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Scattering from a Ring Graph - A~Simple Model for the Study of Resonances

100%
Waltner D., Smilansky U.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1087-1090
EN
Scattering from the very simple ring graph is shown to display several basic features which underlie the complex (chaotic) phenomena observed in scattering from more complex graphs. In particular we demonstrate the appearance of arbitrarily narrow resonances - the "topological resonances" which are directly linked to the existence of cycles. We use the ring graph to study the response of such resonances to perturbations induced by a time-dependent random noise.
2
Content available remote

The Probability Distribution of Spectral Moments for the Gaussian β-Ensembles

81%
Maciążek T., Joyner C., Smilansky U.
Acta Physica Polonica A
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2015
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vol. 128
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issue 6
983-989
EN
We derive the joint probability distribution of the first two spectral moments for the Gaussian β-ensemble random matrix ensembles in N dimensions for any N. This is achieved by making use of two complementary invariants of the domain in ℝ^N where the spectral moments are defined. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when attempting to include higher spectral moments.
3
Content available remote

Note on the Role of Symmetry in Scattering from Isospectral Graphs and Drums

81%
Band R., Sawicki A., Smilansky U.
Acta Physica Polonica A
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2011
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vol. 120
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issue 6A
A-149-A-153
EN
We discuss scattering from pairs of isospectral quantum graphs constructed using the method described in the papers of Parzanchevski, Band and Ben-Shach. It was shown in the paper of Band et al. that scattering matrices of such graphs have the same spectrum and polar structure, provided that infinite leads are attached in a way which preserves the symmetry of isospectral construction. In the current paper we compare this result with the conjecture put forward by Okada et al. that the pole distribution of scattering matrices in the exterior of isospectral domains in ℝ^{2} is different.
4
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Edge Switching Transformations of Quantum Graphs

81%
Aizenman M., Schanz H., Smilansky U., Warzel S.
Acta Physica Polonica A
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2017
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vol. 132
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issue 6
1699-1703
EN
Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by {Eₙ}^{∞}ₙ₌₁ and {Ẽₙ}^{∞}ₙ₌₁ correspondingly, are level-2 interlaced, so that Eₙ-₂ ≤ Ẽₙ ≤ Eₙ₊₂. The proofs are guided by considerations of the quantum graphs' discrete analogs.
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