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- Sven Gnutzmann
- 2008

During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical… (More)

- Galya Blum, Sven Gnutzmann, Uzy Smilansky
- Physical review letters
- 2002

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… (More)

- SVEN GNUTZMANN
- 2001

We generalize the concept of the Wehrl entropy of quantum states which gives a basis–independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi–Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though… (More)

Several types of systems were put forward during the past decades to show that there exist isospectral systems which are metrically different. One important class consists of Laplace Beltrami operators for pairs of flat tori in R n with n ≥ 4. We propose that the spectral ambiguity can be resolved by comparing the nodal sequences (the numbers of nodal… (More)

- G Foltin, S Gnutzmann, U Smilansky
- 2004

In this paper we investigate the properties of nodal structures in random wave fields, and in particular we scrutinize their recently proposed connection with short-range percolation models. We propose a measure which shows the difference between monochromatic random waves, which are characterized by long-range correlations, and Gaussian fields with… (More)

- Sven Gnutzmann, Alexander Altland
- Physical review letters
- 2004

We prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations, as predicted by random-matrix theory. The stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph.

- Sven Gnutzmann, Alexander Altland
- Physical review. E, Statistical, nonlinear, and…
- 2005

We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the energy-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear -model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner-Dyson random… (More)

- Sven Gnutzmann, Panos D Karageorge, Uzy Smilansky
- Physical review letters
- 2006

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… (More)

Let M be a closed Riemannian manifold of dimension n. Let ϕ λ be an eigenfunction of the Laplace–Beltrami operator corresponding to an eigen-value λ. We show that the volume of {ϕ λ > 0} ∩ B is ≥ C|B|/λ n , where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥ C/λ n .

- Yehonatan Elon, Sven Gnutzmann, Christian Joas
- 2006

In an attempt to characterize the distribution of forms and shapes of nodal domains in wave functions, we define a geometric parameter-the ratio ρ between the area of a domain and its perimeter, measured in units of the wavelength 1/ √ E. We show that the distribution function P (ρ) can distinguish between domains in which the classical dynamics is regular… (More)