21 research outputs found
The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions
The Dirichlet Laplacian in a curved three-dimensional tube built along a
spatial (bounded or unbounded) curve is investigated in the limit when the
uniform cross-section of the tube diminishes. Both deformations due to bending
and twisting of the tube are considered. We show that the Laplacian converges
in a norm-resolvent sense to the well known one-dimensional Schroedinger
operator whose potential is expressed in terms of the curvature of the
reference curve, the twisting angle and a constant measuring the asymmetry of
the cross-section. Contrary to previous results, we allow the reference curves
to have non-continuous and possibly vanishing curvature. For such curves, the
distinguished Frenet frame standardly used to define the tube need not exist
and, moreover, the known approaches to prove the result for unbounded tubes do
not work. Our main ideas how to establish the norm-resolvent convergence under
the minimal regularity assumptions are to use an alternative frame defined by a
parallel transport along the curve and a refined smoothing of the curvature via
the Steklov approximation.Comment: 29 pages, 6 figure
Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers
By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators
Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
In a previous paper (arXiv:math-ph/0604055) we introduced a very simple
PT-symmetric non-Hermitian Hamiltonian with real spectrum and derived a closed
formula for the metric operator relating the problem to a Hermitian one. In
this note we propose an alternative formula for the metric operator, which we
believe is more elegant and whose construction -- based on a backward use of
the spectral theorem for self-adjoint operators -- provides new insights into
the nature of the model.Comment: LaTeX, 6 page
PT-symmetric models in curved manifolds
We consider the Laplace-Beltrami operator in tubular neighbourhoods of curves
on two-dimensional Riemannian manifolds, subject to non-Hermitian parity and
time preserving boundary conditions. We are interested in the interplay between
the geometry and spectrum. After introducing a suitable Hilbert space framework
in the general situation, which enables us to realize the Laplace-Beltrami
operator as an m-sectorial operator, we focus on solvable models defined on
manifolds of constant curvature. In some situations, notably for non-Hermitian
Robin-type boundary conditions, we are able to prove either the reality of the
spectrum or the existence of complex conjugate pairs of eigenvalues, and
establish similarity of the non-Hermitian m-sectorial operators to normal or
self-adjoint operators. The study is illustrated by numerical computations.Comment: 37 pages, PDFLaTeX with 11 figure
On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
We consider one-dimensional Schroedinger-type operators in a bounded interval
with non-self-adjoint Robin-type boundary conditions. It is well known that
such operators are generically conjugate to normal operators via a similarity
transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians
in quantum mechanics, we study properties of the transformations in detail. We
show that they can be expressed as the sum of the identity and an integral
Hilbert-Schmidt operator. In the case of parity and time reversal boundary
conditions, we establish closed integral-type formulae for the similarity
transformations, derive the similar self-adjoint operator and also find the
associated "charge conjugation" operator, which plays the role of fundamental
symmetry in a Krein-space reformulation of the problem.Comment: 27 page
Absence of eigenvalues of two-dimensional magnetic Schrödinger operators
By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schrödinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov–Bohm field are also covered