85 research outputs found
Semiclassical Trace Formulae and Eigenvalue Statistics in Quantum Chaos
A detailed discussion of semiclassical trace formulae is presented and it is
demonstrated how a regularized trace formula can be derived while dealing only
with finite and convergent expressions. Furthermore, several applications of
trace formula techniques to quantum chaos are reviewed. Then local spectral
statistics, measuring correlations among finitely many eigenvalues, are
reviewed and a detailed semiclassical analysis of the number variance is given.
Thereafter the transition to global spectral statistics, taking correlations
among infinitely many quantum energies into account, is discussed. It is
emphasized that the resulting limit distributions depend on the way one passes
to the global scale. A conjecture on the distribution of the fluctuations of
the spectral staircase is explained in this general context and evidence
supporting the conjecture is discussed.Comment: 48 pages, LaTeX, uses amssym
A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators
We study the semiclassical time evolution of observables given by matrix
valued pseudodifferential operators and construct a decomposition of the
Hilbert space L^2(\rz^d)\otimes\kz^n into a finite number of almost invariant
subspaces. For a certain class of observables, that is preserved by the time
evolution, we prove an Egorov theorem. We then associate with each almost
invariant subspace of L^2(\rz^d)\otimes\kz^n a classical system on a product
phase space \TRd\times\cO, where \cO is a compact symplectic manifold on
which the classical counterpart of the matrix degrees of freedom is
represented. For the projections of eigenvectors of the quantum Hamiltonian to
the almost invariant subspaces we finally prove quantum ergodicity to hold, if
the associated classical systems are ergodic
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