Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

Pankrashkin, Konstantin

doi:10.1016/j.jmaa.2016.12.039

Cited by 33 publications

(17 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A similar proof as that of point ii) above yields that the second Rayleigh quotient of L Γ is greater than π 2 . This is in the spirit of [31] and provides a more direct proof of the result [28] that L Γ has at most one eigenvalue under its essential spectrum.…”

Section: Broken Guides Of Finite Lengthmentioning

confidence: 66%

“…In fact, the Dirichlet Laplacian on Γ or Γ has exactly one eigenvalue under the threshold of the essential spectrum [22,31] and Theorem 1.2 generalizes to the layers Λ and Λ if we replace the guide Γ by Γ and Γ , respectively, see Section 5 and Appendix B.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Here, once more, the Dirichlet Laplacian on X has exactly one eigenvalue under the essential spectrum [31,Prop. 13] and Theorem 1.2 generalizes to the three-dimensional cross Y if we replace the guide Γ by the two-dimensional cross X , see Section 5.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Dirichlet Spectrum of the Fichera Layer

Dauge

Lafranche

Ourmières-Bonafos

2018

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain can be viewed as an octant from which another "parallel" octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer.We show that the essential spectrum of the Laplacian on such a domain is a half-line and we characterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm-Liouville operator. By finite element computations taking singularities into account, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.2010 Mathematics Subject Classification. 35J05, 35P15, 35Q40, 81Q10, 65M60.

show abstract

Section: Broken Guides Of Finite Lengthmentioning

confidence: 66%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Dirichlet Spectrum of the Fichera Layer

Dauge

Lafranche

Ourmières-Bonafos

2018

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

show abstract

“…in the papers by Grieser [26], Molchanov and Vainberg [48], and in the monograph by Post [59]. Let us recall some basic notions of the theory, mostly following the short presentation given in the paper [55] by Pankrashkin.…”

Section: Discussionmentioning

confidence: 99%

Effective operators for Robin eigenvalues in domains with corners

Khalile,

Ourmières-Bonafos,

Pankrashkin

2018

Preprint

Self Cite

View full text Add to dashboard Cite

We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrödinger-type operator on the boundary of the domain with boundary conditions at the corners.

show abstract

“…Especially inequalities for Laplacian eigenvalues of particular polygonal domains like triangles and rhombi have attracted interest recently due to applications to the hot spots conjecture and other problems, see, e.g., [28,29]. For further literature on mixed elliptic boundary value problems (sometimes also called Zaremba problems) we refer the reader to [1,5,13,20,26,27]. For elliptic boundary value problems on polygonal and polyhedral domains see the monographs [6,12,18].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

Lotoreichik

Rohleder

2017

Journal of Differential Equations

View full text Add to dashboard Cite

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.

show abstract

Eigenvalue inequalities and absence of threshold resonances for waveguide junctions

Cited by 33 publications

References 26 publications

Dirichlet Spectrum of the Fichera Layer

Dirichlet Spectrum of the Fichera Layer

Effective operators for Robin eigenvalues in domains with corners

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

Contact Info

Product

Resources

About