Applications of functional analysis in quantum physics
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
35S30 Fourier integral operators applied to PDEs
35C20 Asymptotic expansions of solutions to PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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