Magnetic spectral bounds on starlike plane domains

R. S. Laugesen; B. A. Siudeja

doi:10.1051/cocv/2014043

Magnetic spectral bounds on starlike plane domains

R. S. Laugesen, B. A. Siudeja

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that σⁿ _j=1 ψ(λ_jA/G) is maximal for a disk whenever ψ is concave increasing, n . 1, the domain has area A, and λj is the jth Dirichlet eigenvalue of the magnetic Laplacian iδ + B/2A(-x2, x1)². Here the flux β is constant, and the scale invariant factor G penalizes deviations from roundness, meaning G > 1 for all domains and G = 1 for disks.

Original language	English (US)
Pages (from-to)	670-689
Number of pages	20
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	21
Issue number	3
DOIs	https://doi.org/10.1051/cocv/2014043
State	Published - Jul 1 2015

Keywords

Heat trace
Isoperimetric
Partition function
Pauli operator
Spectral zeta

ASJC Scopus subject areas

Control and Systems Engineering
Control and Optimization
Computational Mathematics

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T1 - Magnetic spectral bounds on starlike plane domains

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AU - Siudeja, B. A.

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N2 - We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that σn j=1 ψ(λjA/G) is maximal for a disk whenever ψ is concave increasing, n . 1, the domain has area A, and λj is the jth Dirichlet eigenvalue of the magnetic Laplacian iδ + B/2A(-x2, x1)2. Here the flux β is constant, and the scale invariant factor G penalizes deviations from roundness, meaning G > 1 for all domains and G = 1 for disks.

AB - We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that σn j=1 ψ(λjA/G) is maximal for a disk whenever ψ is concave increasing, n . 1, the domain has area A, and λj is the jth Dirichlet eigenvalue of the magnetic Laplacian iδ + B/2A(-x2, x1)2. Here the flux β is constant, and the scale invariant factor G penalizes deviations from roundness, meaning G > 1 for all domains and G = 1 for disks.

KW - Heat trace

KW - Isoperimetric

KW - Partition function

KW - Pauli operator

KW - Spectral zeta

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JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

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Magnetic spectral bounds on starlike plane domains

Abstract

Keywords

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