A spectral isoperimetric inequality for cones

Exner, Pavel; Lotoreichik, Vladimir

doi:10.1007/s11005-016-0917-8

Cited by 15 publications

(9 citation statements)

References 33 publications

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“…1.3] in the setting of δ-interactions supported on conical surfaces. However, the technique employed here is significantly different from the one used in [EL17]. In the latter paper, the spectral problem was first reformulated as a boundary integral equation by means of the Birman-Schwinger-type principle.…”

Section: Resultsmentioning

confidence: 99%

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

Khalile¹,

Lotoreichik²

2019

Preprint

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We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant K• ≥ 0 and under the constraint of fixed perimeter, the geodesic disk of constant curvature K• maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded threedimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.

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Section: Resultsmentioning

confidence: 99%

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

Khalile¹,

Lotoreichik²

2019

Preprint

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“…, N (see [1,2,3,6] and Section 2 for basic definitions). The question of optimization of the principal eigenvalue of self-adjoint Schrödinger Hamiltonians with δ-type or point interactions attracted recently considerable attention especially in a quantum mechanics context [14,17,16,18,36]. This line of research was motivated by the isoperimetric problem posed in [14].…”

Section: Statement Of Problem Motivation and Related Studiesmentioning

confidence: 99%

Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions

Albeverio¹,

Karabash²

2017

Operators and Matrices

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Abstract. Resonances of Schrödinger Hamiltonians with point interactions are considered. The main object under the study is the resonance free region under the assumption that the centers, where the point interactions are located, are known and the associated "strength" parameters are unknown and allowed to bear additional dissipative effects. To this end we consider the boundary of the resonance free region as a Pareto optimal frontier and study the corresponding optimization problem for resonances. It is shown that upper logarithmic bound on resonances can be made uniform with respect to the strength parameters. The necessary conditions on optimality are obtained in terms of first principal minors of the characteristic determinant. We demonstrate the applicability of these optimality conditions on the case of 4 equidistant centers by computing explicitly the resonances of minimal decay for all frequencies. This example shows that a resonance of minimal decay is not necessarily simple, and in some cases it is generated by an infinite family of feasible resonators.

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“…The result of Theorem 5 can be viewed as a kind of isoperimetric inequality: among the conical surfaces with smooth cross-sections of fixed length l ≤ 2π, the circular cones give the highest rate for the accumulation of discrete eigenvalues to the bottom of the essential spectrum for both A S and B S . Remark that the first eigenvalue of B S is also maximized by the circular cones, see [13]. …”

Section: Theorem 2 (Dirichlet Laplacian In a Conical Layer) There Holdsmentioning

confidence: 99%

“…The paper [4] studied general conical surfaces and an expression for the bottom of the essential spectrum of B S was obtained. The paper [13] contains first results on the discrete spectrum of the operator B S for conical surfaces S with arbitrary smooth cross-sections, and the authors showed that there is at least one eigenvalue below the essential spectrum. They also posed an open question on whether or not the discrete spectrum is always infinite.…”

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confidence: 99%

Discrete spectrum of interactions concentrated near conical surfaces

Ourmières-Bonafos

Pankrashkin

2017

Applicable Analysis

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Abstract. We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrödinger operators with attractive δ-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.

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A spectral isoperimetric inequality for cones

Cited by 15 publications

References 33 publications

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions

Discrete spectrum of interactions concentrated near conical surfaces

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