The problem of finding eigenvalue estimates for the Schrodinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. In the paper, these results are discussed, and their counterparts are established for the operator on the combinatorial and metric graphs corresponding to the lattice Z(2).

In the present work we obtain variational Hardy type inequalities with power and logarithmic weights which are generalizations of the corresponding inequalities given earlier in the papers by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom. We formulate and prove inequalities for arbitrary domains, and then we substantially simplify them for the class of convex domains and a special family of nonconvex domains.

The construction of "sparse potentials", suggested in \cite{RS09} for the lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schr\"odinger operator $-\D-\a V$ on such graphs, with a sparse potential $V$, we study the behavior (as $\a\to\infty$) of the number $N_-(-\D-\a V)$ of negative eigenvalues of $-\D-\a V$. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of $N_-(-\D-\a V)$ under very mild regularity assumptions. A similar construction works also for the lattice $\Z^2$, where D=2.

This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {(x, y) : 0 < y < εh(x)}, where h(x) is a positive periodic function. The main assumption is that h(x) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0.

We study a family of differential operators Lα in two variables, depending on the coupling parameter α⩾0 that appears only in the boundary conditions. Our main concern is the spectral properties of Lα, which turn out to be quite different for α<1 and for α>1. In particular, Lα has a unique self-adjoint realization for α<1 and many such realizations for α>1. In the more difficult case α>1 an analysis of non-elliptic pseudodifferential operators in dimension one is involved.

A family $\mathbf{A}_\alpha$ of differential operators depending on a real parameter $\alpha \ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum $\sigma_{a.c.}$ of the operator $\mathbf{A}_\alpha$ and its multiplicity for all values of the parameter. The spectrum of $\mathbf{A}_0$ is purely absolutely continuous and admits an explicit description. It turns out that for $\alpha < \sqrt 2$ one has $\sigma_{a.c.}(\mathbf{A}_\alpha) = \sigma_{a.c.}(\mathbf{A}_0)$, including the multiplicity. For $\alpha \ge \sqrt2$ an additional branch of the absolutely continuous spectrum arises; its source is an auxiliary Jacobi matrix which is related to the operator $\mathbf{A}_\alpha$. This birth of an extra branch of the absolutely continuous spectrum is the exact mathematical expression of the effect that was interpreted by Smilansky as irreversibility.

In the model suggested by Smilansky one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators attached at different points of the graph. This paper is a continuation of our investigation of the case K>1. For the sake of simplicity we consider K=2, but our argument applies to the general situation. In this second paper we apply the variational approach to the study of the point spectrum.

In the model suggested by Smilansky (2004 Waves Random Media 14 143–53) one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators attached at several different points in the graph. The present paper is the first one in which the case K > 1 is investigated. For the sake of simplicity, we consider K = 2, but our argument is of a general character. In this first of two papers on the problem, we describe the absolutely continuous spectrum. Our approach is based upon scattering theory.

A metric tree Gamma is a tree whose edges are viewed as non-degenerate line segments. The Laplacian Delta on such a tree is the operator of second order differentiation on each edge, complemented by the Kirchhoff matching conditions at the vertices. The spectrum of Delta can be quite varied, reflecting the geometry of a tree.We consider a special class of trees, namely the so-called regular metric trees. Any such tree Gamma possesses a rich group of symmetries. As a result, the space L-2 (Gamma) decomposes into the orthogonal sum of subspaces reducing the operator Delta. This leads to detailed spectral analysis of Delta. We survey recent results on this subject.

A partial differential operator depending on the coupling parameter α≥0 is considered. The spectral properties of the operator strongly depend on α. The operator was suggested in Smilansky (2003 Waves Random Media 14 S143–53) as a model of an irreversible physical system.

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the inequalities are sharp.

The Schrödinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an orthogonal decomposition of the space L_2 on the tree which reduces the Schrödinger operator with any symmetric weight. Using this decomposition, we analyse the spectrum of such operators, including the free Laplacian, under various assumptions about the tree and the potential.

The paper studies the spectral properties of the Schrödinger operator A gV = A 0 + gV on a homogeneous rooted metric tree, with a decaying real-valued potential V and a coupling constant g ≥ 0. The spectrum of the free Laplacian A 0 = -Δ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation gV gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of g if the potential V has a fixed sign. Assuming that the latter condition is satisfied and that V is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit g → ∞. Depending on the sign and decay of V, this asymptotics is either of the Weyl type or is completely determined by the behaviour of V at infinity.

Let Ω ⊂R d be an unbounded domain, periodic along a chosen direction (a waveguide-type domain),P be a self-adjoint elliptic second order operator inL 2(Ω) periodic along the same direction, andV be a real-valued decaying potential. We suppose that the bottom of the spectrum ofP is λ=0 and study the asymptotic behaviour of the number of negative eigenvalues of the opeatorP−aV as the parameter α tends to +∞. We show that typically the Weyl asymptotic law for this quantity is violated and find a substitute for this law.

We obtain the sharp order of growth of the eigenvalue distribution function for the operator in the anisotropic Sobolev space\(H^{t_1 ,t_2 } (Q)\), generated by the quadratic form ∫ Q ∣u∣2dμ, whereQ⊂ℝ2 is the unit square and μ is a probability self-affine fractal measure onQ. The geometry of Supp μ should be in a certain way consistent with the parameterst 1 ,t 2 .

Regular trees (see Definition 2.1) form an interesting class of general metric trees, i.e., trees whose edges are regarded as nondegenerate line segments rather than pairs of vertices. The special metric and combinatorial structure of regular trees is reflected by the geometry of the corresponding L-spaces and Sobolev spaces. Revealing this geometry is useful in the study of various analytic problems on such trees. Roughly speaking, this enables one to reduce a problem concerning a tree to a family of more elementary problems on intervals. Certainly, this reduction is possible only if the problem under consideration is in a sense compatible with the geometry of the given tree. In Sections 2 and 3 we present the fundamentals concerning the L-spaces and Sobolev spaces on a regular tree Γ. We construct a special decomposition of these spaces that is simultaneously orthogonal with respect to the inner products in L(Γ), in the Sobolev space H(Γ), and in many other Hilbert function spaces. This decomposition was discovered in our paper [5]. Here we give a new presentation of this material, which is more consistent and detailed, following the approach in [5] and also using some technical tools of the Carlson paper [1] (Carlson found a similar scheme, a bit later and in a different setting). The main difference between our approaches is that Carlson is mainly concentrated on trees of finite total length or at least on those of finite radius (the latter means that the distance between two points is a bounded function on Γ× Γ). On the contrary, a regular tree of infinite radius is the main object of our investigation. The basic material is illustrated by two applications. The first (Section 4) is related to the spectral theory of a class of Schrödinger operators on regular trees of infinite radius. For compatibility with the symmetries of the tree, we assume that the potential is symmetric, i.e., depends only on the distance from a point x ∈ Γ to the root of Γ. We show that any such operator can be decomposed into an orthogonal sum of countably many operators of Sturm–Liouville type, each acting on a semi-infinite interval. The domain of each component is described by means of specific matching conditions on the functions and their derivatives at some points tn, tn →∞. Further applications of this result are given in [8]. The other application (Section 5) is related to Hardy inequalities of the form ∫

Eigenvalue behavior for the equation -\lambda y"=Vu on the edges of a graph G of final total length, with a non-negative weight function V and under the Kirchhoff matching conditions at the vertices and zero boundary condition at at least one point of G, is studied. It is shown that the eigenvalues satisfy an inequality which involves the length |G| and the total mass corresponding to V but otherwise does not depend on the graph. Applications and generalizations of this result are also discussed.