heat release by controlled continuous time markov jump processes we derive the equations governing the protocols minimizing the heat released by a continuous time markov jump process on a one dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon in particular we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the markov jump problem obey a system of differential equations of hamilton jacobi bellman type as the state space mesh tends to zero these equations converge to those satisfied by the diffusion process minimizing the heat released in the langevin formulation of the same problem we also show that in full analogy with the continuum case heat minimization is equivalent to entropy production minimization thus our results may be interpreted as a refined version of the second law of thermodynamics

the evolution operator technique in solving the schrodinger equation and its application to disentangling exponential operators and solving the problem of a mass varying harmonic oscillator the authors present a method for finding the evolution operator for the schrodinger equation for the hamiltonian expressible as h t a1 t j4 a2 t j0 a3 t j where j j0 and j are the su 2 group generators such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary as a demonstration of our general approach they solved the problem of a harmonic oscillator with a varying mass

on the use of stochastic differential geometry for non equilibrium thermodynamic modeling and control we discuss the relevance of geometric concepts in the theory of stochastic differential equations for applications to the theory of non equilibrium thermodynamics of small systems in particular we show how the eells elworthy malliavin covariant construction of the wiener process on a riemann manifold provides a physically transparent formulation of optimal control problems of finite time thermodynamic transitions based on this formulation we turn to an evaluative discussion of recent results on optimal thermodynamic control and their interpretation

procrustes problems in finite dimensional indefinite scalar product spaces abstract some applications from multidimensional scaling in an environment of indefinite scalar products are investigated in particular the construction of points from given distances is considered and two variants of procrustes problems are discussed it is found that both procrustes problems can be solved with the help of h polar decompositions or g h polar decompositions of specified matrices furthermore a method for the numerical computation of h or g h polar decompositions of nonsingular matrices is described

solving balanced procrustes problem with some constraints by eigenvalue decomposition the balanced procrustes problem with some special constraints such as symmetric orthogonality and symmetric idempotence are considered by one time eigenvalue decomposition of the matrix product generated by the matrices a and b the constrained solutions are constructed simply similar strategy is applied to the problem with the corresponding p commuting constraints with a given symmetric matrix p numerical examples are presented to show the efficiency of the proposed methods

j orthogonal matrices properties and generation a real square matrix q is j orthogonal if qtjq j where the signature matrix j diag pm 1 j orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products including in particular the downdating of cholesky factorizations we present techniques and tools useful in the analysis application and construction of these matrices giving a self contained treatment that provides new insights first we define and explore the properties of the exchange operator which maps j orthogonal matrices to orthogonal matrices and vice versa then we show how the exchange operator can be used to obtain a hyperbolic cs decomposition of a j orthogonal matrix directly from the usual cs decomposition of an orthogonal matrix we employ the decomposition to derive an algorithm for constructing random j orthogonal matrices with specified norm and condition number we also give a short proof of the fact that j orthogonal matrices are optimally scaled und

boundary layers in stochastic thermodynamics we study the problem of optimizing released heat or dissipated work in stochastic thermodynamics in the overdamped limit these functionals have singular solutions previously interpreted as protocol jumps we show that a regularization penalizing a properly defined acceleration changes the jumps into boundary layers of finite width we show that in the limit of vanishing boundary layer width no heat is dissipated in the boundary layer while work can be done we further give an alternative interpretation of the fact that the optimal protocols in the overdamped limit are given by optimal deterministic transport burgers equation

generalized frame operator distance problems abstract let s m d c be a positive semidefinite d d complex matrix and let a a i i i k r 0 k indexed by i k 1 k be a k tuple of positive numbers let t d a denote the set of families g g i i i k c d k such that g i 2 a i for i i k thus t d a is the product of spheres in c d endowed with the product metric for a strictly convex unitarily invariant norm n in m d c we consider the generalized frame operator distance function n s a defined on t d a given by n s a g n s s g where s g i i k g i g i m d c in this paper we determine the geometrical and spectral structure of local minimizers g 0 t d a of n s a in particular we show that local minimizers are global minimizers and that these families do not depend on the particular choice of n

factorizations into normal matrices in indefinite inner product spaces abstract we show that any nonsingular real or complex square matrix can be factorized into a product of at most three normal matrices one of which is unitary another is selfadjoint with eigenvalues in the open right half plane and the third one is normal involutory with a neutral negative eigenspace we call the last matrix normal neutral involutory here the words normal unitary selfadjoint and neutral are understood with respect to an indefinite inner product

unitary transformation approach to the exact solutions of time dependent quantum systems with su 1 1 dynamical group a unitary transformation is introduced to determine the exact solution of the time dependent quantum system as examples two quantum systems with hamiltonian including the quadratic term of generators of su 1 1 are solved analytically

local extrema for procrustes problems in the set of positive definite matrices abstract given two positive definite matrices a and b a well known result by gelfand naimark and lidskii establishes a relationship between the eigenvalues of a and b and those of ab by means of majorization inequalities in this work we make a local study focused on the spectrum of the matrices that achieve the equality in those inequalities as an application we complete some previous results concerning procrustes problems for unitarily invariant norms in the manifold of positive definite matrices

analytic controllability of time dependent quantum control systems the question of controllability is investigated for a quantum control system in which the hamiltonian operator components carry explicit time dependence which is not under the control of an external agent we consider the general situation in which the state moves in an infinite dimensional hilbert space a drift term is present and the operators driving the state evolution may be unbounded however considerations are restricted by the assumption that there exists an analytic domain dense in the state space on which solutions of the controlled schrodinger equation may be expressed globally in exponential form the issue of controllability then naturally focuses on the ability to steer the quantum state on a finite dimensional submanifold of the unit sphere in hilbert space and thus on analytic controllability a relatively straightforward strategy allows the extension of lie algebraic conditions for strong analytic controllability derived earlier for the simpler time independent system in which the d

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

generalized adiabatic product expansion a nonperturbative method of solving the time dependent schrodinger equation we outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general possibly non hermitian hamiltonian for a class of such hamiltonians this expansion involves a finite number of terms and our method gives the exact solution of the corresponding time dependent schrodinger equation we apply this method to study the dynamics of a general nondegenerate two level quantum system a time dependent classical harmonic oscillator and a degenerate system consisting of a spin 1 particle interacting with a time dependent electric field e t through the stark hamiltonian h j e 2

on form preserving transformations for the time dependent schrodinger equation in this paper we point out a close connection between the darboux transformation and the group of point transformations which preserve the form of the time dependent schrodinger equation tdse in our main result we prove that any pair of time dependent real potentials related by a darboux transformation for the tdse may be transformed by a suitable point transformation into a pair of time independent potentials related by a usual darboux transformation for the stationary schrodinger equation thus any real potential solvable via a time dependent darboux transformation can alternatively be solved by applying an appropriate form preserving point transformation of the tdse to a time independent potential the pre eminent role of the latter type of transformations in the solution of the tdse is illustrated with a family of quasi exactly solvable time dependent anharmonic potentials

non stochasticity of time dependent quadratic hamiltonians and the spectra of canonical transformations the authors study the quantum mechanical evolution generated by hamiltonians which are quadratic polynomials in q and p with time dependent coefficients this evolution is determined by a unitary implementation of the phase flow of the corresponding classical hamiltonian in the case of quadratic hamiltonians which are periodic in time the floquet operator is shown to have either a pure point spectrum or a purely transient absolutely continuous spectrum thus the motion is non stochastic in a simple model of a quadratic hamiltonian with random time dependence the quantum mechanical motion is shown to be non stochastic almost surely

stability of self adjoint square roots and polar decompositions in indefinite scalar product spaces continuity properties of factors in polar decompositions of matrices with respect to indefinite scalar products are studied the matrix having the polar decomposition and the indefinite scalar product are allowed to vary closely related properties of a self adjoint with respect to an indefinite scalar product perturbed matrix to have a self adjoint square root or to have a representation of the form xx are also

a new polar decomposition in a scalar product space abstract there are various definitions of right and left polar decompositions of an m n matrix f k m n where k c or r with respect to bilinear or sesquilinear products defined by nonsingular matrices m k m m and n k n n the existence and uniqueness of such decompositions under various assumptions on f m and n have been studied here we introduce a new form of right and left polar decompositions f w s and f s w respectively where the matrix w has orthonormal columns w has orthonormal rows with respect to suitably defined scalar products which are functions of m n and f and the matrix s is selfadjoint with respect to the same suitably defined scalar products and has eigenvalues only in the open right half plane we show that our right and left decompositions exist and are unique for any nonsingular matrices m and n when the matrix f satisfies f m n n m f and f m n f f f m n respectively is nonsingular where f m n n 1 f m with f f t for real or complex bilinear forms and f f t for sesquilinear forms when m n our results apply to nonsingular square matrices f our assumptions on f m and n are in some respects weaker and in some respects stronger than those of previous work on polar decompositions

unitary transformation approach to the cyclic evolution of su 1 1 and su 2 time dependent systems and geometrical phases using the time dependent unitary transformation instead of the invariant operator the solutions of su 1 1 and su 2 time dependent quantum systems are obtained it is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the hamiltonian and the other correspond to the floquet operator which gives the cyclic states and their associated phases over one period of the evolution the non adiabatic berry s or aharonov anandan phases are determined totally by such a unitary transformation

symmetric effective mass schrodinger equations we outline a general method of obtaining exact solutions of symmetric schrodinger equations with a position dependent effective mass using this method exact solutions of some symmetric potentials have been obtained we have also discussed the construction of isospectral symmetric potentials

time dependent diffeomorphisms as quantum canonical transformations and the time dependent harmonic oscillator quantum canonical transformations corresponding to time dependent diffeomorphisms of the configuration space are studied a special class of these transformations which correspond to time dependent dilatations is used to identify a previously unknown class of exactly solvable time dependent harmonic oscillators the caldirola kanai oscillator belongs to this class for a general time dependent harmonic oscillator it is shown that choosing the dilatation parameter to satisfy the classical equation of motion one obtains the solution of the schrodinger equation a simple generalization of this result leads to the reduction of the schrodinger equation to a second order ordinary differential equation whose special case is the auxiliary equation of the lewis riesenfeld invariant theory the time evolution operator is expressed in terms of a positive real solution of this equation in a closed form and the time dependent position and momentum operators are calculated

polar decompositions in finite dimensional indefinite scalar product spaces general theory abstract several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied matrices that admit such polar decompositions are described in various ways in particular a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small