13,113 research outputs found
On the rate of quantum ergodicity in Euclidean billiards
For a large class of quantized ergodic flows the quantum ergodicity theorem
due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost
all eigenfunctions become equidistributed in the semiclassical limit. In this
work we first give a short introduction to the formulation of the quantum
ergodicity theorem for general observables in terms of pseudodifferential
operators and show that it is equivalent to the semiclassical eigenfunction
hypothesis for the Wigner function in the case of ergodic systems. Of great
importance is the rate by which the quantum mechanical expectation values of an
observable tend to their mean value. This is studied numerically for three
Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000
eigenfunctions. We find that in configuration space the rate of quantum
ergodicity is strongly influenced by localized eigenfunctions like bouncing
ball modes or scarred eigenfunctions. We give a detailed discussion and
explanation of these effects using a simple but powerful model. For the rate of
quantum ergodicity in momentum space we observe a slower decay. We also study
the suitably normalized fluctuations of the expectation values around their
mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A
version with all figures can be obtained from
http://www.physik.uni-ulm.de/theo/qc/ (File:
http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any
problems contact Arnd B\"acker (e-mail: [email protected]) or Roman
Schubert (e-mail: [email protected]
Nonharmonic analysis of boundary value problems
In this paper we develop the global symbolic calculus of pseudo-differential
operators generated by a boundary value problem for a given (not necessarily
self-adjoint or elliptic) differential operator. For this, we also establish
elements of a non-self-adjoint distribution theory and the corresponding
biorthogonal Fourier analysis. We give applications of the developed analysis
to obtain a-priori estimates for solutions of operators that are elliptic
within the constructed calculus.Comment: 54 pages, updated version, to appear in IMR
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