2,420 research outputs found
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
Optimal control of multiscale systems using reduced-order models
We study optimal control of diffusions with slow and fast variables and
address a question raised by practitioners: is it possible to first eliminate
the fast variables before solving the optimal control problem and then use the
optimal control computed from the reduced-order model to control the original,
high-dimensional system? The strategy "first reduce, then optimize"--rather
than "first optimize, then reduce"--is motivated by the fact that solving
optimal control problems for high-dimensional multiscale systems is numerically
challenging and often computationally prohibitive. We state sufficient and
necessary conditions, under which the "first reduce, then control" strategy can
be employed and discuss when it should be avoided. We further give numerical
examples that illustrate the "first reduce, then optmize" approach and discuss
possible pitfalls
Norm estimates of complex symmetric operators applied to quantum systems
This paper communicates recent results in theory of complex symmetric
operators and shows, through two non-trivial examples, their potential
usefulness in the study of Schr\"odinger operators. In particular, we propose a
formula for computing the norm of a compact complex symmetric operator. This
observation is applied to two concrete problems related to quantum mechanical
systems. First, we give sharp estimates on the exponential decay of the
resolvent and the single-particle density matrix for Schr\"odinger operators
with spectral gaps. Second, we provide new ways of evaluating the resolvent
norm for Schr\"odinger operators appearing in the complex scaling theory of
resonances
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