Fluctuations in the number of nodal domains

Fedor Nazarov; Mikhail Sodin

doi:10.1063/5.0018588

Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution

Journal of Differential Geometry ◽

10.4310/jdg/1452002877 ◽

2016 ◽

Vol 102 (1) ◽

pp. 37-66 ◽

Cited By ~ 8

Author(s):

Junehyuk Jung ◽

Steve Zelditch

Keyword(s):

Singular Points ◽

Curved Surfaces ◽

Nodal Domains ◽

Negatively Curved

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Investigation of nodal domains in a chaotic three-dimensional microwave rough billiard with the translational symmetry

Physics Letters A ◽

10.1016/j.physleta.2007.10.052 ◽

2008 ◽

Vol 372 (11) ◽

pp. 1851-1855 ◽

Cited By ~ 3

Author(s):

Nazar Savytskyy ◽

Oleg Tymoshchuk ◽

Oleh Hul ◽

Szymon Bauch ◽

Leszek Sirko

Keyword(s):

Three Dimensional ◽

Translational Symmetry ◽

Nodal Domains

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Numerical simulations for nodal domains and spectral minimal partitions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv:2008074 ◽

2008 ◽

Vol 16 (1) ◽

pp. 221-246 ◽

Cited By ~ 23

Author(s):

Virginie Bonnaillie-Noël ◽

Bernard Helffer ◽

Gregory Vial

Keyword(s):

Numerical Simulations ◽

Nodal Domains

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On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

Symmetry ◽

10.3390/sym11020185 ◽

2019 ◽

Vol 11 (2) ◽

pp. 185

Author(s):

Ram Band ◽

Sven Gnutzmann ◽

August Krueger

Keyword(s):

Existence Of Solutions ◽

Stationary Waves ◽

Star Graph ◽

Oscillation Theorem ◽

Nodal Domains ◽

Nodal Structure ◽

Star Graphs ◽

Spectral Curves ◽

Nonlinear Quantum ◽

Cubic Nonlinear Schrödinger Equation

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.

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Composites with invisible inclusions: Eigenvalues of ℝ-linear problem

European Journal of Applied Mathematics ◽

10.1017/s0956792516000152 ◽

2016 ◽

Vol 27 (5) ◽

pp. 796-806

Author(s):

V. V. MITYUSHEV

Keyword(s):

Asymptotic Formula ◽

Integral Equations ◽

Linear Problem ◽

Nodal Domains

A new eigenvalue ℝ-linear problem arisen in the theory of metamaterials and neutral inclusions is reduced to integral equations. The problem is constructively investigated for circular non-overlapping inclusions. An asymptotic formula for eigenvalues is deduced when the radii of inclusions tend to zero. The nodal domains conjecture related to univalent eigenfunctions is posed. Demonstration of the conjecture allows to justify that a set of inclusions can be made neutral by surrounding it with an appropriate coating.

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On the existence of nodal domains for elliptic differential operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015651 ◽

1983 ◽

Vol 94 (3-4) ◽

pp. 287-299 ◽

Cited By ~ 4

Author(s):

E. Müller-Pfeiffer

Keyword(s):

Differential Equation ◽

Differential Operator ◽

Quadratic Form ◽

Unbounded Domain ◽

Dirichlet Boundary ◽

Differential Operators ◽

Dirichlet Boundary Conditions ◽

Friedrichs Extension ◽

Nodal Domains ◽

Elliptic Differential Operators

SynopsisWe prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

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