Michael Solomyak

Remarks on the spectral shift function

Novel compounds resulting from the reaction of hindered phenols, such as 3,5-di-t-butyl-4-hydroxybenxyl alcohol, with various aryl amines or carbazole are effective oxidation inhibitors for lubricants.

On the spectrum of the Laplacian on regular metric trees

A metric tree Γ is a tree whose edges are viewed as non-degenerate line segments. The Laplacian Δ 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 Δ 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 Γ possesses a rich group of symmetries. As a result, the space L 2(Γ) decomposes into the orthogonal sum of subspaces reducing the operator Δ. This leads to detailed spectral analysis of Δ. We survey recent results on this subject.

SCHRÖDINGER OPERATORS ON HOMOGENEOUS METRIC TREES: SPECTRUM IN GAPS

The paper studies the spectral properties of the Schrodinger operator AgV = A0 + 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 A0 = -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.

On the spectrum of narrow periodic waveguides

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.

Asymptotic behavior of the spectrum of differential equations

This survey is devoted to an exposition of results on the asymptotics of the discrete spectrum of self-adjoint differential operators, mainly partial differential operators.

Piecewise-polynomial approximation of functions fromHℓ((0, 1)d), 2ℓ=d, and applications to the spectral theory of the Schrödinger operator

For the selfadjoint Schrödinger operator −Δ−αV on ℝ2 the number of negative eigenvalues is estimated. The estimates obtained are based upon a new result on the weightedL2-approximation of functions from the Sobolev spaces in the cases corresponding to the critical exponent in the embedding theorem.

Two-sided estimates on singular values for a class of integral operators on the semi-axis

Abstract(Quasi)-norms inCp andCp, w of “weighted operators of the integration of (fractional) order ν” are estimated. It is shown that, in most cases, the estimates obtained are sharp both in order and in function classes for the weight function involved.

Counting Schrödinger Boundstates: Semiclassics and Beyond

This is a survey of the basic results on the behavior of the num ber of the eigenvalues of a Schrodinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassical rules are violated.

On the negative discrete spectrum of a preiodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential

Let Ω ⊂Rd be an unbounded domain, periodic along a chosen direction (a waveguide-type domain),P be a self-adjoint elliptic second order operator inL2(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.

On a differential operator appearing in the theory of irreversible quantum graphs

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.