Biomedical subjects

Uzy Smilansky

Publications and source records attributed to Uzy Smilansky.

5 recordsLinked to original sources

PubMed2006

Can one count the shape of a drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study. This sequence is expressed as a trace formula, which consists of a smooth (Weyl-like) part which depends on global geometrical parameters, and a fluctuating part, which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical content of the nodal sequence is thus explicitly revealed.

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PubMed2002

Nodal domains statistics: a criterion for quantum chaos.

We consider the distribution of the (properly normalized) numbers of nodal domains of wave functions in 2D quantum billiards. We show that these distributions distinguish clearly between systems with integrable (separable) or chaotic underlying classical dynamics, and for each case the limiting distribution is universal (system independent). Thus, a new criterion for quantum chaos is provided by the statistics of the wave functions, which complements the well-established criterion based on spectral statistics.

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PubMed2001

Spectral cross correlations of magnetic edge states.

We observe strong, nontrivial cross correlations between the edge states found in the interior and the exterior of magnetic quantum billiards. Our analysis is based on a novel definition of the edge state spectral density which is rigorous, practical, and semiclassically accessible.

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PubMed1993

A scattering approach to the quantization of billiards- The inside-outside duality.

We present some recent results on the semiclassical quantization of billiards using an approach which is based on the strong link between the billiard interior and exterior problems. That is, the spectrum of the interior problem is extracted from the scattering matrix of the exterior problem. Once this is put on a rigorous basis, the semiclassical approximation is used to derive the semiclassical zeta function and the spectral density. The duality between the inside and outside problems prevails also in the classical description and offers new insight into this quantization procedure. The relation between the present approach and the more standard quantization methods is also discussed and illustrated with some numerical results.

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PubMed1992

Chaotic spectroscopy.

The spectra of quantized chaotic billiards from the point of view of scattering theory are discussed. It is shown how the spectral and resonance density functions both fluctuate about a common mean. A semiclassical treatment explains this in terms of classical scattering trajectories and periodic orbits of the Poincare scattering map. It is shown that this formalism provides an alternative derivation and a new interpretation of Gutzwiller's periodic orbits sum for the spectral density. Moreover, it is a convenient starting point for a derivation of a Riemann-Siegel "look alike" expression for the secular equation in terms of periodic orbits of finite length.

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