arXiv subjects

Vadim Kostrykin

Publications and source records attributed to Vadim Kostrykin.

At least 37 records · Page 2Linked to original sources

arXiv2007

Heat kernels on metric graphs and a trace formula

We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.

math-ph↗

arXiv2006

The Birman-Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein formula in the abstract scattering theory, we represent the de la Harpe-Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.

math.SP↗

arXiv2006

On Krein's Example

In his 1953 paper [Matem.~Sbornik \textbf{33} (1953), 597 -- 626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval $[-1,1]$ and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.

math.SP↗

arXiv2006

On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials

The present paper is devoted to the study of spectral properties of random Schroedinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the corresponding discrete model, too. In certain disorder regimes we are able to prove the Lipschitz continuity of the integrated density of states and/or localization near spectral edges.

math-ph↗

arXiv2006

The inverse scattering problem for metric graphs and the traveling salesman problem

We present a solution to the inverse scattering problem for differential Laplace operators on metric noncompact graphs. We prove that for almost all boundary conditions (i) the scattering matrix uniquely determines the graph and its metric structure, (ii) the boundary conditions are determined uniquely up to trivial gauge transformations. The main ingredient of our approach is a combinatorial Fourier expansion of the scattering matrix which encodes the topology of the graph into analytic properties of the scattering matrix. Using the technique developed in this work, we also propose an analytic approach to solving some combinatorial problems on graphs, in particular, the Traveling Salesman Problem.

math-ph↗

arXiv2006

Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups

The main objective of the present work is to study the negative spectrum of (differential) Laplace operators on metric graphs as well as their resolvents and associated heat semigroups. We prove an upper bound on the number of negative eigenvalues and a lower bound on the spectrum of Laplace operators. Also we provide a sufficient condition for the associated heat semigroup to be positivity preserving.

math-ph↗

arXiv2005

Wegner Estimate for Indefinite Anderson Potentials: Some Recent Results and Applications

We review recent and give some new results on the spectral properties of Schroedinger operators with a random potential of alloy type. Our point of interest is the so called Wegner estimate in the case where the single site potentials change sign. The indefinitness of the single site potential poses certain difficulties for the proof of the Wegner estimate which are still not fully understood. The Wegner estimate is a key ingredient in an existence proof of pure point spectrum of the considered random Schroedinger operators. Under certain assumptions on the considered models additionally the existence of the density of states can be proven.

math-ph↗

arXiv2004

The singularly continuous spectrum and non-closed invariant subspaces

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\mathfrak{H}_0$ is of codimension 1, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ of $\mathbf{A}$ that are off-diagonal with respect to the decomposition $\mathfrak{H}= \mathfrak{H}_0\oplus\mathfrak{H}_1$. In particular, we prove the existence of a one-parameter family of dense non-closed invariant subspaces of the operator $\mathbf{A}+\mathbf{V}$ provided that this operator has a nonempty singularly continuous spectrum. We show that such subspaces are related to non-closable densely defined solutions of the operator Riccati equation associated with generalized eigenfunctions corresponding to the singularly continuous spectrum of $\mathbf{B}$.

math.SP↗

arXiv2004

Generating Functions of Random Walks on Graphs

The article provides an explicit algebraic expression for the generating function of walks on graphs. Its proof is based on the scattering theory for the differential Laplace operator on non-compact graphs.

math.CO↗

arXiv2003

A Random Necklace Model

We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.

math-ph↗

arXiv2003

On the existence of solutions to the operator Riccati equation and the tanΘtheorem

Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d>0 be the distance between the spectra of A and C. We prove that under these assumptions the sharp value of the constant c in the condition ||B||<cd guaranteeing the existence of a (bounded) solution to the operator Riccati equation XA-CX+XBX=B^* is equal to \sqrt{2}. We also prove an extension of the Davis-Kahan \tanΘtheorem and provide a sharp estimate for the norm of the solution to the Riccati equation. If C is bounded, we prove, in addition, that the solution X is a strict contraction if B satisfies the condition ||B||<d, and that this condition is sharp.

math.SP↗

arXiv2003

Quantum wires with magnetic fluxes

In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given self-adjoint Laplacian by perturbing it by magnetic fields. In particular, it is shown that generically this set is isomorphic to a torus. We also describe the conditions under which the operator is unambiguously (up to unitary equivalence) defined by prescribing the magnetic fluxes through all loops of the graph.

math-ph↗

arXiv2003

A generalization of the $tan 2Θ$ Theorem

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$, where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces $\mathfrak{H}_0$ and $\mathfrak{H}_1=\mathfrak{H}_0^\perp$, respectively, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to the decomposition $\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$. We obtain sharp two-sided estimates on the norm of the difference of the orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$ and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated Davis-Kahan $\tan 2Θ$ Theorem. On this basis we also prove new existence and uniqueness theorems for contractive solutions to the operator Riccati equation, thus, extending recent results of Adamyan, Langer, and Tretter.

math.SP↗

arXiv2002

Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach

We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator $\mathbf{B}$. In this case we establish a criterion for the uniqueness of contractive solutions.

math.SP↗

arXiv2002

On a Subspace Perturbation Problem

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let $A$ and $V$ be bounded self-adjoint operators. Assume that the spectrum of $A$ consists of two disjoint parts $σ$ and $Σ$ such that $d=\text{dist}(σ, Σ)>0$. We show that the norm of the difference of the spectral projections $\EE_A(σ)$ and $\EE_{A+V}\big (\{λ| \dist(λ, σ)$ $<d/2\}\big)$ for $A$ and $A+V$ is less then one whenever either (i) $\|V\|<\frac{2}{2+π}d$ or (ii) $\|V\|<{1/2}d$ and certain assumptions on the mutual disposition of the sets $σ$ and $Σ$ are satisfied.

math.SP↗

arXiv2002

Perturbation Theory for the Quantum Time-Evolution in Rotating Potentials

The quantum mechanical time-evolution is studied for a particle under the influence of an explicitly time-dependent rotating potential. We discuss the existence of the propagator and we show that in the limit of rapid rotation it converges strongly to the solution operator of the Schrödinger equation with the averaged rotational invariant potential.

math-ph↗

arXiv2002

Statistical ensembles and density of states

We propose a definition of microcanonical and canonical statistical ensembles based on the concept of density of states. This definition applies both to the classical and the quantum case. For the microcanonical case this allows for a definition of a temperature and its fluctuation, which might be useful in the theory of mesoscopic systems. In the quantum case the concept of density of states applies to one-particle Schroedinger operators, in particular to operators with a periodic potential or to random Anderson type models. In the case of periodic potentials we show that for the resulting $n$-particle system the density of states is $[(n-1)/2]$ times differentiable, such that like for classical microcanonical ensembles a (positive) temperature may be defined whenever $n\geq 5$. We expect that a similar result should also hold for Anderson type models. We also provide the first terms in asymptotic expansions of thermodynamic quantities at large energies for the microcanonical ensemble and at large temperatures for the canonical ensemble. A comparison shows that then both formulations asymptotically give the same results.

cond-mat.stat-mech↗

arXiv2002

Energy Transfer in Scattering by Rotating Potentials

Quantum mechanical scattering theory is studied for time-dependent Schroedinger operators, in particular for particles in a rotating potential. Under various assumptions about the decay rate at infinity we show uniform boundedness in time for the kinetic energy of scattering states, existence and completeness of wave operators, and existence of a conserved quantity under scattering. In a simple model we determine the energy transfered to a particle by a collision with a rotating blade.

math-ph↗