MathSciDoc: An Archive for Mathematician ∫

We consider the Laplacian in curved tubes of arbitrary crosssection rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.

We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 smooth loops of a fixed length L, formally given by (x) with >0. It is shown that the ground state of this operator is locally maximized by a circular . We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of .

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit N. We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart from the resonances coming from the decoupled-surface eigenvalues, such scatterers exhibit the high-energy behavior typical for the interaction for the physically interesting couplings.

We discuss the Nckel model of an open quantum dot, i.e. a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, there are embedded eigenvalues which turn into resonances if the symmetry is violated, either by applying a magnetic field or by deformation of the well. For a weak symmetry breaking we derive the perturbative expansion of these resonances. We also deduce a sufficient condition under which the discrete spectrum of such a system (without any symmetry) survives the presence of a strong magnetic field. It is satisfied, in particular, if the dot potential is purely attractive.

The most general admissible boundary conditions are derived for an idealized AharonovBohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; the other two parameters of the model are the AharonovBohm flux and the homogeneous magnetic field. The generalized boundary conditions may be regarded as a combination of the AharonovBohm effect with a point interaction. Spectral properties of the derived Hamtonians are studied in detail.

We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.

We discuss approximations of the vertex coupling on a star-shaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the CheonShigehara technique using interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges, one can approximate the {n+1\choose 2}-parameter family of all time-reversal invariant couplings.

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles that the waveguide can bind is given by a one-particle BirmanSchwinger 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.

A two-dimensional model of electron motion in a curved quantum wire of length 2<i>D</i> attached to a pair of macroscopic electrodes is studied. The wire is regarded as infinitely thin and supports a potential which is a combination of a constant transversal-mode energy and an attractive curvature-induced term. The system is shown to have at least one resonance which exhibits spectral concentration as <i>D</i>.

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrdinger operator in L^2(\mathbb{R}^2) with an attractive interaction supported on a closed curve , formally given by (<i>x</i>); we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread in L^2(\mathbb{R}^2), homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of . We prove an isoperimetric theorem for <i>p</i>-means of chords of curves when <i>p</i> 2, which implies in particular that the global extrema for the physical problems are always attained when is a circle. The letter concludes with a discussion of the <i>p</i>-means of chords when <i>p</i> &gt; 2.