Vadim Kostrykin

• Prof. Dr.
• Professor (Full) at Johannes Gutenberg University Mainz

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes operator from flui...

The main aims are to study the existence of stationary periodic solutions , called 1-bump periodic solutions, of the Amari model and to analyse their linear stability.

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the exis...

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the exis...

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes Operator from flui...

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating t...

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating t...

A version of the Davis–Kahan Tan 2Θ theorem for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a result by Motovilov and Selin.

We consider a monotone increasing operator in an ordered Banach space having and as a strong super- and subsolution, respectively. In contrast with the well-studied case , we suppose that . Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval . MSC: 47H0...

We study the neuronal field equation, a nonlinear integro-differential equation of Hammerstein type. By means of the Amann three fixed point theorem we prove the existence of bump solutions to this equation. Using the Krein-Rutman theorem we show their Lyapunov instability.

We provide a class of self-adjoint Laplace operators on metric graphs with the property that the solutions of the associated wave equation satisfy the finite propagation speed property. The proof uses energy methods, which are adaptions of corresponding methods for smooth manifolds.

Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary co...

Pathwise constructions of Brownian motions which satisfy all possible boundary conditions at the vertex of star graphs are given.

Various equivalent conditions for a semigroup or a resolvent generated by a Markov process to be of Feller type are given.

Brownian motions on a metric graph are defined, their Feller property is proved, and their generators are characterized. This yields a version of Feller's theorem for metric graphs. Comment: version 02: references updated, font problem fixed

Pathwise constructions of Brownian motions which satisfy all possible boundary conditions at the vertex of single vertex graphs are given. Comment: version 02: references updated, font problem fixed

Consider a metric graph G with set of vertices V. Assume that for every vertex in V one is given a Wentzell boundary condition. It is shown how one can construct the paths of a Brownian motion on G such that its generator - viewed as an operator on the space of continuous functions vanishing at infinity - has a domain consisting of twice continuous...

The construction of the paths of all possible Brownian motions (in the sense of Knight) on a half line or a finite interval is reviewed. Comment: 61 pages, 7 figures

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operat...

In his 1953 paper [Matem. Sbornik 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 i...

The main objective of the present work is to study contraction semigroups generated by Laplace operators on metric graphs, which are not necessarily self-adjoint. We prove criteria for such semigroups to be continuity and positivity preserving. Also we provide a characterization of generators of Feller semigroups on metric graphs.

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 characteri...

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 t...

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 characteri...

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...

We outline a new approach to the proof of the Adiabatic Theorem of Quantum Mechanics. The key idea of this approach is to parametrize the subspaces involved as graphs of some operators. This leads to a singular perturbation problem for the differential Riccati equation with operator valued coefficients.

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...

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...

The industrial application of laser welding implies a reliable and efficient production process. Thus, modelling and simulation is used to reveal the crucial points of the welding process. The welding process is described by transport phenomena for mass, momentum and energy. The three involved phases (solid, liquid, gaseous) interact over their fre...

IntroductionThe Free Boundary ProblemFinite-dimensional ApproximationsConclusion Bibliography

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. Under these assumptions we prove that the best possible value of the constant c in the condition || B || < cd\left\| B \right\| < cd guaranteeing the existence of a (bounded) solution to t...

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 regim...

The present paper is devoted to the study of spectral properties of random Schrödinger 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 regime...

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.

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...

Let 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 A. Assuming that sup spec(A0) ≤ inf spec(A1), where A0 and A1 are restrictions of A onto the subspaces \({{\mathfrak{H}}_{0}}\) and \({{\mathfrak{H}}_{1}} = \mathfrak{H}_{0}^{ \bot }...

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 d...

We consider the problem of variation of spectral subspaces for linear self-adjoint operators under off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections.

 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 t...

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=\mathfr...

Summary form only given. Laser cutting is an established separation process. Reliable machining at high productivity and quality as well as simplified machine operation are the actual goals. Monitoring and control of the dynamical process is investigated. This work describes the advances in fundamental physical modeling of time dependent processes...

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 th...

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 implying the solvability of the operator Riccati equation XA-CX+XBX = B # is equal to # 2. We als...

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+X...

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 o...

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 Schrodinger equation with the averaged rotational invariant potential...

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 Schr\"odinger equation with the averaged rotational invariant potenti...

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 $\sigma$ and $\Sigma$ such that $d=\text{dist}(\sigma, \Sigma)>0$. We show that the norm of the difference of t...

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 c...

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...

Methods from scattering theory are introduced to analyze random Schroedinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynam...

In this article we continue our analysis of Schroedinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defin...

We prove that the integrated surface density of states of continuous or discrete Anderson-type random Schrödinger operators is a measurable locally integrable function rather than a signed measure or a distribution. This generalizes our recent results on the existence of the integrated surface density of states in the continuous case and those of A...

Methods from scattering theory are introduced to analyze random Schroedinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynam...

We prove that the integrated density of surface states of continuous or discrete Anderson-type random Schroedinger operators is a measurable locally integrable function rather than a signed measure or a distribution. This generalize our recent results on the existence of the integrated density of surface states in the continuous case and those of A...

In this article we continue our analysis of Schr\"odinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to...

It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift func...

Drillings in zirconia coated Ni-superalloys is done by melt extraction with pulsed laser radiation provided by a Nd:YAG slab laser with microsecond pulse duration. This laser system distinguishes itself by a high beam quality and offers the possibility to investigate drilling of holes with a diameter of 200 micrometer by percussion drilling and tre...

In this article we continue our analysis of Schr\"odinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to...

In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a two-sided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schr\"odinger operators in one dimension. These Schr\"odinger operators are given by potentials of identical shape centered at every lattice site but with non-overlapp...

In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local...

The dynamical behaviour of the laser beam fusion cutting process of metals is investigated. Integral methods such as the variational formulation are applied to the partial differential equations for the free boundary problem and a finite dimensional approximation of the dynamical system is obtained. The model describes the shape of the evolving cut...

In this Letter we show how the scattering amplitudes of nonrelativistic one-particle Schrdinger operators with a scalar (not necessarily rotation invariant) potential may be obtained from the scattering cross-sections for the system where a scalar potential is added and whose scattering amplitudes are known explicitly.

15A15 Determinants, permanents, other special matrix functions (See also 19B10, 19B14) 35J10 Schrödinger operator (See also 35Pxx) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations 81U20 S-matrix theory, etc.

We consider a Rydberg atom under the influence of a short, strong laser pulse within the theoretical context of a Stark Hamiltonian. We prove an upper bound on the ionization probability involving the electric field only through the pulse duration tau and its total energy. In particular, for fixed energy this bound tends to zero with tau to 0.

The interaction of picosecond and sub-picosecond laser pulses with metals is investigated both, theoretically and experimentally. Analyzing the Boltzmann equation for electrons and phonons the hyperbolic two-temperature model of heat conduction in metals is obtained. In particular the parameter range for which the hyperbolic effects are significant...

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by...

We continue the study of cluster properties of spectral and scattering characteristics of Schrödinger operators with potentials given as a sum of two wells, begun in our preceding article [Rev. Math. Phys. 6 (1994) 833-853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separation of...

Methods from scattering theory are introduced to analyze random Schrodinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynami...

© 1998 Optical Society of America

We continue our investigation concerning the question of whether atomic bound states begin to stabilize in the ultra-intense field limit. The pulses considered are essentially arbitrary, but we distinguish between three situations. First the total classical momentum transfer is non-vanishing, second not both the total classical momentum transfer an...

We continue the study of cluster properties of spectral and scattering characteristics of Schrodinger operators with potentials given as a sum of two wells, begun in our preceeding article [Rev. Math. Phys. 6 (1994) 833 -- 853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separatio...

This article provides an extension to multiparticle systems (like atoms and molecules with many electrons) of nonperturbative results obtained previously by the authors in collaboration with A. Fring on the ionization of atomic bound states under the influence of short, ultra-intense laser pulses. We give upper and lower bounds which in particular...

Cutting materials with laser beams is a widely used technology. We develop a mathematical model in order to give a sufficiently accurate description of the main physical processes and their mutual interaction. This should contribute eventually to a reliable control of the technical process and enhancements of the quality of laser cuts. The full pro...

We address the question of whether atomic bound states begin to stabilize in the short ultra-intense field limit. We provide a general theory of ionization probability and investigate its gauge invariance. For a wide range of potentials we find an upper and lower bound by non-perturbative methods, which clearly exclude the possibility that the ultr...

We address the question of whether atomic bound states begin to stabilize in the short ultra-intense field limit. We provide a general theory of ionization probability and investigate its gauge invariance. For a wide range of potentials we find an upper and lower bound by non-perturbative methods, which clearly exclude the possibility that the ultr...

We provide a rigorous lower bound for the ionization probability of a Rydberg atom under the perturbation by a time-dependent electric field in the form of an ultrashort pulse idealized as a Dirac ffi -function. This estimate is of the form 1 Gamma O(F Gamma4 0 ) for large values of the electric field F 0 . For the hydrogen atom we also prove the s...

We prove that the integrated Krein's spectral shift function for one particle Schrödinger operators in R3 is concave with respect to the perturbation potential. The proof is given by showing that the spectral shift function is the limit in the distributional sense of the difference of the counting functions for the given Hamiltonian and the free Ha...

We continue the study of cluster properties of spectral and scattering characteristics of Schrodinger operators with potentials given as a sum of two wells, begun in our preceding article [Rev. Math. Phys. 6 (1994) 833-853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separation of...

We provide a rigorous lower bound for the ionization probability of a Rydberg atom under the perturbation by a time-dependent electric field in the form of an ultrashort pulse idealized as a Dirac delta function. This estimate is of the form 1-O(F-40) for large values of the electric field F0. For the hydrogen atom we also prove the scaling behavio...

Exact crossing of potential curves related to the adiabatic hyperspherical (AH) approach to a three-body Coulomb system is studied. Analytic struc- ture of the AH potential curves, harmonics, and coupling matrix elements near crossing points in the complex plane of hyperradius is investigated. Results are applied to derive some basic features of av...

As an approach to the highly excited states of a Coulomb three-body system, a Schro¨dinger operator on a hypersphere of radius &rgr;, the hyperradius of the system, is considered. A corresponding spectral problem is studied in the limit &rgr;↠∞, which is interpreted as semiclassical. For two particular models, the semiclassical quantization rules f...

The adiabatic hyperspherical (AH) approach to the quantum three-body problem is considered. It is proven that the AH harmonics are complete and differentiable with respect to the hyperradius for a large class of two-body potentials. For the case of short-range potentials, the scattering theory is studied in the framework of the AH approach. The asy...

The scattering theory for the Hamiltonian of the Stark effect is considered. A partial decomposition of the S-matrix is derived corresponding to separation of variables in the parabolic coordinates, and the analytic structure of the partial Jost functions and S-matrices are studied.

The adiabatic hyperspherical (AH) approach to the three-body Coulomb bound-state problems is considered. The variational method of computation of the AH harmonics potential curves and coupling matrix elements is developed. The method takes into account the asymptotic behaviour of the AH harmonics at large and small values of the hyperradius . The d...

For a system of three charged particles the Faddeev equations are derived in the total-angular-momentum representation. They have the form of coupled sets of partial differential equations in three-dimensional space and can be used to develop new efficient numerical procedures to tackle the three-body Coulomb problem. The asymptotic conditions at l...

The local structure of the Faddeev components of the wave function of an arbitrary system of three particles is studied in the vicinity of the ternary and binary collision points. The asymptotic expansions of the components are derived in the bispherical basis and in the representation of the total arbital angular momentum. These results can be use...

The problem of potential scattering in a homogeneous external electrostatic field is considered. The asymptotic behavior of the continuum wave functions in the configuration space is investigated. The scattering process is interpreted in terms of the trajectories of the asymptotic motion of the particles. The singularities of the scattering amplitu...

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