31 research outputs found
Dirichlet eigenvalues of asymptotically flat triangles
This paper is devoted to the study of the eigenpairs of the Dirichlet Laplacian on a family of triangles where two vertices are fixed and the altitude associated with the third vertex goes to zero. We investigate the dependence of the eigenvalues on this altitude. For the first eigenvalues and eigenfunctions, we obtain an asymptotic expansion at any order at the scale cube root of this altitude due to the influence of the Airy operator. Asymptotic expansions of the eigenpairs are provided, exhibiting two distinct scales when the altitude tends to zero. In addition, we generalize our analysis to the case of a shrinking symmetric polygon and we quantify the corresponding tunneling effect
On the Bound States of Schr\"odinger Operators with -interactions on Conical Surfaces
In dimension greater than or equal to three, we investigate the spectrum of a
Schr{\"o}dinger operator with a -interaction supported on a cone whose
cross section is the sphere of co-dimension two. After decomposing into fibers,
we prove that there is discrete spectrum only in dimension three and that it is
generated by the axisymmetric fiber. We get that these eigenvalues are
non-decreasing functions of the aperture of the cone and we exhibit the precise
logarithmic accumulation of the discrete spectrum below the threshold of the
essential spectrum
Dirichlet eigenvalues of cones in the small aperture limit
We are interested in finite cones of fixed height 1 parametrized by their opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when their apertures tend to 0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues of each fiber of the Dirichlet Laplacian. In order to do this, we investigate the family of their one-dimensional Born-Oppenheimer approximations. The eigenvalue asymptotics involves powers of the cube root of the aperture, while the eigenfunctions include simultaneously two scales
Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors
This paper deals with the study of the two-dimensional Dirac operatorwith
infinite mass boundary condition in a sector. We investigate the question
ofself-adjointness depending on the aperture of the sector: when the sector is
convexit is self-adjoint on a usual Sobolev space whereas when the sector is
non-convexit has a family of self-adjoint extensions parametrized by a complex
number of theunit circle. As a byproduct of this analysis we are able to give
self-adjointnessresults on polygones. We also discuss the question of
distinguished self-adjointextensions and study basic spectral properties of the
operator in the sector
Dirac operators with Lorentz scalar shell interactions
This paper deals with the massive three-dimensional Dirac operator coupled
with a Lorentz scalar shell interaction supported on a compact smooth surface.
The rigorous definition of the operator involves suitable transmission
conditions along the surface. After showing the self-adjointness of the
resulting operator we switch to the investigation of its spectral properties,
in particular, to the existence and non-existence of eigenvalues. In the case
of an attractive coupling, we study the eigenvalue asymptotics as the mass
becomes large and show that the behavior of the individual eigenvalues and
their total number are governed by an effective Schr\"odinger operator on the
boundary with an external Yang-Mills potential and a curvature-induced
potential
Dirac operators on hypersurfaces as large mass limits
We show that the eigenvalues of the intrinsic Dirac operator on the boundary
of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean
Dirac operators, either in the domain with a MIT-bag type boundary condition or
in the whole space, with a suitably chosen zero order mass term.Comment: 39 page
Spectral transitions for Aharonov-Bohm Laplacians on conical layers
We consider the Laplace operator in a tubular neighbourhood of a conical
surface of revolution, subject to an Aharonov-Bohm magnetic field supported on
the axis of symmetry and Dirichlet boundary conditions on the boundary of the
domain. We show that there exists a critical total magnetic flux depending on
the aperture of the conical surface for which the system undergoes an abrupt
spectral transition from infinitely many eigenvalues below the essential
spectrum to an empty discrete spectrum. For the critical flux we establish a
Hardy-type inequality. In the regime with infinite discrete spectrum we obtain
sharp spectral asymptotics with refined estimate of the remainder and
investigate the dependence of the eigenvalues on the aperture of the surface
and the flux of the magnetic field.Comment: 27 pages, 4 figure
Effective operators for Robin eigenvalues in domains with corners
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 Schr\"odinger-type operator on the
boundary of the domain with boundary conditions at the corners.Comment: 84 pages. 23 figures. To appear in Annales de l'Institut Fourier
(Grenoble