2,295 research outputs found
Geometric coupling thresholds in a two-dimensional strip
We consider the Laplacian in a strip with the
boundary condition which is Dirichlet except at the segment of a length of
one of the boundaries where it is switched to Neumann. This operator is known
to have a non-empty and simple discrete spectrum for any . There is a
sequence of critical values at which new eigenvalues emerge
from the continuum when the Neumann window expands. We find the asymptotic
behavior of these eigenvalues around the thresholds showing that the gap is in
the leading order proportional to with an explicit coefficient
expressed in terms of the corresponding threshold-energy resonance
eigenfunction
On the number of particles which a curved quantum waveguide can bind
We discuss the discrete spectrum of N particles in a curved planar waveguide.
If they are neutral fermions, the maximum number of particles which the
waveguide can bind is given by a one-particle Birman-Schwinger bound in
combination with the Pauli principle. On the other hand, if they are charged,
e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial
role. We prove a sufficient condition under which the discrete spectrum of such
a system is empty.Comment: a LateX file, 12 page
Recommended from our members
Laser Micro Sintering – A Quality Leap through Improvement of Powder Packing
Laser micro sintering, a modification of selective laser sintering for freeform fabrication
of micro-parts, was continuously upgraded since its first application. Poor density of the powder
layers has been a persisting problem that had to be dealt with from the beginning. One solution
was the application of high intensity q-switched laser pulses. Compaction of the material and
improvement of the sinter resolution was achieved. But with these pulse-regimes only limited
density of the sintered body has been achievable. Recently special efforts have been made to get
rid of or at least reduce these drawbacks by markedly higher compaction of the respective powder
layers. There is clear evidence that with sufficiently compacted powder layers even laser micro
sintering with continuous radiation should be feasible. Till recently laser sintering of metal had
been applied mainly to produce monolithic components. With the upgraded technique direct
generation of micro devices with freely movable subassemblies can be possible.Mechanical Engineerin
Point interactions in a strip
We study the behavior of a quantum particle confined to a hard--wall strip of
a constant width in which there is a finite number of point
perturbations. Constructing the resolvent of the corresponding Hamiltonian by
means of Krein's formula, we analyze its spectral and scattering properties.
The bound state--problem is analogous to that of point interactions in the
plane: since a two--dimensional point interaction is never repulsive, there are
discrete eigenvalues, , the lowest of which is
nondegenerate. On the other hand, due to the presence of the boundary the point
interactions give rise to infinite series of resonances; if the coupling is
weak they approach the thresholds of higher transverse modes. We derive also
spectral and scattering properties for point perturbations in several related
models: a cylindrical surface, both of a finite and infinite heigth, threaded
by a magnetic flux, and a straight strip which supports a potential independent
of the transverse coordinate. As for strips with an infinite number of point
perturbations, we restrict ourselves to the situation when the latter are
arranged periodically; we show that in distinction to the case of a
point--perturbation array in the plane, the spectrum may exhibit any finite
number of gaps. Finally, we study numerically conductance fluctuations in case
of random point perturbations.Comment: a LaTeX file, 38 pages, to appear in Ann. Phys.; 12 figures available
at request from [email protected]
Sufficient conditions for the anti-Zeno effect
The ideal anti-Zeno effect means that a perpetual observation leads to an
immediate disappearance of the unstable system. We present a straightforward
way to derive sufficient conditions under which such a situation occurs
expressed in terms of the decaying states and spectral properties of the
Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno
effects is in fact very narrow.Comment: LatEx2e, 9 pages; a revised text, to appear in J. Phys. A: Math. Ge
On the discrete spectrum of Robin Laplacians in conical domains
We discuss several geometric conditions guaranteeing the finiteness or the
infiniteness of the discrete spectrum for Robin Laplacians on conical domains.Comment: 12 page
Lieb-Thirring inequalities for geometrically induced bound states
We prove new inequalities of the Lieb-Thirring type on the eigenvalues of
Schr\"odinger operators in wave guides with local perturbations. The estimates
are optimal in the weak-coupling case. To illustrate their applications, we
consider, in particular, a straight strip and a straight circular tube with
either mixed boundary conditions or boundary deformations.Comment: LaTeX2e, 14 page
Weakly coupled states on branching graphs
We consider a Schr\"odinger particle on a graph consisting of links
joined at a single point. Each link supports a real locally integrable
potential ; the self--adjointness is ensured by the type
boundary condition at the vertex. If all the links are semiinfinite and ideally
coupled, the potential decays as along each of them, is
non--repulsive in the mean and weak enough, the corresponding Schr\"odinger
operator has a single negative eigenvalue; we find its asymptotic behavior. We
also derive a bound on the number of bound states and explain how the
coupling constant may be interpreted in terms of a family of
squeezed potentials.Comment: LaTeX file, 7 pages, no figure
- …