Geometric coupling thresholds in a two-dimensional strip

We consider the Laplacian in a strip $\mathbb{R}\times (0,d)$ with the boundary condition which is Dirichlet except at the segment of a length $2a$ 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 $a>0$. There is a sequence $0<a_1<a_2<...$ 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 $(a-a_n)^2$ 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

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 $N$ 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 $m$ discrete eigenvalues, $1\le m\le N$, 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 $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as $\,x^{-1-\epsilon}$ 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 $\,\delta\,$ coupling constant may be interpreted in terms of a family of squeezed potentials.Comment: LaTeX file, 7 pages, no figure