We consider a model for the distribution of a long homopolymer with an attractive zero-range pote... more We consider a model for the distribution of a long homopolymer with an attractive zero-range potential at the origin in R 3 (polymer pinning and de-pinning). The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length T of the polymer and a parameter γ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point γ cr = 0 the transition occurs from the globular phase (positive recurrent behavior of the polymer, γ > 0) to the extended phase (Brownian type behavior, γ < 0). The main result of the paper is a detailed analysis of the behavior of the polymer when γ is near γ cr. Our approach is based on analyzing the semigroups generated by the self-adjoint extensions L γ of the Laplacian on C ∞ 0 (R 3 \{0}) parametrized by γ, which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator L γ .
Page 181. Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Volume 42, 2007 On... more Page 181. Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Volume 42, 2007 On the Influence of Random Perturbations on the Propagation of Waves Described by a Periodic Schrodinger Operator ...
The aim of this paper is to investigate the distribution of a continuous polymer in the presence ... more The aim of this paper is to investigate the distribution of a continuous polymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously and independently vary two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which tends to infinity. We describe how the typical size of the polymer depends on the two parameters.
Integral Methods in Science and Engineering, Volume 1, 2009
The paper contains a simplified and improved version of the results obtained by the authors earli... more The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph. Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it.
Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymp... more Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in . The present paper contains a detailed review of [21] under some assumptions which make the results much more transparent. It also contains several new theorems on the structure of the spectrum near the threshold. small diameter asymptotics of the resolvent, and solutions of the evolution equation.
The paper contains mathematical justification of basic facts concerning the Brownian motor theory... more The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.
ABSTRACT This book gives a self-contained and up-to-date account of mathematical results in the l... more ABSTRACT This book gives a self-contained and up-to-date account of mathematical results in the linear theory of water waves. The study of waves has many applications, including the prediction of behavior of floating bodies (ships, submarines, tension-leg platforms etc.), the calculation of wave-making resistance in naval architecture, and the description of wave patterns over bottom topography in geophysical hydrodynamics. The first section deals with time-harmonic waves. Three linear boundary value problems serve as the approximate mathematical models for these types of water waves. The next section uses a plethora of mathematical techniques in the investigation of these three problems. The techniques used in the book include integral equations based on Green&#39;s functions, various inequalities between the kinetic and potential energy and integral identities which are indispensable for proving the uniqueness theorems. The so-called inverse procedure is applied to constructing examples of non-uniqueness, usually referred to as &#39;trapped nodes.&#39;
The problem of creating compact optical delay devices is studied. One of possible applications of... more The problem of creating compact optical delay devices is studied. One of possible applications of such a device is synchronizing the work of fast optical elements and much slower electronics. The goal is to find optical media where delay coexists with transparency. It will be shown that an optical line which can be approximated by a necklace-type quantum graph has this property. The work is joint with S. Molchanov.
The influence of disorder on the transmission through periodic waveguides is studied. Using a can... more The influence of disorder on the transmission through periodic waveguides is studied. Using a canonical form of the transfer matrix we investigate dependence of the Lyapunov exponent γ on the frequency ν and magnitude of the disorder σ. It is shown that in the bulk of the bands γ ∼ σ 2 , while near the band edges it has the order γ ∼ σ 2/3. This dependence is illustrated by numerical simulations.
We study the variance of the solution of a periodic randomly perturbed one-dimensional Schrödinge... more We study the variance of the solution of a periodic randomly perturbed one-dimensional Schrödinger operator after propagation through N periods. It is shown that if the frequency of propagation lies inside the band, then the total variance is proportional to N σ 2 , where σ is the intensity of the white noise. However, if the wave frequency is close to the band edge (where the transfer matrix has a Jordan block structure), the resulting variance is proportional to N σ 2/3. Thus, propagation becomes highly sensitive to random perturbations. Numerical simulations reveal that even low noise in a periodic potential can suppress transmission near the band edges and make it strongly irregular inside the band. Further increase of the noise amplitude leads to intermittent behaviour of the transmission coefficient, and makes transmission possible only for a few random frequencies in the band.
We will discuss a one-dimensional approximation for the problem of wave propagation in networks o... more We will discuss a one-dimensional approximation for the problem of wave propagation in networks of thin fibers. The main objective here is to describe the boundary (gluing) conditions at branching points of the limiting one-dimensional graph. The results will be applied to Mach-Zehnder interferometers on chips and to periodic chains of the interferometers. The latter allows us to find parameters which guarantee the transparency and slowing down of wave packets.
We consider the Helmholtz equation in the half space and suggest two methods for determining the ... more We consider the Helmholtz equation in the half space and suggest two methods for determining the boundary impedance from knowledge of the far field pattern of the time-harmonic incident wave. We introduce a potential for which the far field patterns in specially selected directions represent its Fourier coefficients. The boundary impedance is then calculated from the potential by an explicit formula or from the WKB approximation. Numerical examples are given to demonstrate efficiency of the approaches. We also discuss the validity of the WKB approximation in determining the impedance of an obstacle.
The inverse scattering problem of determining the boundary impedance from knowledge of the time h... more The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We solve the finite difference Helmholtz equation subject to the exact radiation condition for the discrete problem. The approach is decomposed into two steps. First, we reduce the problem to a well-posed system of linear equations for a modified potential. We next find the boundary impedance using the modified potential through an explicit formula. Thus, the computational part of the nonlinear problem of reconstruction of the boundary impedance is reduced to the solution of a linear system. Numerical examples are given to demonstrate efficiency of the new method.
This paper is devoted to the spectral theory of the Schrödinger operator on the simplest fractal:... more This paper is devoted to the spectral theory of the Schrödinger operator on the simplest fractal: Dyson's hierarchical lattice. An explicit description of the spectrum, eigenfunctions, resolvent and parabolic kernel are provided for the unperturbed operator, i.e., for the Dyson hierarchical Laplacian. Positive spectrum is studied for the perturbations of the hierarchical Laplacian. Since the spectral dimension of the operator under consideration can be an arbitrary positive number, the model allows a continuous phase transition from recurrent to transient underlying Markov process. This transition is also studied in the paper.
We consider wave propagation and scattering governed by one dimensional Schrödinger operators wit... more We consider wave propagation and scattering governed by one dimensional Schrödinger operators with truncated periodic potentials. The propagation of wave packets with narrow frequency supports is studied. The goal is to describe potentials for which the group velocity (for periodic problem) is small and the transmission coefficient for the truncated potential is not too small, i.e. to find media where a slowing down of the wave packets coexists with a transparency.
Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The ... more Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ. We calculate the Lagrangian gluing conditions at vertices v ∈ Γ for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.
We study the influence of disorder on propagation of waves in one-dimensional structures. Transmi... more We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.
We consider a model for the distribution of a long homopolymer with an attractive zero-range pote... more We consider a model for the distribution of a long homopolymer with an attractive zero-range potential at the origin in R 3 (polymer pinning and de-pinning). The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length T of the polymer and a parameter γ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point γ cr = 0 the transition occurs from the globular phase (positive recurrent behavior of the polymer, γ > 0) to the extended phase (Brownian type behavior, γ < 0). The main result of the paper is a detailed analysis of the behavior of the polymer when γ is near γ cr. Our approach is based on analyzing the semigroups generated by the self-adjoint extensions L γ of the Laplacian on C ∞ 0 (R 3 \{0}) parametrized by γ, which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator L γ .
Page 181. Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Volume 42, 2007 On... more Page 181. Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Volume 42, 2007 On the Influence of Random Perturbations on the Propagation of Waves Described by a Periodic Schrodinger Operator ...
The aim of this paper is to investigate the distribution of a continuous polymer in the presence ... more The aim of this paper is to investigate the distribution of a continuous polymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously and independently vary two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which tends to infinity. We describe how the typical size of the polymer depends on the two parameters.
Integral Methods in Science and Engineering, Volume 1, 2009
The paper contains a simplified and improved version of the results obtained by the authors earli... more The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph. Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it.
Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymp... more Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in . The present paper contains a detailed review of [21] under some assumptions which make the results much more transparent. It also contains several new theorems on the structure of the spectrum near the threshold. small diameter asymptotics of the resolvent, and solutions of the evolution equation.
The paper contains mathematical justification of basic facts concerning the Brownian motor theory... more The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.
ABSTRACT This book gives a self-contained and up-to-date account of mathematical results in the l... more ABSTRACT This book gives a self-contained and up-to-date account of mathematical results in the linear theory of water waves. The study of waves has many applications, including the prediction of behavior of floating bodies (ships, submarines, tension-leg platforms etc.), the calculation of wave-making resistance in naval architecture, and the description of wave patterns over bottom topography in geophysical hydrodynamics. The first section deals with time-harmonic waves. Three linear boundary value problems serve as the approximate mathematical models for these types of water waves. The next section uses a plethora of mathematical techniques in the investigation of these three problems. The techniques used in the book include integral equations based on Green&#39;s functions, various inequalities between the kinetic and potential energy and integral identities which are indispensable for proving the uniqueness theorems. The so-called inverse procedure is applied to constructing examples of non-uniqueness, usually referred to as &#39;trapped nodes.&#39;
The problem of creating compact optical delay devices is studied. One of possible applications of... more The problem of creating compact optical delay devices is studied. One of possible applications of such a device is synchronizing the work of fast optical elements and much slower electronics. The goal is to find optical media where delay coexists with transparency. It will be shown that an optical line which can be approximated by a necklace-type quantum graph has this property. The work is joint with S. Molchanov.
The influence of disorder on the transmission through periodic waveguides is studied. Using a can... more The influence of disorder on the transmission through periodic waveguides is studied. Using a canonical form of the transfer matrix we investigate dependence of the Lyapunov exponent γ on the frequency ν and magnitude of the disorder σ. It is shown that in the bulk of the bands γ ∼ σ 2 , while near the band edges it has the order γ ∼ σ 2/3. This dependence is illustrated by numerical simulations.
We study the variance of the solution of a periodic randomly perturbed one-dimensional Schrödinge... more We study the variance of the solution of a periodic randomly perturbed one-dimensional Schrödinger operator after propagation through N periods. It is shown that if the frequency of propagation lies inside the band, then the total variance is proportional to N σ 2 , where σ is the intensity of the white noise. However, if the wave frequency is close to the band edge (where the transfer matrix has a Jordan block structure), the resulting variance is proportional to N σ 2/3. Thus, propagation becomes highly sensitive to random perturbations. Numerical simulations reveal that even low noise in a periodic potential can suppress transmission near the band edges and make it strongly irregular inside the band. Further increase of the noise amplitude leads to intermittent behaviour of the transmission coefficient, and makes transmission possible only for a few random frequencies in the band.
We will discuss a one-dimensional approximation for the problem of wave propagation in networks o... more We will discuss a one-dimensional approximation for the problem of wave propagation in networks of thin fibers. The main objective here is to describe the boundary (gluing) conditions at branching points of the limiting one-dimensional graph. The results will be applied to Mach-Zehnder interferometers on chips and to periodic chains of the interferometers. The latter allows us to find parameters which guarantee the transparency and slowing down of wave packets.
We consider the Helmholtz equation in the half space and suggest two methods for determining the ... more We consider the Helmholtz equation in the half space and suggest two methods for determining the boundary impedance from knowledge of the far field pattern of the time-harmonic incident wave. We introduce a potential for which the far field patterns in specially selected directions represent its Fourier coefficients. The boundary impedance is then calculated from the potential by an explicit formula or from the WKB approximation. Numerical examples are given to demonstrate efficiency of the approaches. We also discuss the validity of the WKB approximation in determining the impedance of an obstacle.
The inverse scattering problem of determining the boundary impedance from knowledge of the time h... more The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We solve the finite difference Helmholtz equation subject to the exact radiation condition for the discrete problem. The approach is decomposed into two steps. First, we reduce the problem to a well-posed system of linear equations for a modified potential. We next find the boundary impedance using the modified potential through an explicit formula. Thus, the computational part of the nonlinear problem of reconstruction of the boundary impedance is reduced to the solution of a linear system. Numerical examples are given to demonstrate efficiency of the new method.
This paper is devoted to the spectral theory of the Schrödinger operator on the simplest fractal:... more This paper is devoted to the spectral theory of the Schrödinger operator on the simplest fractal: Dyson's hierarchical lattice. An explicit description of the spectrum, eigenfunctions, resolvent and parabolic kernel are provided for the unperturbed operator, i.e., for the Dyson hierarchical Laplacian. Positive spectrum is studied for the perturbations of the hierarchical Laplacian. Since the spectral dimension of the operator under consideration can be an arbitrary positive number, the model allows a continuous phase transition from recurrent to transient underlying Markov process. This transition is also studied in the paper.
We consider wave propagation and scattering governed by one dimensional Schrödinger operators wit... more We consider wave propagation and scattering governed by one dimensional Schrödinger operators with truncated periodic potentials. The propagation of wave packets with narrow frequency supports is studied. The goal is to describe potentials for which the group velocity (for periodic problem) is small and the transmission coefficient for the truncated potential is not too small, i.e. to find media where a slowing down of the wave packets coexists with a transparency.
Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The ... more Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ. We calculate the Lagrangian gluing conditions at vertices v ∈ Γ for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.
We study the influence of disorder on propagation of waves in one-dimensional structures. Transmi... more We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.
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