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

a modified brent s method for finding zeros of functions brent s method also known as zeroin has been the most popular method for finding zeros of functions since it was developed in 1972 this method usually converges very quickly to a zero for the occasional difficult functions encountered in practice it typically takes o n iterations to converge where n is the number of steps required for the bisection method to find the zero to approximately the same accuracy while it has long been known that in theory brent s method could require as many as o n 2 iterations to find a zero such behavior had never been observed in practice in this paper we first show that brent s method can indeed take o n 2 iterations to converge by explicitly constructing such worst case functions in particular for double precision accuracy brent s method takes 2 914 iterations to find the zero of our function compared to the 77 iterations required by bisection secondly we present a modification of brent s method that places a stricter complexity bound of o n on the search for a zero in our extensive testing this modification appears to behave very similarly to brent s method for all the common functions yet it remains at worst five times slower than the bisection method for all difficult functions in sharp contrast to brent s method

alternative descriptions in quaternionic quantum mechanics we characterize the quasianti hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra moreover we state a necessary and sufficient condition for a set of quasianti hermitian quaternionic operators to be anti hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional right quaternionic hilbert space according to such results we obtain two alternative descriptions of a quantum optical physical system in the realm of quaternionic quantum mechanics while no alternative can exist in complex quantum mechanics and we discuss some differences between them

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

embedding compact three dimensional cr manifolds of finite type in mathbf c n an abstract three dimensional cr manifold x is a smooth manifold without boundary of real dimension three equipped with a smooth one dimensional complex sub bundle t0 of its complexified tangent bundle which for each x e x satisfies txol n txo 1 o canonically associated to the cr structure is a first order differential operator db which maps functions on x to sections of the bundle b 1 dual to to and is defined by

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

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

estimates for the bar partial neumann problem in pseudoconvex domains of finite type in c2problem in pseudoconvex domains of finite type in c2

fundamental solutions for second order subelliptic operators estimation de la solution fondamentale d un operateur aux derivees partielles lineaire du second ordre sur une variete compacte avec une mesure reguliere

optimal hoelder and l p estimates for b on the boundaries of real ellipsoids in c n let d be a real ellipsoid in cn n 3 with defining function p z en x k yk2k 1 zk xk iyk where nk mk e n in this paper we study the sharp holder and lp estimates for the solutions of the tangential cauchy riemann equations 0b on the boundary od of d using the integral kernel method in particular we proved that if a e l o j od such that oba 0 on ad in the distribution sense then there exists a u e a112m od satisfying obu a and iiuiia 2 9d 0 independent of a where al 2m od is the lipschitz space with exponent 21 and 2m maxl k f min 2nk 2mk is the type of the m k domain d

time evolution of quantum systems with time dependent hamiltonian and the invariant hermitian operator we study the time evolution of a class of exactly solvable time dependent quantum systems with a time dependent hamiltonian given by a linear combination of su 1 1 and su 2 generators with the help of the invariant hermitian operator the exact common solutions of the schrodinger equations for both the su 1 1 and su 2 systems are obtained in terms of eigenstates of the invariant operator the adiabatic and non adiabatic berry phases are calculated with the exact solutions moreover we derive an explicit time evolution operator which is used to investigate the time dependent two photon squeezing states and su 2 squeezing states the squeezing properties of the time dependent su 1 1 coherent states are also discussed

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

estimates for the parametrix of the kohn laplacian on certain domains soit le demi espace superieur degenere generalise dans c 3 d p w 1 w 2 w imw p w 1 w 2 2 ou p c p 0 0 p est reel p 0 p 0 0 k p 0 0 pour k 2 on etudie la taille de la solution fondamentale de b sur des formes 0 1 sur la frontiere de d p

on geometric phases and dynamical invariants a formalism is developed for calculating berry phases for nonadiabatic time periodic quantum systems when a dynamical invariant is known it is found that when the invariant is periodic and has a nondegenerate spectrum this method allows a convenient way to obtain generalized berry phases and the proper cyclic initial states the method is applied to the generalized harmonic oscillator and the two level system where the invariant operators are explicitly constructed formulae for the conventional berry s phases are readily obtained by taking the adiabatic limit of the exact results

an implementation of the dqds algorithm positive case abstract the dqds algorithm was introduced in 1994 to compute singular values of bidiagonal matrices to high relative accuracy but it may also be used to compute eigenvalues of tridiagonal matrices this paper discusses in detail the issues that have to be faced when the algorithm is to be realized on a computer criteria for accepting a value for splitting the matrix and for choosing a shift to reduce the number of iterations as well as the relative advantages of using ieee arithmetic when available ways to avoid unnecessary over underflows are described in addition some new formulae are developed to approximate the smallest eigenvalue from a twisted factorization of a matrix the results of extensive testing are presented at the end the list of contents is a valuable guide to the reader interested in specific features of the algorithm

an improved dqds algorithm in this paper we present an improved dqds algorithm for computing all the singular values of a bidiagonal matrix to high relative accuracy there are two key contributions a novel deflation strategy that improves the convergence for badly scaled matrices and some modifications to certain shift strategies that accelerate the convergence for most bidiagonal matrices these techniques together ensure linear worst case complexity of the improved algorithm denoted by v5 our extensive numerical experiments indicate that v5 is typically 1 2x 4x faster than dlasq the lapack 3 4 0 implementation of dqds without any degradation in accuracy on matrices for which dlasq shows very slow convergence v5 can be 3x 10x faster we develop a hybrid algorithm hdlasq by combining our improvements with the aggressive early deflation strategy aggdef2 in y nakatsukasa k aishima and i yamazaki siam j matrix anal appl 33 2012 pp 22 51 numerical results show that hdlasq is the fastest among these dif

analysis of quantum time dependent evolution by the averaged variational principle in this paper we give a general method to solve a time dependent schrodinger equation the formalism developed is based on the quantum variational principle conveniently averaged in addition we apply our method to study the time dependent systems in which the hamiltonians are linear functions of su 1 1 and su 2 lie algebras