Heat transfer in a complex medium

The heat equation is considered in the complex medium consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies an impedance boundary condition is imposed. An equation for the limiting field is derived when the characteristic size $a$ of the small bodies tends to zero, their total number $\mathcal{N}(a)$ tends to infinity at a suitable rate, and the distance $d = d(a)$ between neighboring small bodies tends to zero: $a << d$, $\lim_{a\to 0}\frac{a}{d(a)}=0$. No periodicity is assumed about the distribution of the small bodies. These results are basic for a method of creating a medium in which heat signals are transmitted along a given line. The technical part for this method is based on an inverse problem of finding potential with prescribed eigenvalues.Comment: arXiv admin note: text overlap with arXiv:1207.056

Vortex structure in exponentially shaped Josephson junctions

We report the numerical calculations of the static vortex structure and critical curves in exponentially shaped long Josephson junctions for in-line and overlap geometries. Each solution of the corresponding boundary value problem is associated with the Sturm-Liouville problem whose minimal eigenvalue allows to make a conclusion about the stability of the vortex. The change in width of the junction leads to the renormalization of the magnetic flux in comparison to the case of a linear one-dimensional model. We study the influence of the model's parameters and, particularly, the shape parameter on the stability of the states of the magnetic flux. We compare the vortex structure and critical curves for the in-line and overlap geometries. Our numerically constructed critical curve of the Josephson junction matches well with the experimental one.Comment: 8 pages, 10 figures, NATO Advanced Research Workshop on "Vortex dynamics in superconductors and other complex systems" Yalta, Crimea, Ukraine, 13-17 September 200

On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the theorems, and we updated two reference

The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result

Inverse spectral problems for Dirac operators with summable matrix-valued potentials

We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the potential from the corresponding spectral data.Comment: 32 page

Localization on quantum graphs with random vertex couplings

We consider Schr\"odinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of self-adjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges

L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case

We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $\int_{\gamma} |\phi_{j}^{\hbar}|^2 ds = {\mathcal O}(|\log \hbar|)$ for generic curves $\gamma$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions.Comment: Correct some typos and added some more detail in section

Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4, and 3.12 extended, details added in subsections 2.3 and 4.2, section 6 rearranged, typos corrected, references adde