MathSciDoc: An Archive for Mathematician ∫

This paper is mainly inspired by the conjecture about the existence of bound states for magnetic Neumann Laplacians on planar wedges of any aperture <i></i> (0, <i></i>). So far, a proof was only obtained for apertures <i></i> 0.511. The conviction in the validity of this conjecture for apertures <i></i> 0.511<i></i> mainly relied on numerical computations. In this paper we succeed to prove the existence of bound states for any aperture <i></i> 0.583 using a variational argument with suitably chosen test functions. Employing some more involved test functions and combining a variational argument with computer assistance, we extend this interval up to any aperture <i></i> 0.595<i></i>. Moreover, we analyse the same question for closely related problems concerning magnetic Robin Laplacians on wedges and for magnetic Schrdinger operators in the plane with <i></i>-interactions supported on broken lines.

We discuss a modification of Smilansky model in which a singular potential channel is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential channels.

In this note we prove that the essential spectrum of a Schrdinger operator with potential supported on a finite number of compact Lipschitz hypersurfaces is given by [0, +). We emphasize that the union of a family of Lipschitz hypersurfaces is in general not Lipschitz. ( 2013 WileyVCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Resonance and decay phenomena are ubiquitous in the quantum world. To understand them in their complexity it is useful to study solvable models in a wide sense, that is, systems which can be treated by analytical means. The present review offers a survey of such models startingt he classical Friedrichs result and carryingfu rther to recent developments in the theory of quantum graphs. Our attention concentrates on dynamical mechanism underlyingre sonance effects and at time evolution of the related unstable systems.

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a hybrid surface consisting on a halfline attached by its endpoint to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term.

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.

We discuss differences between the exact<i>S</i>-matrix for scattering on serial structures and a known factorized expression constructed of single-element<i>S</i>-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities.

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a <i></i> type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.

The subject of the paper is Schrdinger operators on tree graphs which are radial, having the branching number bn at all the vertices at the distance tn from the root. We consider a family of coupling conditions at the vertices characterized by (bn1)2+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of {tn+1tn} growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrdinger operator can be purely absolutely continuous.