674 research outputs found
Multiple Aharonov--Bohm eigenvalues: the case of the first eigenvalue on the disk
It is known that the first eigenvalue for Aharonov--Bohm operators with
half-integer circulation in the unit disk is double if the potential's pole is
located at the origin. We prove that in fact it is simple as the pole
Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles
We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm
operator with half-integer circulation and Dirichlet boundary conditions in a
planar domain. We provide sharp asymptotics for eigenvalues as the pole is
moving in the interior of the domain, approaching a zero of an eigenfunction of
the limiting problem along a nodal line. As a consequence, we verify
theoretically some conjectures arising from numerical evidences in preexisting
literature. The proof relies on an Almgren-type monotonicity argument for
magnetic operators together with a sharp blow-up analysis
On multiple eigenvalues for Aharonov-Bohm operators in planar domains
We study multiple eigenvalues of a magnetic Aharonov-Bohm operator with
Dirichlet boundary conditions in a planar domain. In particular, we study the
structure of the set of the couples position of the pole-circulation which keep
fixed the multiplicity of a double eigenvalue of the operator with the pole at
the origin and half-integer circulation. We provide sufficient conditions for
which this set is made of an isolated point. The result confirms and validates
a lot of numerical simulations available in preexisting literature.Comment: 33 pages, 4 figure
On the leading term of the eigenvalue variation for Aharonov-Bohm operators with a moving pole
We study the behavior of eigenvalues for magnetic Aharonov-Bohm operators
with half-integer circulation and Dirichlet boundary conditions in a planar
domain. We analyse the leading term in the Taylor expansion of the eigenvalue
function as the pole moves in the interior of the domain, proving that it is a
harmonic homogeneous polynomial and detecting its exact coefficients.Comment: 26 pages, 1 figur
Solutions to nonlinear Schr\"odinger equations with singular electromagnetic potential and critical exponent
We investigate existence and qualitative behaviour of solutions to nonlinear
Schr\"odinger equations with critical exponent and singular electromagnetic
potentials. We are concerned with magnetic vector potentials which are
homogeneous of degree -1, including the Aharonov-Bohm class. In particular, by
variational arguments we prove a result of multiplicity of solutions
distinguished by symmetry properties
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