sufficient conditions for conservativity of quantum dynamical semigroups abstract we give sufficient conditions for the conservativity of a quantum dynamical semi group these conditions are expressed as upper bounds of quadratic forms

on three dimensional quasi stackel hamiltonians a three dimensional integrable generalization of the stackel systems is proposed a classification of such systems is obtained which results in two families the first family is the direct sum of the two dimensional system which is equivalent to the representation of the schottky manakov top in the quasi stackel form and a stackel one dimensional system the second family is probably a new three dimensional system the system of hydrodynamic type which we get from this family in the usual way is a three dimensional generalization of the gibbons tsarev system a generalization of the quasi stackel systems to the case of any dimension is discussed

sufficient conditions for conservativity of minimal quantum dynamical semigroups abstract the conservativity of a minimal quantum dynamical semigroup is proved whenever there exists a generalized subharmonic operator bounded from below by the dissipative part of the infinitesimal generator we discuss applications of this criteria in mathematical physics and quantum probability

quasi lagrangian systems of newton equations systems of newton equations of the form q 12a 1 q k with an integral of motion quadratic in velocities are studied these equations generalize the potential case when a i the identity matrix and they admit a curious quasi lagrangian formulation which differs from the standard lagrange equations by the plus sign between terms a theory of such quasi lagrangian newton qln systems having two functionally independent integrals of motion is developed with focus on two dimensional systems such systems admit a bi hamiltonian formulation and are proved to be completely integrable by embedding into five dimensional integrable systems they are characterized by a linear second order partial differential equation pde which we call the fundamental equation fundamental equations are classified through linear pencils of matrices associated with qln systems the theory is illustrated by two classes of systems separable potential systems and driven systems new separation variables for driven systems are foun

on dissipative stochastic equations in a hilbert space a general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a hilbert space is proved the dual equation is introduced and the duality relation is established proofs take inspirations from quantum stochastic calculus however without using it solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup describing quantum markovian evolution the problem of the mean square norm conservation closely related to the unitality non explosion of the quantum dynamical semigroup is considered and a hyperdissipativity condition ensuring such conservation is discussed comments are given on the existence of solutions of a nonlinear stochastic differential equation introduced and discussed recently in physical literature in connection with continuous quantum measurement processes

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

heisenberg evolution of quantum observables represented by unbounded operators this paper deals with open quantum systems in particular we focus on the adjoint quantum master equations with initial conditions given by unbounded operators examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many body particle systems the article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the born markov approximations to this end we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of schrodinger type furthermore we examine quantum dynamical semigroup properties of the heisenberg evolutions of general classes of observables

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

the construction of canonical separated variables for binary constrained akns flow abstract for separation of variables of a binary constrained flow with 2 n degrees of freedom 2 n pairs of canonical variables need to be constructed the method available can lead to n pairs of canonical separated variables by using its lax matrix we propose a new method for introducing n pairs of additional canonical separated variables and additional separated equations furthermore the jacobi inversion problem for the binary constrained flow is presented

separation of variables for soliton equations via their binary constrained flows binary constrained flows of soliton equations admitting 2 2 lax matrices have 2n degrees of freedom which is twice as many degrees of freedom than in the case of monoconstrained flows by using the normal method their lax matrices directly give rise to first n pairs of canonical separated variables for their separation of variables we propose a new method to introduce the other n pairs of canonical separated variables and additional separated equations the jacobi inversion problems for binary constrained flows are established finally the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the jacobi inversion problems for some soliton hierarchies

bi quasi hamiltonian systems a general notion of bi quasi hamiltonian systems is introduced and is related to previous work on various special cases of such systems

quantum dynamical semigroups for diffusion models with hartree interaction we consider a class of evolution equations in lindblad form which model the dynamics of dissipative quantum mechanical systems with mean field interaction particularly this class includes the so called quantum fokker planck poisson model the existence and uniqueness of global in time mass preserving solutions is proved thus establishing the existence of a nonlinear conservative quantum dynamical semigroup the mathematical difficulties stem from combining an unbounded lindblad generator with the hartree nonlinearity

separation of variables for the ruijsenaars system we construct a separation of variables for the classical n particle ruijsenaars system the relativistic analog of the elliptic calogero moser system the separated coordinates appear as the poles of the properly normalised eigenvector baker akhiezer function of the corresponding lax matrix two different normalisations of the ba functions are analysed the canonicity of the separated variables is verified with the use of the r matrix technique the explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n 2 and n 3 taking the nonrelativistic limit we also construct a separation of variables for the elliptic calogero moser system

conductance and resistance jumps in finite size random resistor networks when bonds are removed one by one and at random from a finite size resistor network the conductance or resistance does not change continuously but rather a sequence of conductance resistance jumps for various sizes occurs the larger jumps arise from those bonds which carry a relatively large current just before they are cut the authors report on numerical simulations of these jumps of a random resistor network on the square lattice they also give a scaling argument to account for this phenomenon which yields the number of conductance jumps larger than delta g scaling as delta g lambda g with lambda g d nu d nu t where t is the conductivity exponent and nu is the percolation correlation length exponent equivalently the number of resistance jumps greater than delta r scales as delta r lambda r with lambda r d nu d nu t these predictions account for the data on a qualitative level only however and they discuss some possible mechanisms for the quantitative discrepancies

separation of variables in quasi potential systems of bi cofactor form we perform variable separation in the quasi potential systems of equations of the form q a 1 k a 1 k w herea and a are killing tensors by embedding these systems into a bi hamiltonian chain and by calculating the corresponding darboux nijenhuis coordinates on the symplectic leaves of one of the hamiltonian structures of the system we also present examples of the corresponding separation coordinates in two and three dimensions

quasi bi hamiltonian structures complex functions and superintegrability the tremblay turbiner winternitz ttw and the post winternitz pw systems the existence of quasi bi hamiltonian structures for the tremblay turbiner winternitz ttw and the post winternitz pw systems is studied we first recall that the superintegrability of these two systems is related with the existence of certain complex functions endowed with interesting poisson bracket properties and then we prove the existence of several quasi bi hamiltonian structures making use of these complex functions this is done in two steps first with complex 2 forms wedge product of the differentials of the complex functions and then with several real 2 forms the properties of these geometric structures and the associated recursion operators are also analyzed the paper can be considered as divided in two parts in the first part a we study the ttw system related to the harmonic oscillator and in the second part we study the pw system related to the kepler problem

bi hamiltonian representation of stackel systems it is shown that linear separation relations are fundamental objects for integration by quadratures of stackel separable liouville integrable systems the so called stackel systems these relations are further employed for the classification of stackel systems moreover we prove that any stackel separable liouville integrable system can be lifted to a bi hamiltonian system of gel fand zakharevich type in conjunction with other known result this implies that the existence of bi hamiltonian representation of liouville integrable systems is a necessary condition for stackel separability

a class of nonconservative lagrangian systems on riemannian manifolds we generalize results of rauch wojciechowski marciniak and lundmark concerning a class of nonconservative lagrangian systems from the euclidean to the riemannian case

generators of quantum markov semigroups quantum markov semigroups qmss originally arose in the study of the evolutions of irreversible open quantum systems mathematically they are a generalization of classical markov semigroups where the underlying function space is replaced by a non commutative operator algebra in the case when the qms is uniformly continuous theorems due to the works of lindblad commun math phys 48 119 130 1976 stinespring proc am math soc 6 211 216 1955 and kraus ann phys 64 311 335 1970 imply that the generator of the semigroup has the form l a n 1 vn avn ga ag where vn and g are elements of the underlying operator algebra in the present paper we investigate the form of the generators of qmss which are not necessarily uniformly continuous and act on the bounded operators of a hilbert space we prove that the generators of such semigroups have forms that reflect the results of lindblad and stinespring we also make some progress towards forms reflecting kraus result finally we look at