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M. Zworski

Publications and source records attributed to M. Zworski.

10 recordsLinked to original sources
arXiv2013

Experimental Observation of the Spectral Gap in Microwave n-Disk Systems

Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions.

cond-mat.mes-hall↗
arXiv2012

Weyl asymptotics: From closed to open systems

We present microwave experiments on the symmetry reduced 5-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to an non-integer value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.

cond-mat.mes-hall↗
arXiv2009

Probabilistic Weyl laws for quantized tori

For the Toeplitz quantization of complex-valued functions on a $2n$-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for ``false'' eigenvalues created by pseudospectral effects.

math.SP↗
arXiv2009

A mathematical formulation of the Mahaux-Weidenmüller formula for the scattering matrix

The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes referred to as the Mahaux-Weidenmüller formula. We show that a version of this formula given below gives the standard scattering matrix used in the mathematics literature. We also show that the finite rank approximation of the interaction matrix gives an approximation of the scattering matrix with errors inversely proportional to a fixed dimension-dependent power of the rank. A simple example shows that this estimate is optimal.

math-ph↗
arXiv2003

Elementary linear algebra for advanced spectral problems

We discuss the general method of Grushin problems, closely related to Shur complements, Feshbach projections and effective Hamiltonians, and describe various appearances in spectral theory, pdes, mathematical physics and numerical problems.

math.SP↗
arXiv2003

Eigenfunctions for partially rectangular billiards

In this note we further develop the idea of using a ``black box'' point of view (see our previous work) to study eigenfunctions for billiards which have rectangular components: they include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards

math.SP↗
arXiv2003

Bouncing ball modes and quantum chaos

Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics, and in physics. Despite this long tradition we are able to present a new rigorous result using only elementary calculus. In the case of the famous Bunimovich billiard table we prove that the wave functions have to spread into any neighbourhood of the wings.

math.AP↗