correspondence between berry s phase and lewis s phase for quadratic hamiltonians the authors show the existence of a relation between berry s phase and lewis s phase for the case of quadratic hamiltonians

time dependent invariants of motion for complete sets of non commuting observables abstract using the generalized ehrenfest theorem the dynamics of the mean values of a complete set of non commuting observables csnco associated to a given hamiltonian is expressed we found refined time dependent invariants of motion tdim for the csnco and associated them with different lie algebras we also show that when the csnco are related through an antisymmetric matrix then the characteristic polynomial of the covariance matrix for the csnco is also a tdim examples for the heisemberg su 2 and so 2 groups are outlined

gauge transformation approach to the exact solution of a generalized harmonic oscillator the exact solution and the invariant operator of a generalized harmonic oscillator are obtained by performing three consecutive gauge transformations on the time dependent schrodinger equation in contrast to previous results they depend only on solutions of a linear differential equation for the associated classical harmonic oscillator non adiabatic and adiabatic berry phases for the system are calculated on the basis of the exact solution

quasi coherent states for harmonic oscillator with time dependent parameters in this study we discuss the harmonic oscillator with the time dependent frequency t and the mass m t by generalizing the holomorphic coordinates for the harmonic oscillator in general cases we solve the schrodinger equation by reducing it into the riccati equation and discuss the uncertainties for the quasi coherent states of the time dependent harmonic oscillator in special cases we find the following results first for a time dependent harmonic oscillator if t m t is constant then the coherent states will evolve as the coherent states second for the driven harmonic oscillator the coherent states will evolve as the coherent states with new eigenvalues third we derive quasi coherent states for the caldirola kanai hamiltonian and show that the product of uncertainties x p is larger than minimum value however it is constant we also discuss the classical equations of motion for the system

the linearized system for isometric embeddings and its characteristic variety abstract in this paper we prove a conjecture of bryant griffiths and yang concerning the characteristic variety for the determined isometric embedding system in particular we show that the characteristic variety is not smooth for any dimension greater than 4 this is accomplished by introducing a smaller yet equivalent linearized system in an appropriate way which facilitates analysis of the characteristic variety

exact schrodinger wavefunctions of n coupled time dependent harmonic oscillators for n coupled time dependent harmonic oscillators with any couplings coordinates and momentum we have derived the primary invariant and the generalized lewis and riesenfeld invariant represented in terms of the classical solution matrix of hamilton s oscillator equation using the lewis and riesenfeld invariant method and exploiting the properties of the symplectic matrix we have obtained the exact schrodinger wavefunctions for the oscillators

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

noncyclic geometric phase and its non abelian generalization we use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the abelian and non abelian geometric phases this derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the abelian noncyclic geometric phase we also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non abelian noncyclic geometric phase for a spin 1 magnetic or electric quadrupole interacting with a precessing magnetic electric field

exact time dependence of solutions to the time dependent schrodinger equation solutions of the schrodinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time independent operators using time dependent unitary transformations exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time dependent hamiltonians in d dimensions including non hermitean symmetric hamiltonians hamiltonians are constructed using time dependent unitary spatial transformations comprising dilatations translations and rotations and solutions are found in several forms as eigenfunctions of a quadratic invariant as coherent state eigenfunctions of boson operators as plane wave solutions from which the general solution is obtained as an integral transform by means of the fourier transform and as distributional solutions for which the initial wavefunction is the dirac function for the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d 1 including barut girardello and perelomov coherent states i e vector coherent states which are shown to be related to eigenfunctions of the quadratic invariant by the transformation this transformation which leaves the ermakov equation invariant implements su 1 1 transformations on linear dynamical invariants coherent states are derived also for the time dependent linear potential exact solutions are found for hamiltonians with electromagnetic interactions in which the time dependent magnetic and electric fields are not necessarily spatially uniform as an example it is shown how to find exact solutions of the time dependent schrodinger equation for the dirac magnetic monopole in the presence of time dependent magnetic and electric fields of a specified form

on monge ampere equations with homogenous right hand sides we study the regularity and behavior at the origin of solutions to the two dimensional degenerate monge ampere equation det d2u x with 2 we show that when 0 solutions admit only two possible behaviors near the origin radial and nonradial which in turn implies c2 regularity we also show that the radial behavior is unstable for 0 we prove that solutions admit only the radial behavior near the origin 2008 wiley periodicals inc

thermodynamics of the standard quantum harmonic oscillator of time dependent frequency with and without inverse quadratic potential making use of a dynamical invariant operator we obtain a quantum mechanical solution of a damped harmonic oscillator having time dependent frequency with and without inverse quadratic potential we confirm that the uncertainty relation is always larger than 2 in both the number and thermal state we obtain a density operator satisfying the liouville von neumann equation and use its diagonal elements to calculate various expectation values in the thermal state

statistical solutions of differential equations with non uniquely solvable cauchy problems in both cases the existence of global weak solutions is known for large classes of initial values 4 5 however the existence of classical and unique solutions even for very smooth initial values has only been proved locally and bifurcations of the solutions after some time interval cannot be excluded more specifically such bifurcations could e g for the vlasov equation occur at points t x u where the right side of the characteristic system

unitary relations in time dependent harmonic oscillators for a harmonic oscillator with time dependent positive mass and frequency a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time dependent frequency as well as operators for a driven harmonic oscillator a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given as a generalization of previous results in terms of the solution of classical equation of motion of the driven system thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system provided the quantum states of the corresponding system of unit mass are given

berry s phase and wavefunctions for time dependent hamilton systems an alternative derivation of berry s phase 1985 to that given by morales 1988 is given for non autonomous hamiltonian systems which admit an energy like first integral the generality of the problems treated is greater

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

existence of c local solutions for the monge ampere equation on considere l equation de monge ampere det u ij f y u u dans un ouvert de r n on demontre l existence de solutions c locales

coherent states and their time dependence in fractional dimensions we construct representations of the lie algebra su 1 1 using representations of the momentum and position operators satisfying the r deformed heisenberg relations in which the fractional dimension d and angular momentum l appear as parameters the bargmann index k which characterizes representations of the positive discrete series of su 1 1 can take any positive value we construct coherent states in fractional dimensions in particular we extend the two well known analytic representations of coherent states for su 1 1 perelomov and barut girardello states from dimension one to any dimension d we generalize this construction to time dependent coherent states by means of the su 1 1 symmetries of the quantum time dependent harmonic oscillator in fractional dimensions we investigate the uncertainty relations of the momentum and position operators with respect to these coherent states and their dependence on the dimension

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

existence of local sufficiently smooth solutions to the complex monge ampere equation we prove the c local solvability of the n dimensional complex monge ampere equation det u ij f z u u f 0 in a neighborhood of any point z 0 where f z 0 0

the generalized harmonic oscillator and the infinite square well with a moving boundary for the one dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in i the algebraic properties of the dynamical symmetry of the system and ii the construction of an invariant operator the total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion two possible reductions for the system exist the static harmonic oscillator and the free particle in the latter case the quantum system becomes a fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules the time dependent boundary conditions in the well play the role of an effective interaction acting on the particle the formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed

the time evolution and the quantum fluctuation for two forced quantum oscillators with mixing of two modes by using the lewis riesenfeld quantum invariant theory and properly choosing hermitian invariant operator a closed solution of the schrodinger equation is derived for two forced quantum oscillators with mixing of two modes and the quantum fluctuations in the output fields are evaluated for the initial two mode squeezed number or squeezed coherent state in some particular conditions the time evolution of the oscillators can not only preserve the initial two mode squeezing but also produce squeezing in the individual modes and exhibit a periodical squeezing behaviour for the initial two mode number state or coherent state there is no squeezing in the individual and mutual quadrature phases of the two mode fields furthermore regardless of which state above being initially considered the quantum fluctuations of all the quadrature phases in the output fields are all independent of the driving parameters in particular for the initial two mode coherent state the variances of the output fields are also independent of other parameters in the hamiltonian and always preserve their initial values 1 4

unitary transformation approach to the exact solution for the singular oscillator by performing unitary transformations the exact solution of the time dependent singular oscillator is obtained the invariant operator and the auxiliary equation are rigorously established the non adiabatic berry s phase is calculated

existence of distributional solutions of a closed dirichlet problem for an elliptic hyperbolic equation abstract the closed dirichlet problem for an elliptic hyperbolic equation arising in linearizing the monge ampere type equation is shown to admit a distributional solution by using an a b c integral method

invariants and lie algebraic solutions of schrodinger equation a representation of invariants in quantum mechanics for a general time dependent hamiltonian is provided the main properties of these operators and their relevance for solving the schrodinger equation are discussed in light of this representation in particular the well known lewis and riesenfeld s method is shown to be recovered quite directly by the present approach a detailed analysis is concerned with hamiltonians which at each instant are elements of a finite dimensional lie algebra exploiting the wei and norman s theory for such hamiltonians it is found that the evolution of a quantum system is always determined in part or fully by an invariant which belongs to the same algebra

hannay s angle and berry s phase for su 1 1 systems the classical and quantum systems related to the group su 1 1 are considered the hannay angle is calculated and its relation to berry s phase established

weyl ordered polynomials in fractional dimensional quantum mechanics we develop algebraic properties of weyl ordered polynomials in the momentum and position operators p q which satisfy the r deformed heisenberg algebra representations of which describe quantum mechanics in fractional dimensions by viewing weyl ordered polynomials as tensor operators with respect to the lie algebra we derive a specific form for these polynomials including an expression in terms of hypergeometric functions and determine various algebraic properties such as recurrence relations symmetries and also a general product formula from which all commutators and anti commutators may be calculated we briefly discuss several applications to quantum mechanics in fractional dimensions

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

classical non adiabatic angles if a family of tori in phase space is driven by a time dependent hamiltonian flow in such a way as to return after some time to the original family there generally results a shift in the angle variables one realisation of this process is in the cyclic adiabatic change of a classical hamiltonian and the angle change has previously been shown to separate naturally into a dynamical part and a geometrical part here the same geometrical angle change is extracted when the return is achieved non adiabatically and the dynamical remainder calculated two examples are given the precession of a spin and the rotation of phase space ellipses

time evolution cyclic solutions and geometric phases for the generalized time dependent harmonic oscillator the generalized time dependent harmonic oscillator is studied though several approaches to the solution of this model have been available yet a new approach is presented here which is very suitable for the study of cyclic solutions and geometric phases in this approach finding the time evolution operator for the schrodinger equation is reduced to solving an ordinary differential equation for a c number vector which moves on a hyperboloid in a three dimensional space cyclic solutions do not exist for all time intervals a necessary and sufficient condition for the existence of cyclic solutions is given there may exist some particular time interval in which all solutions with definite parity or even all solutions are cyclic criteria for the appearance of such cases are given the known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical hannay angle is reestablished however this is valid only for special cyclic solutions for more general ones the nonadiabatic geometric phase may contain an extra term several cases with relatively simple hamiltonians are solved and discussed in detail cyclic solutions exist in most cases the pattern of the motion say finite or infinite cannot be simply determined by the nature of the hamiltonian elliptic or hyperbolic etc for a hamiltonian with a definite nature the motion can change from one pattern to another that is some kind of phase transition may occur if some parameter in the hamiltonian goes through some critical value

exact evolution of the generalized damped harmonic oscillator in this paper we use the algebraic and the invariant method to study the time dependent damped harmonic oscillator from classical and quantum points of view the solution of the classical equation of motion and the wavefunction solving the time dependent schrodinger equation are found explicitly we show that the original time dependent quantum mechanical problem is completely related to the well known time independent harmonic oscillator in addition we elucidate the intimate connection between the damped harmonic oscillator dho and the generalized harmonic oscillator gho more importantly the evolution of the states of the dho cannot be cyclic in contradistinction with the states of the gho explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit the coherent states describing the invariant angle variables of the classical dho are constructed they allow us to recover the classical evolution and invariant from the quantum evolution

exactly solvable three body systems with internal degrees of freedom the approach of multi dimensional susy quantum mechanics is used in an explicit construction of exactly solvable three body and quasi exactly solvable n body matrix problems on a line from intertwining relations with time dependent operators we build exactly solvable non stationary scalar and 2 2 matrix three body models which are time dependent extensions of the calogero model finally we investigate the invariant operators associated with these systems

measure valued solutions and the phenomenon of blow down in logarithmic diffusion abstract the existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory a prominent example being the fundamental solutions of the linear theory in nonlinear equations the existence of such solutions may find special obstacles that can be either essential or otherwise they may lead to more general concepts of solution we give a particular review of results in the field of nonlinear diffusion as a new contribution we study in detail the case of logarithmic diffusion associated with ricci flow in the plane where we can prove existence of measure valued solutions the surprising thing is that these solutions become classical after a finite time in that general setting the standard concept of weak solution is not adequate but we can solve the initial value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions we prove that the problem is well posed the phenomenon of blow down in finite time is precisely described initial point masses diffuse into the medium and eventually disappear after a finite time t i m i 4

on statistical and measure valued solutions of differential equations abstract we discuss the relation between statistical measure valued and weak solutions for typical cases of nonuniquely solvable systems of ordinary differential equations and for the case of hyperbolic conservation laws

geometric phases symmetries of dynamical invariants and exact solution of the schrodinger equation we introduce the notion of the geometrically equivalent quantum systems geqss as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors we give a characterization of the geqss these systems have a common dynamical invariant and their hamiltonians and evolution operators are related by symmetry transformations of the invariant if the invariant is t periodic the corresponding class of geqss includes a system with a t periodic hamiltonian we apply our general results to study the classes of geqss that include a system with a cranked hamiltonian h t e ikt h0e ikt we show that the cranking operator k also belongs to this class hence in spite of the fact that it is time independent it leads to nontrivial cyclic evolutions and geometric phases our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time independent simple harmonic oscillator the period of these cyclic states is half the characteristic period of the oscillator

statistical solutions of hyperbolic conservation laws foundations we seek to define statistical solutions of hyperbolic systems of conservation laws as time parametrized probability measures on p integrable functions to do so we prove the equivalence between probability measures on l p spaces and infinite families of correlation measures each member of this family termed a correlation marginal is a young measure on a finite dimensional tensor product domain and provides information about multi point correlations of the underlying integrable functions we also prove that any probability measure on a l p space is uniquely determined by certain moments correlation functions of the equivalent correlation measure we utilize this equivalence to define statistical solutions of multi dimensional conservation laws in terms of an infinite set of equations each evolving a moment of the correlation marginal these evolution equations can be interpreted as augmenting entropy measure valued solutions with additional information about the evolution of all possible multi point correlation functions our concept of statistical solutions can accommodate uncertain initial data as well as possibly non atomic solutions even for atomic initial data for multi dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting entropy statistical solutions exist are unique and are stable with respect to the 1 wasserstein metric on probability measures on l 1

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

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

quantum mechanical models in fractional dimensions we formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators p q satisfy the r deformed heisenberg relations which depend on an operator we find representations of p q in which the dimension d and angular momentum l appear as parameters related to the eigenvalues of we analyse the domain of p and find conditions which ensure that p is hermitian we investigate plane wave solutions and also free particle wavefunctions in fractional dimensions and show that as a consequence of wavefunction continuity l is quantized the representations of p q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d both algebraically and analytically we demonstrate that the formalism extends also to time dependent hamiltonians by solving the time dependent harmonic oscillator in any dimension d 0 using the method of lewis and riesenfeld

compactness of alexandrov nirenberg surfaces we study a class of compact surfaces in 3 introduced by alexandrov and generalized by nirenberg and prove a compactness result under suitable assumptions on induced metrics and gauss curvatures 2017 wiley periodicals inc

universal integrals of motion and universal invariants of quantum systems universal quantum integrals of motion are introduced and their relation with the universal quantum invariants is established the invariants concerned are certain combinations of the second and higher order moments variances of quantum mechanical operators which are preserved in time independently of the concrete form of the coefficients of the schr dinger equation provided the hamiltonian is either a generic quadratic form of the coordinate and momenta operators or a linear combination of generators of some finite dimensional algebra in particular any semisimple lie algebra using the phase space representation of quantum mechanics in terms of the wigner function the relations between the quantum invariants and the classical universal integral invariants by poincar and cartan are elucidated examples of the universal invariant solutions of the schr dinger equation i e self consistent eigenstates of the universal integrals of motion are given applications to the physics of optical and particle beams are discussed

the local isometric embedding in mathbb r 3 of two dimensional riemannian manifolds with gaussian curvature changing sign to finite order on a curve we consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of monge ampere type these two problems are the local isometric embedding problem for two dimensional riemannian manifolds and the problem of locally prescribed gaussian curvature for surfaces in r3 we prove a general local existence result for a large class of monge ampere equations in the plane and obtain as corollaries the existence of regular solutions to both problems in the case that the gaussian curvature vanishes to arbitrary finite order on a single smooth curve

so 2 1 invariant systems and the berry phase the general quadratic time dependent quantum hamiltonian is analysed using a time dependent realisation of its so 2 1 invariance lie algebra several interesting features of this procedure are pointed out as an example the authors easily calculate dynamical and geometrical berry phases using only an algebraic procedure an incorrect result previously published by another author is also pointed out

unitary transformation approach to the exact solution for a class of time dependent nonlinear hamiltonian systems by performing unitary transformations on the time dependent schrodinger equation the exact solution for a class of nonlinear hamiltonian systems is obtained and it is shown that these time dependent problems are related to the associated time independent problems in addition the evolution operator is derived nonadiabatic berry s phase is calculated on the basis of the exact solution the theory is applied to some illustrative examples

non abelian geometric phase floquet theory and periodic dynamical invariants for a periodic hamiltonian periodic dynamical invariants may be used to obtain non degenerate cyclic states this observation is generalized to the degenerate cyclic states and the relation between the periodic dynamical invariants and the floquet decompositions of the time evolution operator is elucidated in particular a necessary condition for the occurrence of cyclic non adiabatic non abelian geometrical phase is derived degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field