135 research outputs found
Stability and control of a 1D quantum system with confining time dependent delta potentials
The evolution problem for a quantum particle confined in a 1D box and
interacting with one fixed point through a time dependent point interaction is
considered. Under suitable assumptions of regularity for the time profile of
the Hamiltonian, we prove the existence of strict solutions to the
corresponding Schr\"odinger equation. The result is used to discuss the
stability and the steady-state local controllability of the wavefunction when
the strenght of the interaction is used as a control parameter.Comment: latex, 21 pages, title change
On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region
We investigate relations between spectral properties of a single-centre
point-interaction Hamiltonian describing a particle confined to a bounded
domain , with Dirichlet boundary, and the
geometry of . For this class of operators Krein's formula yields an
explicit representation of the resolvent in terms of the integral kernel of the
unperturbed one, . We use a moving plane
analysis to characterize the behaviour of the ground-state energy of the
Hamiltonian with respect to the point-interaction position and the shape of
, in particular, we establish some conditions showing how to place the
interaction to optimize the principal eigenvalue.Comment: LaTeX, 15 page
Time dependent delta-prime interactions in dimension one
We solve the Cauchy problem for the Schr\"odinger equation corresponding to
the family of Hamiltonians in which
describes a -interaction with time-dependent strength .
We prove that the strong solution of such a Cauchy problem exits whenever the
map belongs to the fractional Sobolev space
, thus weakening the hypotheses which would be required by
the known general abstract results. The solution is expressed in terms of the
free evolution and the solution of a Volterra integral equation.Comment: minor changes, 10 page
Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces
We provide a limiting absorption principle for the self-adjoint realizations
of Laplace operators corresponding to boundary conditions on (relatively open
parts of) compact hypersurfaces ,
. For any of such self-adjoint operators we also
provide the generalized eigenfunctions and the scattering matrix; both these
objects are written in terms of operator-valued Weyl functions. We make use of
a Krein-type formula which provides the resolvent difference between the
operator corresponding to self-adjoint boundary conditions on the hypersurface
and the free Laplacian on the whole space . Our results apply
to all standard examples of boundary conditions, like Dirichlet, Neumann,
Robin, and -type, either assigned on or on
.Comment: Final revised version, to appear in Journal of Spectral Theor
Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
The abstract theory of self-adjoint extensions of symmetric operators is used
to construct self-adjoint realizations of a second-order elliptic operator on
with linear boundary conditions on (a relatively open part of)
a compact hypersurface. Our approach allows to obtain Krein-like resolvent
formulas where the reference operator coincides with the "free" operator with
domain ; this provides an useful tool for the scattering
problem from a hypersurface. Concrete examples of this construction are
developed in connection with the standard boundary conditions, Dirichlet,
Neumann, Robin, and -type, assigned either on a
dimensional compact boundary or on a relatively open
part . Schatten-von Neumann estimates for the difference
of the powers of resolvents of the free and the perturbed operators are also
proven; these give existence and completeness of the wave operators of the
associated scattering systems.Comment: Final revised version, to appear in Journal of Differential Equation
Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach
Artificial interface conditions parametrized by a complex number
are introduced for 1D-Schr\"odinger operators. When this complex parameter
equals the parameter of the complex deformation which unveils
the shape resonances, the Hamiltonian becomes dissipative. This makes possible
an adiabatic theory for the time evolution of resonant states for arbitrarily
large time scales. The effect of the artificial interface conditions on the
important stationary quantities involved in quantum transport models is also
checked to be as small as wanted, in the polynomial scale as
, according to .Comment: 60 pages, 13 figure
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