21 research outputs found
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
Nonlinear Schr{\"o}dinger equation: concentration on circles driven by an external magnetic field
In this paper, we study the semiclassical limit for the stationary magnetic
nonlinear Schr\"odinger equation \begin{align}\label{eq:initialabstract}\left(
i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in
\mathbb{R}^{3},\end{align}where p\textgreater{}2, is a vector potential
associated to a given magnetic field , i.e and is a
nonnegative, scalar (electric) potential which can be singular at the origin
and vanish at infinity or outside a compact set.We assume that and
satisfy a cylindrical symmetry. By a refined penalization argument, we prove
the existence of semiclassical cylindrically symmetric solutions of upper
equation whose moduli concentrate, as , around a circle. We
emphasize that the concentration is driven by the magnetic and the electric
potentials. Our result thus shows that in the semiclassical limit, the magnetic
field also influences the location of the solutions of
(\ref{eq:initialabstract}) if their concentration occurs around a locus, not
a single point
Estimates for eigenvalues of Aharonov-Bohm operators with varying poles and non-half-interger circulation
We study the behavior of eigenvalues of a magnetic Aharonov-Bohm operator
with non-half-integer circulation and Dirichlet boundary conditions in a planar
domain. As the pole is moving in the interior of the domain, we estimate the
rate of the eigenvalue variation in terms of the vanishing order of the limit
eigenfunction at the limit pole. We also provide an accurate blow-up analysis
for scaled eigenfunctions and prove a sharp estimate for their rate of
convergence.Comment: 35 page
On the eigenvalues of Aharonov-Bohm operators with varying poles
We consider a magnetic operator of Aharonov-Bohm type with Dirichlet boundary
conditions in a planar domain. We analyse the behavior of its eigenvalues as
the singular pole moves in the domain. For any value of the circulation we
prove that the k-th magnetic eigenvalue converges to the k-th eigenvalue of the
Laplacian as the pole approaches the boundary. We show that the magnetic
eigenvalues depend in a smooth way on the position of the pole, as long as they
remain simple. In case of half-integer circulation, we show that the rate of
convergence depends on the number of nodal lines of the corresponding magnetic
eigenfunction. In addition, we provide several numerical simulations both on
the circular sector and on the square, which find a perfect theoretical
justification within our main results, together with the ones in
[Bonnaillie-No\"el, V., Helffer, B., Exp. Math. 20 (2011), no. 3, 304-322;
MR2836255 (2012i:35274)].Comment: 39 pages, 28 figure
Schrödinger equations with an external magnetic field: Spectral problems and semiclassical states
In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential.Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe
Schrödinger equations with an external magnetic field: Spectral problems and semiclassical states
In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential.Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe