geometrical phase in the cyclic evolution of non hermitian systems the authors derive by a biorthonormal state approach the analogy of berry s phase factor for open non conservative systems for both adiabatic and non adiabatic evolution in the latter case a non unitary evolution operator method is exploited an application is given to the optical supermode propagation in the free electron laser

a new concept of geometric phase in parameter space coupling as a parameter we have obtained the geometric phase of the tripartite spin 1 2 system with spin spin coupling and discussed the influence of this coupling on the geometric phase we have discovered the coupling independent geometric phase and explained it based on this work we propose the concept of quasimagnetic field and assimilate the coupling into the parameter space by this theory the relation between the geometric phase and the solid angle in parameter space still holds for a system with spin spin coupling if the concept of the parameter is properly revised in the way we have elucidated

a bogolyubov type theorem with a nonconvex constraint in banach spaces we prove an analogue of the classical bogolyubov theorem with a nonconvex constraint in the case we consider the constraint is the solution set of a cauchy problem for a differential inclusion with a nonconvex right hand side satisfying a lipschitz condition our approach is based on a relaxation argument as in the filippov wazewski theorem

semilinear elliptic equations on unbounded domains on presente une nouvelle approche variationnelle a l existence des solutions pour le probleme aux valeurs limites u u r x f u x x x h 0 1 sur des domaines non bornes

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

a comparative study of the hannay s angles associated with a damped harmonic oscillator and a generalized harmonic oscillator in this paper we show that a damped harmonic oscillator the simplest dissipative physical system is canonically equivalent to a generalized harmonic oscillator a conservative system even for time dependent parameters as a consequence the hannay s angles and the adiabatic invariants of the two systems appear to be the same modulo this equivalence this raises the question of whether this analogy can be extended to other nonlinear dissipative systems

a topological investigation of the quantum adiabatic phase using algebraic topology the appearance of the quantum adiabatic phase over various parameter manifolds is investigated the relation with nontrivial gauge bundles both abelian and non abelian is studied and it is shown that the phase appears as a result of homotopically non trivial mappings induced by the hamiltonian in the space of wave functions the cohomological picture is developed and some topological considerations concerning field theory anomalies in the hamiltonian picture are presented a proof of the nielsen ninomiya theorem is given inspired from the notion of the adiabatic phase

hannay berry phase and the restricted three vortex problem abstract asymptotic expansions for the hannay berry phase are computed for a class of hamiltonian systems in perturbed action angle form with slowly varying parameters where the parameters recur on a slow time scale a multi scale perturbation method is used which in principle can be carried to any desired order of accuracy as a specific application we use the method to examine the restricted three vortex problem a completely integrable configuration of two co rotating point vortices a distance d apart and a tracer particle rotating around one of the vortices with distance r in the asymptotic limit r d e we compute the particle trajectory through o e and find that the hannay berry phase is given by 2 3 r 0 d o 1 d 2 where r 0 is the conserved action

the degenerate parametric oscillator and ince s equation we construct green s function for the quantum degenerate parametric oscillator in the coordinate representation in terms of standard solutions of ince s equation in a framework of a general approach to variable quadratic hamiltonians exact time dependent wavefunctions and their connections with dynamical invariants and su 1 1 group are also discussed an extension to the degenerate parametric oscillator with time dependent amplitude and phase is also mentioned

hamilton s turn and geometric phase for two level systems hamilton in the course of his studies on quaternions introduced an elegant geometric representation for the composition of su 2 elements in terms of turns on the unit sphere s2 the authors use these turns to study two level systems with particular reference to geometric phase the special roles played by piecewise geodesic circuits in the state space and evolution under constant hamiltonian are recognized

berry s phase as the asymptotic limit of an exact evolution an example the authors exhibit for arbitrary time variations of the parameters of the generalised harmonic oscillator hamiltonian a canonical transformation which facilitates an exact analysis of quantal phase and classical angle formulae for berry s phase and hannay s angle are readily obtained by taking the adiabatic limit of the exact theory

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

berry s phase and hannay s angle from quantum canonical transformations using an action angle coherent states formalism introduced by the authors in a preceding paper it is shown that berry s phase and hannay s angle can both be derived from the same quantum unitary transformation their relationship is easily established in this framework

a factorization property of the berry phase it is shown that for systems with hamiltonians expressible in terms of the generators of a lie group the berry phase appears in the form of a product of a representation dependent term and a purely geometrical factor

invariance of the generalised berry phase under unitary transformations application to the time dependent generalised harmonic oscillator a generalisation of the berry phase which reduces to the usual berry phase in the adiabatic and cyclic case is derived in a gauge invariant way this generalised berry phase unlike the berry phase is invariant under unitary transformations a time dependent generalised harmonic oscillator with a cross term in the hamiltonian is taken as an example the hamiltonian of this system is not in general the energy an energy the time derivative of which is the power is obtained from the equation of motion when the system is quantised the generalised berry phase is zero and remains invariant under unitary transformations on the other hand if the energy is chosen to be the hamiltonian a non zero generalised berry phase is obtained but the power is given incorrectly in this case the total phase which includes the dynamical phase and the generalised berry phase is equal in the adiabatic limit to the correct total phase

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

the generalized strauss compactness lemma and its applications on nonlinear elliptic problems to prove the existence of solutions of partial differential equations usually it needs some compactness properties for the bounded domain case there is a useful one rellich kondrakov theorem however for the unbounded domain case we know quite a few of such results strauss i i proved a compactness lemma for the entire space rn in this article we generalized strauss result to the case of an infinite strip q rn x 0 l n l 2 several applications on elliptic equations are also given in this section we introduce some notations concepts and equations which will be used later in section 2 we prove the compactness lemma in section 3 we give some applications on nonlinear elliptic equations as one of the continuing studies of lions 8 and berestycki lions 41 consider 0 1 as the circle group r z and q rn x 0 i the locally compact abelian group with the haar measure dxdt where dx and dt are lebesque measure on r and 0 l respectively we therefore may consider the convolution of two functions or generalized functions on the group a let f r r be a measurable function the distribution function f rf r off is defined by for cy 0 f ilx e 4lf l 41

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

extensions of clarke s proximal characterization for reachable mappings of differential inclusions in this paper we show that clarke s proximal characterization for reachable mappings of lipschitz continuous differential inclusions is valid for a larger class of continuous and locally one side kamke continuous inclusions we also give a new proximal characterization for reachable mappings of upper semi continuous differential inclusions

on the removability of berry s phase the authors present a thorough analysis of the removability of the berry phase they show that for any system they can by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one however this extra shift in the dynamical phase retains its truly geometrical character the removability is thus only apparent and the geometrical phase can still be measured in interference experiments as indeed has already been done several times

generic properties of multifunctions application to differential inclusions abstract we prove that almost all in baire sense caratheodory multifunctions in finite dimensional space are kamke continuous further the main properties of differential inclusions with kamke and one sided kamke right hand sides are studied as a corollary we prove that for almost all optimal control problems the relaxation and relaxation stability properties hold

geometric amplitude adiabatic invariants quantization and strong stability of hamiltonian systems considered is a linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters asymptotic solutions have been constructed as has been shown an important characteristic determining the qualitative portrait of the system is the real part of berry s complex geometric phase which we call the geometric amplitude for systems with purely imaginary eigenvalues the equivalence has been proven in the adiabatic approximation of system sets without geometric amplitude hamiltonian systems quantizable systems and strongly stable systems classification of the systems without geometric amplitude is given with respect to the kind of matrix of the initial system

berry s transport and minimization of averaged distances in this paper we consider the metric approach to the berry s transport geometrical part of the adiabatic evolution of any pure or mixed state for possibly degenerate hamiltonians we emphasize that explicit formulae for the transport of such states need the introduction of an averaging procedure in analogy with the classical case this quantum averaging involves multiplying each energy eigenstate by a different phase and then integrating over these phases we show in particular how the transport of non stationary pure states rays and density matrices arise from the minimization of hilbert fubini study and bures averaged distances respectively

the removability of the hopf term in the 2 1d model and aharonov bohm effect in this paper the removability of the hopf term in the 2 1d model is studied a quantum unitary transformation is found and used to remove the hopf term from the hamiltonian for the model it is then shown that there is an intimate connection between the hopf term and the multivaluedness of the wavefunctional describing the model

berry s phase hannay s angle and coherent states through the use of coherent states describing the action angle variables the authors establish a strict parallelism between berry s phase and hannay s angle in particular they define a geometrical transport for classical tori which leads to hannay s angle as in the case of berry s phase the transport can be associated with the minimization of a distance

geometric phase decomposition for exactly solvable cases with interaction in the paper a problem of geometric phase decomposition for general evolutions in the hilbert space is addressed the decomposition of total phase into dynamical and geometrical parts employs a state representation in the noninteraction picture the noninteraction picture is introduced with the help of decomposition of the system evolution operator into two parts the one pertaining to free evolution and the other to interaction the procedure requires the problem to be an exactly solvable case of the schrodinger equation the most common class of such problems includes dipole hamiltonians for which the evolution operator can be decomposed into a combination of unitary operators geometric phase decomposition in the noninteraction picture can be applied to general noncyclic evolutions but for the cyclic states it reduces to the floquet decomposition defined this way geometric phase possesses characteristic features of geometric phase for the free system but extends some stationary properties to temporal dependences as a case the time dependent relationship between geometric phase and the nonstationarity of a quantum state is illustrated by an example of a spin in a rotating magnetic field it is shown that geometric phase reaches a maximum when the spin state becomes completely nonstationary

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

adiabatic invariance in the standard map adiabatic invariance of the action is investigated in the standard map under slow changes of the stochasticity parameter k a fixed action representation of the rotational tori is developed perturbatively in k and is connected with the usual kam representation at fixed winding number the notion of adiabatic invariance to a given order in k k 1 is then introduced it involves an approximation to exact dynamics by essentially a power series expansion in both k and a slowness parameter adiabatic invariance is explicitly verified to second order and the dependence of the nonadiabaticity on the form of the switching function and the slowness of the change is investigated the case of adiabatic switching to larger values of k k 1 is approached phenomenologically taking into account the problem of separatrix crossing it is shown that this crossing leads in general to nonadiabatic effects in the limit of infinitely slow change these reflect the finite widths or intrinsic action uncertainties associated with the main island chains crossed the latter are determined from the farey tree in the range of variation of the winding number an estimate of the critical slowness parameter corresponding to the onset of the intrinsic nonadiabatic effects is derived it involves typical time scales associated with the main island chains crossed

near optimal nonholonomic motion planning for a system of coupled rigid bodies how does a falling cat change her orientation in midair without violating angular momentum constraint this has become an interesting problem to both control engineers and roboticists in this paper we address this problem together with a constructive solution first we show that a falling cat problem is equivalent to the constructive nonlinear controllability problem thus the same principle and algorithm used by a falling cat can be used for space robotic applications such as reorientation of a satellite using rotors and attitude control of a space structure using internal motion and other robotic tasks such as dextrous manipulation with multifingered robotic hands and nonholonomic motion planning for mobile robots then using ideas from ritz approximation theory we develop a simple algorithm for motion planning of a falling cat finally we test the algorithm through simulation on two widely accepted models of a falling cat it is interesting to note that one set of simulation results closely resembles the real trajectories employed by a falling cat

the geometric phase for chaotic systems the geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two form through a surface in the system s parameter space we obtain the classical limit of this two form in a form applicable to systems whose classical dynamics is chaotic for integrable systems the expression is equivalent to the hannay two form we discuss various properties of the classical two form derive semiclassical corrections to it associated with classical periodic orbits and consider implications for the semiclassical density of degeneracies

berry s phase in the floquet representation by means of the floquet theory the investigation of berry s phase for the quantum system with time dependent periodic hamiltonian is transformed into the study of a time independent eigenvalue equation an expression for the infinite dimensional matrix is obtained describing the berry s phase in terms of the floquet state

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results

berry phases and unitary transformations berry phases for spin are defined for any closed loop made by a vector changing direction in three dimensional space a sequence of rotations moves the vector along the loop each rotation is around the axis perpendicular to the moving vector the berry phases come from the eigenvalues of the unitary operator for the product of these rotations the angle of the product rotation is shown to be the solid angle enclosed by the loop the proof uses the ordinary language of quantum mechanics the product is calculated from the commutation relations for spin a general framework is set up to define berry phases for other transformations and states like those for rotations and spin the integral formula is derived alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics an example is used to show how one hamiltonian may be simpler than others adiabatic evolution is obtained in the limit as a coupling constant goes to zero so the adiabatic changes are m