Search

Asymptotic synchronization in n-dimensional dissipative lattice of coupled oscillators with Neumann boundary condition was considered by decomposing phase space into two orthogonal subspaces.

Consider an attractor of a dissipative non-autonomous system with one angle coordinate. We give conditions for this attractor to be homeomorphic to the circle where we find connections with the work of R. A. Smith. Several applications are studied, such as: the forced pendulum, discretizations of the sine-Gordon equation, n'th order equations, among others.

By employing Chebotarev and Fagnola's sufficient conditions for conservativity of minimal quantum dynamical semigroups [7, 8], we construct the conservative minimal quantum dynamical semigroups generated by noncommutative elliptic operators in the sense of [2]. We apply our results to concrete examples.

A space of boundary values is constructed for the minimal symmetric operator generated by an infinite Jacobi matrix in the limit-circle case. A description of all maximal dissipative, accretive and selfadjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and define its characteristic function. We prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative operators.

In a totally dissipative behavior, all non-trivial trajectories dissipate energy. A characterization of such behaviors is given in terms of properties of the one- and two-polynomial matrices associated with the supply rate and with their kernel- and image representation. (c) 2004 Elsevier B.V. All rights reserved.

We consider nonself-adjoint singular Sturm–Liouville boundary-value problems in the limit-circle case with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary-value problem. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations that make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and specify its characteristic function in terms of solutions of the corresponding Sturm–Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and the Sturm–Liouville boundary-value problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Dissipative singular Sturm–Liouville operators are studied in the Hilbert space L w 2 [a,b) (−∞

A space of boundary values is constructed for minimal symmetric operator, generated by discrete Hamiltonian system, acting in the Hilbert space l^2"A (@? E @? E) (@?={0, +/-1, +/-2, ...}, dim E = n < ~) with deficiency indices (n, n) (in limit-circle case at +/-~ and limit point case at @?~). A description of all maximal dissipative, maximal accretive, and self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at +/-~. We investigate two classes of maximal dissipative operators with boundary conditions, called 'dissipative at -~' and 'dissipative at ~'. In each of these cases we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation in terms of the Titchmarsh-Weyl matrix-valued function of the self-adjoint operator. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the scattering matrix of dilation. Finally, we prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

A space of boundary values is constructed for minimal symmetric Dirac operator in the Hilbert space \(L_A^2 (( - \infty ,\infty );\mathbb{C}^2 )\) with defect index (2,2) (in Weyl’s limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with separated boundary conditions, called ‘dissipative at −∞’ and ‘dissipative at +∞’. In each of these cases we construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix. We construct a functional model of the maximal dissipative operator and define its characteristic function. We prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

Dissipative Schrodinger operators with a matrix potential are studied in L 2 ( ( 0 , ∞ ) ; E ) ( dim E = n ∞ ) which are extension of a minimal symmetric operator L 0 with defect index ( n , n ) . A selfadjoint dilation of a dissipative operator is constructed, using the Lax–Phillips scattering theory, the spectral analysis of a dilation is carried out, and the scattering matrix of a dilation is founded. A functional model of the dissipative operator is constructed and its characteristic function's analytic properties are determined, theorems on the completeness of eigenvectors and associated vectors of a dissipative Schrodinger operator are proved.

A one-dimensional coupled drift-diffusion dissipative Schrodinger model (hybrid model) is mathematically analyzed. The device domain is split into two parts: one in which the transport is well described by the drift-diffusion equations (classical zone) and another in which a quantum description via dissipative Schrodinger equations (quantum zone) is used. Both system are coupled such that the continuity of the current densities is guaranteed. The electrostatic potential is self-consistently determined by Poisson's equation on the whole device domain. We show that the hybrid model is well posed, and we prove existence of solutions and show their uniform boundedness, provided the distribution functions satisfy a so-called balance condition. The current densities are different from zero in the nonequilibrium case and are uniformly bounded.

Theory of the Krein string is considered in the framework of a continual analog of the theory of unitary operator nodes. On this basis, a generalization of the M. G. Krein-A. A. Nudelman result on the completeness of the string operator is obtained. Bibliography: 28 titles.

We consider the equation \(\ddot x + B\dot x + Ax = 0\) in a Hilbert space ℋ, where A is a uniformly positive self-adjoint operator and B is a dissipative operator. The main result is the proof of a theorem stating the exponential energy decay for solutions of this equation (or the exponential stability of the semigroup associated with the equation) under the additional assumption that B is sectorial and is subordinate to A in the sense of quadratic forms.

We give necessary and sufficient conditions under which a C-0-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded. (C) 2004 Elsevier Inc. All rights reserved.

Maximal dissipative Schrodinger operators are studied in L2((−∞,∞);E) (dimE=n

The mechanism of generation and annihilation of attractors during transition from a Hamiltonian system to a dissipative system is studied numerically using the dissipative standard map. The transient process related to the formation of attracting basins of periodic attractors is studied by discussing the evolution of the KAM tori of the standard map. The result shows that as damping increases, attractors are mainly generated from elliptic orbits of the Hamiltonian system and annihilated by colliding with unstable periodic orbits originating from the corresponding hyperbolic orbits of the Hamiltonian system. The transient process also exhibits the general feature of bifurcation.

Dissipative singular Dirac operators are studied in the space L A 2 ([a,b); C 2 ) (−∞ , that the extensions of a minimal symmetric operator in Weyl's limit-point case. We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and specify its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator. We prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative Dirac operators.

We study a stationary Schrödinger–Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows us to model a non-zero current through the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.

Let X be a uniformly smooth real Banach space and T : X → X be a generalized Lipschitzian and m-accretive operator. It is proved that under suitable conditions, the Ishikawa iterative schemes with errors both converges strongly to the unique solution of the nonlinear operator equation x + Tx = f and is almost stable. A few related results deal with the operator equations for the dissipative type operator. The results presented in this paper extend substantially the corresponding results in [1,2].

x0. Introduction In this paper we construct a new linearly similar functional model for linear operators and study its elementary properties. This model generalizes the Sz.-Nagy{Foia s model for C0-contractions and also forC0-dissipative operators. We shall not restrict ourselves to the disk or the half-plane: the model will be constructed in a fairly arbitrary domain. The reduction of an operator to an \almost diagonal" model form will be written directly via the resolvent of the operator. Attention will be focused on the case of C00-operators. The main results of this paper were announced in [76]. Let be a union of nitely many piecewise-smooth contours in the complex plane C. Suppose that the sets int ,ext are open and have empty intersection,C =int[ [ext, and = @int = @ext (here we omit a certain technical condition on ). The main objects of this paper are model spacesH() that will be associated with operator-valued bounded analytic functions dened on int and having some special properties. Consider the special case where 2 H 1 int;L(R;R) , 1 is a meromorphic function on int ,a ndk 1 () kC a.e. on . Here R and R are Hilbert spaces, and L(R;R) denotes the space of all bounded linear operators from R to R .W e always consider only separable Hilbert spaces. Let E 2 (ext;R) denote the Smirnov class of functions on ext whose values are vectors in the space R (seex2). Suppose that int and ext are unbounded. In this special case the model spaceH() consists of all functions f meromorphic in int, holomorphic in ext, and satisfying fjext2 E 2 (ext;R) ;fjint2 E 2 (int;R) ;f int = fext a.e. on : This is a Hilbert space. The general denition ofH() will be given inx2. Ap air (A;J) of linear operators (possibly, unbounded) will be called a 2-system if 1) A is a closed operator in a Hilbert space H with nonempty eld of regularity (A )= Cn(A) and with domainD(A )( here(A) is the spectrum of the operator A); 2) J : D(J)! R ,w hereD(J )= D(A) H and J is bounded in the graph norm kxkG def = kxk 2 +kAxk 2 1=2 inD(A). Here R is a Hilbert space.

The baker map is investigated by two different theories of irreversibility by Prigogine and his colleagues, namely, theΛ-transformation and complex spectral theories, and their structures are compared. In both theories, the evolution operatorU†of observables (the Koopman operator) is found to acquire dissipativityby restrictingobservables to an appropriate subspaceΦof the Hilbert spaceL2of square integrable functions. Consequently, its spectral set contains an annulus in the unit disc. However, the two theories are not equivalent. In theΛ-transformation theory, a bijective mapΛ†−1:Φ→L2is looked for and the evolution operatorUof densities (the Frobenius-Perron operator) is transformed to a dissipative operatorW=ΛUΛ−1. In the complex spectral theory, the class of densities is restricted further so that most values in the interior of the annulus are removed from the spectrum, and the relaxation of expectation values is described in terms of a few point spectra in the annulus (Pollicott-Ruelle resonances) and faster decaying terms.

A space of boundary values is constructed for minimal symmetric second-order difference operator in the Hilbert space I_{w}^{2}(Z) {\rm (Z := \{ 0, \pm 1, \pm 2, \ldots \} )} with defect index (2,2) (in Weyl's limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. In particular, if we consider separated boundary conditions, that at -∞ and ∞ nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of maximal dissipative operator and determine its characteristic function. We prove a theorem on completeness of the system...

Earlier results 1. , 2. , 3. , 4. are developed in application to a certain special class of non-conservative mechanical systems in which the matrices of dissipative and non-conservative forces are singular. For this class of systems, necessary and sufficient conditions are formulated for reducing the initial matrix equation to a form that admits of direct application of the Kelvin-Chetayev theorems. An example is presented.

We propose a new notion of weak solutions (dissipative solutions) for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations.

We consider the maximal dissipative operators icting in the Hilbert space l"c^2(N;E) (N = {0,1, 2, ... ~, dim E = n < ~) that the extensions of a minimal symmetric operator with maximal deficiency indices (n, n) (in completely indeterminate case or limit-circle case) generated by an infinite Jacobi matrix with matrix entries. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the scattering matrix of the dilation. We prove the theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators. 2003 Elsevier Ltd. All rights reserved.

We study the exterior initial–boundary value problem for the linear dissipative wave equation ( □ + ∂ t ) u = 0 in Ω × ( 0 , ∞ ) with ( u , ∂ t u ) | t = 0 = ( u 0 , u 1 ) and u | ∂ Ω = 0 , where Ω is an exterior domain in N-dimensional Euclidean space R N . We first show higher local energy decay estimates of the solution u ( t ) , and then, using the cut-off technique together with those estimates, we can obtain the L 1 estimate of the solution u ( t ) when N ⩾ 3 , that is, ‖ u ( t ) ‖ L 1 ( Ω ) ⩽ C ( ‖ u 0 ‖ H n ( Ω ) + ‖ u 1 ‖ H n − 1 ( Ω ) + ‖ u 0 ‖ W n , 1 ( Ω ) + ‖ u 1 ‖ W n − 1 , 1 ( Ω ) ) for t ⩾ 0 , where n = [ N / 2 ] is the integer part of N / 2 . Moreover, by induction argument, we derive the higher energy decay estimates of the solution u ( t ) for t ⩾ 0 .

We consider a semilinear differential stochastic equation in a Hilbert space $H$ with a dissipative and Lipschitz nonlinearity. We study the corresponding transition semigroup in a space $L^2(H,\nu)$ where $\nu$ is an invariant measure.

A space of boundary values is constructed for the minimal symmetric singular Sturm-Liouville operator in the Hilbert space \(L_{w}^{2}(a,b)(-\infty \le a < b \le \infty )\) with defect index (2,2) (in Weyl’s limit-circle cases at singular points a and b). A description of all maximal dissipative, maximal accretive, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at a and b. We investigate maximal dissipative operators with, generally speaking, nonseparated boundary conditions. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We also construct a functional model of a dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.

This paper is devoted to a nonlinear averaging principle for periodic solutions of a class of second order inclusions. In addition an existence theorem for periodic solutions of such inclusions is established. This work which complements the abstract nonlinear averaging principle worked out in Couchouron and Kamenski (Nonlin. Anal. 42 (2000) 1101) makes a synthesis of the methods contained in Couchouron and Kamenski and Couchouron and Precup (Electron. J. Differential, Equations 4 (2002) 1) and represents a (nonvariational) topological approach for boundary values problems.

A space of boundary values is constructed for minimal symmetric Dirac operator in L A 2 ((−∞,∞); C 2 ) with defect index (2,2) (in Weyl's limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint, and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate maximal dissipative operators with, generally speaking, nonseparated (nondecomposed) boundary conditions. In particular, if we consider separated boundary conditions, at ±∞ the nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Dirac operators.

A space of boundary values is constructed for symmetric discrete Dirac operators in l 2 A (Z: C 2 )(Z := {0, ±1, ±2,...}) with defect index (1, 1) (in Weyl's limit-circle case at ±∞ and limit-point case at ∓∞). A description of all maximal dissipative (accretive), self-adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with boundary conditions, called 'dissipative at -∞ and 'dissipative at ∞'. In each of these cases we construct a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh-Weyl function of the self-adjoint operator. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative operators.

Generation of nonlinear operator semigroups and nonlinear evolution operators are proved by a different method, based on the theory of difference equations.

We study in detail Schrodinger–type operators on a bounded interval of R with dissipative boundary conditions. The characteristic function of this operator is computed, its minimal self–adjoint dilation is constructed and the generalized eigenfunction expansion for the dilation is developed. The problem is motivated by semiconductor physics.

In this paper, we consider equations of the form \(\user1{\ddot x}\user2{ + }B\user1{\dot x}\user2{ + }A\user1{x} = 0\), where \(\user1{x}\user2{ = }\user1{x}\left( \user1{t} \right)\) is a function with values in the Hilbert space \(\mathcal{H}\), the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in \(\mathcal{H}\). The linear operator \(\mathcal{T}\) generating the C0-semigroup in the energy space \({\mathcal{H}}_1 \times {\mathcal{H}}\) is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.

We present a new variational method, based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show that indeed the stationary state is often well described by a MPO of very moderate dimensions., Accepted version

The dissipative Zakharov system which models the propagation of Langmuir waves in plasmas is considered on the interval [0, L]. We are interested in the case of large ion acoustic speed λ. After the formal limiting transition λ → ∞ this system turns into the coupling system of the parabolic and Schrödinger equations. We prove that this limit system has a solution and generates a dissipative dynamical system possessing a global compact attractor. Our main result is the upper semicontinuity of the attractor as λ → ∞. Рассмотрена диссипативная система уравнений Захарова на промежутке [0, L], которая моделирует распространение ленгмюровских волн в плазме. Исследован случай большой акустической скорости ионов λ. После формального предельного перехода λ → ∞ система Захарова превращается в новую систему, которая состоит из параболического уравнения и уравнения Шредингера. Доказывается, что полученная система имеет глобальное решение и порождает диссипативную динамическую систему, которая обладает компактным глобальным аттрактором. Основным результатом является доказательство верхней полунепрерывности аттрактора при λ → ∞ . Розглянуто дисипативну систему рівнянь Захарова на проміжку [0, L], яка моделює розповсгодження ленгмюрівських хвиль у плазмі. Досліджено випадок великої акустичної швидкості іонів λ. Після формального граничного переходу λ → ∞ система Захарова перетворюється у нову систему, яка складається з параболічного рівняння та рівняння Шредінгера. Доведено, що отримана система має глобальний розв'язок та породжує дисипативну динамічну систему, яка має компактний глобальний атрактор. Головним результатом є доведення верхньої напівнеперервності атрактора при λ → ∞.

Complex biological systems, like the photosynthetic units of bacteria or plants, usually consist of several pigment molecules, which form aggregates, embedded in a protein environment. Within a quantum mechanical treatment it is practically impossible to take all degrees of freedom (DOFs) of such large systems explicitly into account. Note that there exist highly efficient methods, e.g., the multiconfiguration time-dependent Hartree method (MCTDH) [38], to treat the quantum dynamics of isolated systems, e.g., the intra-molecular dynamics of the aforementioned pigment molecules.

In the present paper, existence of dissipative solutions of the Jeffreys-Oldroyt-alpha equation is studied. The result is obtained by means of the approximation-topological approach for hydrodynamics problems.

In this paper we analyse the spectrum of the dissipative Schrodinger operator on binary tree-shaped networks. As applications, we study the stability of the Schrodinger system using a Riesz basis as well as the transfer function associated to the system.

We discuss the classes C,M, and S of analytic functions that can be realized as the Livsic characteristic functions of a symmetric densely defined operator \( \dot{A} \) with deficiency indices (1, 1), the Weyl–Titchmarsh functions associated with the pair (A,\( \dot{A} \)) where A is a self-adjoint extension of \( \dot{A} \), and the characteristic function of a maximal dissipative extension  of \( \dot{A} \), respectively. We show that the class M is a convex set, both of the classes S and C are closed under multiplication and, moreover, C ⊂ S is a double-sided ideal in the sense that S•C = C•S ⊂ S. The goal of this paper is to obtain these analytic results by providing explicit constructions for the corresponding operator realizations. In particular, we introduce the concept of an operator coupling of two unbounded maximal dissipative operators and establish an analog of the Livsic–Potapov multiplication theorem [14] for the operators associated with the function classes C and S. We also establish that the modulus of the von Neumann parameter characterizing the domain of  is a multiplicative functional with respect to the operator coupling.

We recall some basic definitions and properties of functional analysis: unbounded dissipative operators, the Hille-Yosida semi-group generation theorem, local existence and uniqueness for semilinear problems, with applications to the heat equation and the wave equation in a bounded spatial domain. These preliminaries are useful to define the dynamical systems associated to semilinear PDE of parabolic or hyperbolic type.

We regard a current flow through an open one-dimensional quantum system which is determined by a dissipative Schrodinger operator. The imaginary part of the corresponding form originates from Robin boundary conditions with certain complex valued coefficients imposed on Schrodinger’s equation. This dissipative Schrodinger operator can be regarded as a pseudo-Hamiltonian of the corresponding open quantum system. The dilation of the dissipative operator provides a (self-adjoint) quasi-Hamiltonian of the system, more precisely, the Hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system. The carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. Finally a rigorous setup of a dissipative Schrodinger–Poisson system is outlined.

Generation of C0 semigroups and nonlinear operator semigroups are proved by a different method, based on the theory of difference equations.

Conditions for nonlocal existence of a continuous storage function for nonlinear dissipative system are presented. More precisely, it is shown that under the local /spl omega/-uniform reachability assumption at one point x/sub */ the required supply function is continuous on the set of points reachable from x. Conditions for the local /spl omega/-uniform reachability based on the local controllability properties of the system are provided.

Numerically exact solutions for the quantum rate of potential barrier crossing in dissipative systems are only possible for highly idealized systems. It is, therefore, of interest to develop approximate theories of more general applicability. In this paper we formulate a mixed quantum classical thermodynamical rate theory for dissipative systems. The theory consists of two parts. The evaluation of a thermal flux and the computation of the classically evolved product projection operator. Since the dividing surface is perpendicular to the unstable normal mode of the dissipative system, we reformulate the theory in terms of the unstable normal mode and a collective bath mode. The influence functional for the thermal flux matrix elements in this representation is derived. The classical mechanics are reformulated in terms of the same two degrees of freedom. The one-dimensional Langevin equation for the system coordinate is replaced by a coupled set of Langevin equations for the unstable normal mode and the collective bath mode. The resulting rate expression is given in the continuum limit, so that computation of the rate does not necessitate a discretization of the bath modes. To overcome the necessity of computing a multidimensional Fourier transform of the matrix elements of the thermal flux operator, we adapt, as in previous studies, a method of Creswick [Mod. Phys. Lett. B 9, 693 (1995)], by which only a one-dimensional Fourier transform is needed. This transform is computed by quadrature. The resulting theory is tested against the landmark numerical results of Topaler and Makri [J. Chem. Phys. 101, 7500 (1994)] obtained for barrier crossing in a symmetric double well potential. We find that mixed quantum classical rate theory (MQCLT) provides a substantial improvement over our previous quantum transition state theory as well as centroid transition state theory computations and is in overall good agreement with the exact results.

Some classes of dissipative and Hamiltonian distributed systems are described. The dynamics of these systems is effectively reduced to finite-dimensional dynamics which can be “unboundedly complex” in a sense. Yarying the parameters of these systems, we can obtain an arbitrary (to within the orbital topological equivalence) structurally stable attractor in the dissipative case and an arbitrary polynomial weakly integrable Hamiltonian in the conservative case. As examples, we consider Hopfield neural networks and some reaction–diffusion systems in the dissipative case and a nonlinear string in the Hamiltonian case.