Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for ``decorated'' quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (``scars'').Comment: 4 eps figures, LATEX file, 21 pages Revised form: a cut-and-paste blooper fixe

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Index theorems for quantum graphs

In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a "Laplacian") with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe

Absence of bound states for waveguides in 2D periodic structures

We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting models a so-called "soft-wall" waveguide problem. We show that there are no bound states, i.e., the spectrum of the operator under study contains no point spectrum.Comment: This is an updated version of our paper (in slightly different form in Journal of Mathematical Physics). An anonymous reviewer kindly made us aware that ref. 10 is not applicable in our situation. An application of the theorem in ref. 10 would have proved the absence of singular continuous spectrum also. Our result on the absence of point spectrum is not affected by thi

On occurrence of spectral edges for periodic operators inside the Brillouin zone

The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of ``corner'' high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a ``generic'' case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in the new versio

A series solution and a fast algorithm for the inversion of the spherical mean Radon transform

An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo- and photo- acoustic tomography. Closed-form inversion formulae are currently known only for the case when the centers of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly known - such as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centers of the integration spheres) lie on a surface of a cube. This algorithm reconsrtucts 3-D images thousands times faster than backprojection-type methods

Essential spectra of difference operators on \sZ^n-periodic graphs

Let (\cX, \rho) be a discrete metric space. We suppose that the group \sZ^n acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a \sZ^n-periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on $l^p(X)$ where $X$ is a \sZ^n-periodic discrete metric space. Our approach is based on the theory of band-dominated operators on \sZ^n and their limit operators. In case $X$ is the set of vertices of a combinatorial graph, the graph structure defines a Schr\"{o}dinger operator on $l^p(X)$ in a natural way. We illustrate our approach by determining the essential spectra of Schr\"{o}dinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures

Effective Dielectric Constants of Photonic Crystal of Aligned Anisotropic Cylinders: Application to the Optical Response of Periodic Array of Carbon Nanotubes

We calculate the static dielectric tensor of a periodic system of aligned anisotropic dielectric cylinders. Exact analytical formulas for the effective dielectric constants for the E- and H- eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylinders. It is shown that depending on the symmetry of the unit cell photonic crystal of anisotropic cylinders behaves in the low-frequency limit like uniaxial or biaxial natural crystal. The developed theory of homogenization of anisotropic cylinders is applied for calculations of the dielectric properties of photonic crystals of carbon nanotubes

Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography

The paper contains a simple approach to reconstruction in Thermoacoustic and Photoacoustic Tomography. The technique works for any geometry of point detectors placement and for variable sound speed satisfying a non-trapping condition. A uniqueness of reconstruction result is also obtained