Complex patterns in wave functions: drums, graphs and disorder

Gnutzmann, Sven; Smilansky, Uzy

doi:10.1098/rsta.2013.0264

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…Studies of nodal domains have yielded many new and surprising insights into various branches of physics and mathematics (see e.g. [5] for a collection of relevant papers). In particular, it became apparent that the nodal sequence {ν n } ∞ n=1 stores information about the domain on which the Laplacian is defined, such as its boundaries or metric, which does not overlap with the information stored in the spectrum [6].…”

Section: Introductionmentioning

confidence: 99%

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

Juul¹,

Joyner²

2018

J. Phys. A: Math. Theor.

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The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count) [1,2]. Recently examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by K. Ammann by utilising Godsil-McKay switching [3]. Here we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts. ‡ The term 'cospectral' is used in place of 'isospectral' by some other authors. arXiv:1801.05246v1 [math-ph]

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Section: Introductionmentioning

confidence: 99%

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

Juul¹,

Joyner²

2018

J. Phys. A: Math. Theor.

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“…Topological properties of Laplacian eigenfunctions on domains and manifolds are of essential interest to mathematical physics in recent years [17,37]. Nodal patterns of eigenfunctions are a major and well developed research area in this field.…”

Section: Introductionmentioning

confidence: 99%

Topological Properties of Neumann Domains

Band

Fajman

2016

Ann. Henri Poincaré

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Abstract. A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction -those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry.

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Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

Robert

Thomann

2015

Commun. Math. Phys.

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This paper is a continuation of Poiret et al. (Ann Henri Poincaré 16:651-689, 2015), where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential V on R d . We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on R d . This construction relies on accurate estimates on the spectral function in a noncompact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all bases of Hermite functions are quantum uniquely ergodic.

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Complex patterns in wave functions: drums, graphs and disorder

Cited by 4 publications

References 12 publications

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

Topological Properties of Neumann Domains

Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

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