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

berry s phase and the planar three body problem berry s angular 2 form in seen as a correction to the symplectic structure in a separation of variables type scheme where the variables are canonically non separable this view is applied to the planar three body problem where the rotational and vibrational motions are not separable it is shown that the corrected symplectic structure gives the correct quantisation

on b function spectrum and rational singularity by duality 18 because r tr ccy 0 for i 0 by 6 31 this follows also from 13 21 23 where rc is assumed projective here coy denotes the dualizing sheaf i e the dualizing complex 18 shifted by the dimension to the right the trace morphism 0 2 is injective and its image is independent of the choice of resolution because 0 2 is an isomorphism if y is smooth we will denote by csy the image of 0 2 see 2 4 8 below now assume y is a reduced divisor d on a complex manifold x of dimension n let f be a reduced defining equation of d on a neighborhood of a c d the b function i e bernstein polynomial of f is a monic polynomial bf s in s with rational coefficients and is a generator of the ideal whose element b s satisfies the relation

hamiltonian and linear space structure for damped oscillators i general theory the phase space of n damped linear oscillators is endowed with a bilinear map under which the evolution operator is symmetric this analogue of self adjointness allows properties familiar from conservative systems to be recovered e g eigenvectors are orthogonal under the bilinear map and obey sum rules initial value problems are readily solved and perturbation theory applies to the complex eigenvalues these concepts are conveniently represented in a biorthogonal basis

logarithmic perturbation theory for quasinormal modes logarithmic perturbation theory lpt is developed and applied to quasinormal modes qnms in open systems qnms often do not form a complete set so lpt is especially convenient because summation over a complete set of unperturbed states is not required attention is paid to potentials with exponential tails and the example of a poschl teller potential is briefly discussed a numerical method is developed that handles the exponentially large wavefunctions which appear in dealing with qnms

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

on the real and complex geometric phases some of the existing definitions of the real and complex geometric phases are unified and generalized the relationship of the real to the complex geometric phase is discussed it is shown why these two phases are in general completely independent in particular a simple explicit example is given which shows that the real geometric phase is not in general the real part of the complex geometric phase

homotopy perturbation method in quantum mechanical problems abstract we apply homotopy perturbation method hpm to study nonlinear quantum mechanical systems it is shown that this technique can be used efficiently to deal with quantum mechanical bound state as well as the continuum state problems to illustrate the capability and reliability of the method few examples of bound state and scattering state problems are discussed

two component eigenfunction expansion for open systems described by the wave equation i completeness of expansion the concept of eigenfunction expansions for the wave equation is generalized to open systems in which waves escape to the outside these non conservative systems are non hermitian in the usual sense it is shown that the natural framework is an eigenfunction expansion within a two component formalism that treats the wavefunction and its conjugate momentum together provided the system approaches spatial infinity rapidly without tails and possesses spatial discontinuities the expansion in terms of the eigenfunctions which are now quasinormal modes is shown to be valid

on the l 2 overline partial cohomology of certain complete kahler metrics let v be a compact and irreducible complex space of complex dimension v whose regular part is endowed with a complete hermitian metric h let pi m rightarrow v be a resolution of v under suitable assumptions on h we prove that begin aligned h v q 2 overline partial mathrm reg v h cong h v q overline partial m quad q 0 ldots v end aligned then we show that the previous isomorphism applies to the case of saper type kahler metrics as introduced by grant melles and milman and to the case of complete kahler metrics with finite volume and pinched negative sectional curvatures

unbounded derivations tangential to compact groups of automorphisms ii let g be a compact abelian group and an action of g on a c algebra u such that u u u 0 u for all g where u is the spectral subspace of u corresponding to the character on g derivations which are defined on the algebra uf of g finite elements are considered in the special case u 0 these derivations are characterized by a cocycle on g with values in the relative commutant of u in the multiplier algebra of u and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded under various restrictions on g and properties of the cocycle are deduced which again give characterizations of in terms of decompositions into generators of one parameter subgroups of g and approximately inner derivations finally a perturbation technique is devised to reduce the case uf uf to the case uf uf and u 0 this is used to show that any derivation with d uf is wellbehaved and if furthermore g t1 and uf uf the closure of generates a one parameter group of automorphisms of u in the case g td d 2 3 finite and uf uf it is shown that extends to a generator of a group of automorphisms of the weak closure of u in any g covariant representation

compact symmetry groups and generators for sub markovian semigroups we consider closed infinitesimal operators l with dense domain d in the algebra of all continuous functions vanishing at infinity on a given locally compact hausdorff space x necessary and sufficient conditions are given for l to be the infinitesimal generator of a strongly continuous sub markovian semigroup on c0 x the conditions involve 1 commutativity with a compact group g acting continuously on x i e spatial g symmetry 2 a tangential notion defined in terms of the g action and finally 3 a dispersive estimate for l the dispersive estimate is satisfied on a subalgebra which is a core for l and contained in the domain d if g is also a lie group we explicitly construct such a core algebra in the domain

generalized coherent states and berry s phase the quantum dynamics of time dependent hamiltonians of the kind h t u t h 0 u t is studied in the framework of generalized coherent states it is shown that the quantum and the classical dynamics are isomorphic and that the phase of t is simply the classical action in adiabatic approximation it is easy to extract berry s phase from this a discussion of generalized coherent states cs s necessary to deal with excited states is given in particular for su 2 su 1 1 and e 2 their structure is completely characterized showing that the deep connection with geometric quantization extends to these generalized cs s this technique may be employed to estimate higher order correction to the adiabatic approximation

two component wave formalism in spherical open systems we study wave evolution in open dielectric spheres by expanding the wave field and its conjugate momentum the two components in terms of relevant quasi normal modes qnms which are complete under appropriate conditions we first establish a novel outgoing boundary condition at the surface of a sphere for waves emanating from its interior a proper definition of inner product for two component outgoing wavefunctions involving only the waves inside the sphere and a surface term can then be defined in general the orthogonality relation of qnms and hence a unique expansion in terms of the qnm basis are found which can be applied to solve for the evolution of waves inside open dielectric cavities furthermore a time independent perturbation for qnms can also be developed

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

two component eigenfunction expansion for open systems described by the wave equation ii linear space structure for a broad class of open systems described by the wave equation the eigenfunctions which are quasinormal modes provide a complete basis for simultaneously expanding outgoing wavefunctions in this paper the linear space structure associated with this expansion is developed under a modified inner product the time evolution operator is self adjoint even though energy is not conserved for the system alone thus the eigenfunctions are mutually orthogonal consequently the usual tools of eigenfunction expansions can be transcribed to these open systems as an example the time independent perturbation theory is developed in straightforward analogy with quantum mechanics giving the shift in both the real part and the imaginary part of the eigenvalues

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

l2 overline partial cohomology groups of some singular complex spaces let x be a pure n dimensional where n 2 complex analytic subset in n with an isolated singularity at 0 in this paper we express the l 2 0 q overline partial cohomology groups for all q with 1 q n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data we also obtain identifications of the l 2 0 q overline partial cohomology groups of the smooth points of x in terms of resolution data when x is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities

a new efficient method for calculating perturbative energies using functions which are not square integrable regularization and justification the method recently proposed by skala and cizek for calculating perturbation energies in a strict sense is ambiguous because it is expressed as a ratio of two quantities which are separately divergent even though this ratio comes out finite and gives the correct perturbation energies the calculational process must be regularized to be justified we examine one possible method of regularization and show that the proposed method gives traditional quantum mechanical results

analysis of berry s phase by the evolution operator method berry 1984 discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase aharonov and anandan 1987 on the other hand showed that an eigenstate transporting round a closed circuit in the projective hilbert space suffices to generate the topological phase the authors employ the evolution operator method to study the propagation of an eigenstate they show that berry s phase can be represented as a closed path integral in the hilbert space and is independent of the choice for the base mod n t the treatments of berry and aharonov and anandan are shown to correspond to two different choices of the base therefore their two approaches are unified they have acquired a more general viewpoint on the origin of berry s phase

berry phase of a resonant state we derive closed analytical expressions for the complex berry phase of an open quantum system in a state which is a superposition of resonant states and evolves irreversibly due to the spontaneous decay of the metastable states the codimension of an accidental degeneracy of resonances and the geometry of the energy hypersurfaces close to a crossing of resonances differ significantly from those of bound states we discuss some of the consequences of these differences for the geometric phase factors for example instead of a diabolical point singularity there is a continuous closed line of singularities formally equivalent to a continuous distribution of magnetic charge on a diabolical circle there are different classes of topologically inequivalent non trivial closed paths in parameter space the topological invariant associated with the sum of the geometric phases dilations of the wavefunction due to the imaginary part of the berry phase and others

jordan blocks and generalized bi orthogonal bases realizations in open wave systems dissipation can sometimes be described by a non hermitian hamiltonian h whose left and right eigenvectors f j fj form a bi orthogonal basis bb for waves in a class of open systems this is known to lead to exact complete bb expansions if f j fj 0 for all j if not normalization seems impossible and many familiar formulae fail examples are given the problem is related to the merging of eigenmodes so that h can only be diagonalized to jordan blocks the resolution involves a generalized bb containing extra vectors whose dynamics are modified by polynomials in the time t the splitting of merged modes under a perturbation is also treated one thus obtains a non trivial extension of the bb formalism for open systems

excited bound state logarithmic perturbation theory without nodes the logarithmic perturbation theory is modified slightly in order to deal with excited states instead of considering a real wavefunction describing the physical stationary state the authors consider a complex wavefunction at the same energy by mixing in the ghost state for excited bound states the former has nodes while the latter is guaranteed not to have any nodes and can be represented simply as exp g to which the logarithmic perturbation method can be applied in a straightforward manner the physical entities the energy corrections are independent of the amount of mixing of the ghost state the connection to the green function method is also shown the freedom to mix in the ghost state allows the authors to justify an ad hoc approach whereby the simple version of the logarithmic perturbation theory is applied to excited bound states the formalism is illustrated with simple examples

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 hartogs extension theorem on n 1 complete complex spaces let x be a connected normal complex space of dimension n 2 which is cohomologically n 1 complete and let m x be a resolution of singularities by use of takegoshi s generalization of the grauertriemenschneider vanishing theorem we deduce h cpt m o 0 which in turn implies hartogs extension theorem on x by the technique of ehrenpreis

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

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

numerical algorithm for the determination of the potential of a conservative system from its normal mode spectra an efficient numerical algorithm is developed for determining the potential of a conservative system described by a klein gordon or schrodinger equation from its normal mode spectra the algorithm is based on first solving the linear problem of perturbative inversion by means of tikhonov regularization and then using a series of perturbative steps to approach the full potential by newton s method the algorithm is stable and convergence is rapid

susy transformations for quasinormal modes of open systems supersymmetry susy in quantum mechanics is extended from square integrable states to those satisfying the outgoing wave boundary condition in a klein gordon formulation this boundary condition allows both the usual normal modes and quasinormal modes with complex eigenvalues the simple generalization leads to three features the counting of eigenstates under susy becomes more systematic the linear space structure of outgoing waves nontrivially different from the usual hilbert space of square integrable states is preserved by susy and multiple states at the same frequency not allowed for normal modes are also preserved the existence or otherwise of susy partners is furthermore relevant to the question of inversion are open systems uniquely determined by their complex outgoing wave spectra

unbounded derivations commuting with compact group actions let be a closed derivation in ac algebra mathfrak a which commutes with an ergodic action of a compact group on mathfrak a then generates ac dynamics of mathfrak a similar results are obtained for non ergodic actions on abelianc algebras and on the algebra of compact operators

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

berry s phase and aharonov anandan s phase abstract the aharonov anandan geometric phase is generalized to non unitary evolution and is shown to be always real by using a counter example which is exactly solvable it is shown that berry s geometric phase is not always the adiabatic limit of aharonov anandan s geometric phase for a non hermitian driving hamiltonion the reason is then given

some comments on the kim sukhatme alternative approach to nonrelativistic perturbation theory the new perturbation theory expressions obtained by kim and sukhatme 1992 are commented on and some new results are obtained starting from their expressions

comments on the dalgarno lewis method as a perturbation theory by nandi et al 1996 j phys a math gen 29 1101 and on alternate approach to non relativistic perturbation theory by kim and sukhatme 1992 j phys a math gen 25 l647 i point out the connection between the above papers and some earlier work appearing in 1984 phys rev a 29 1034 and in 1979 phys rev a 20 2245