periodic potentials and pt symmetry we enlarge the limited class of analytically solvable one dimensional periodic potentials with a finite number of band gaps by considering the complex pt symmetric generalized associated lame gal potential with four parameters and applying supersymmetry we obtain three duality relations which relate the band edge eigenstates of the twenty four gal potentials obtained by permutation of the four parameters

equivalence of unstable anharmonic oscillators and double wells we give a direct and general proof of the equality of the energy levels of a pair of different quantum problems the double well and the unstable anharmonic oscillator defined by complex translation the resonances of the two problems are also equal and are the analytic continuations of the anharmonic oscillator energy levels

a single saddle model for the β relaxation in supercooled liquids we analyse the relaxational dynamics of a system close to a saddle of the potential energy function within an harmonic approximation our main aim is to relate the topological properties of the saddle as encoded in its spectrum to the dynamical behaviour of the system in the context of the potential energy landscape approach this represents a first formal step to investigate the belief that the dynamical slowing down at tc is related to the vanishing of the number of negative modes found at the typical saddle point in our analysis we keep the description as general as possible using the spectrum of the saddle as an input we prove the existence of a timescale t which is uniquely determined by the spectrum but is not simply related to the fraction of negative eigenvalues the mean square displacement develops a plateau of length t such that a two step relaxation is obtained if t diverges at tc we analyse different spectral shapes and outline the conditions under which the mean square displacement exhibits a dynamical scaling identical to the relaxation regime of mode coupling theory with a power law approach to the plateau and power law divergence of t at tc

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

on the shape of spectra for non self adjoint periodic schrodinger operators the spectra of the schrodinger operators with periodic potentials are studied when the potential is real and periodic the spectrum consists of at most countably many line segments energy bands on the real line while when the potential is complex and periodic the spectrum consists of at most countably many analytic arcs in the complex plane in some recent papers such operators with complex symmetric periodic potentials have been studied in particular the authors argued that some energy bands would appear and disappear under perturbations here we show that the appearance and disappearance of such energy bands imply existence of nonreal spectra this is a consequence of a more general result describing the local shape of the spectrum

the cal pt symmetric rosen morse ii potential effects of the asymptotically non vanishing imaginary potential component bound and scattering solutions of the symmetric rosen morse ii potential are investigated the energy eigenvalues and the corresponding wavefunctions are written in a closed analytic form and it is shown that this potential always supports at least one bound state it is found that with increasing non hermiticity the real bound state energy spectrum does not turn into complex conjugate pairs i e the spontaneous breakdown of symmetry does not occur rather the energy eigenvalues remain real and shift to positive values closed expression is found for the pseudo norm of the bound states and its sign is found to follow the 1 n rule similarly to the known scattering examples the reflection coefficients exhibit a handedness effect while the transmission coefficient picks up a complex phase factor when the direction of the incoming wave is reversed it is argued that the unusual findings might be caused by the asymptotically non vanishing though finite imaginary potential component comparison with the real rosen morse ii potential is also made

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

exact solutions for the statistics of extrema of some random 1d landscapes application to the equilibrium and the dynamics of the toy model abstract the real space renormalization group rsrg method introduced previously for the brownian landscape is generalized to obtain the joint probability distribution of the subset of the important extrema at large scales of other one dimensional landscapes for a large class of models we give exact solutions obtained either by the use of constrained path integrals in the continuum limit or by solving the rsrg equations via an ansatz which leads to the liouville equation we apply in particular our results to the toy model energy landscape which consists in a quadratic potential plus a brownian potential which describes among others the energy of a single domain wall in a 1d random field ising model rfim in the presence of a field gradient the measure of the renormalized landscape is obtained explicitly in terms of airy functions and allows to study in detail the boltzmann equilibrium of a particle at low temperature as well as its non equilibrium dynamics for the equilibrium we give results for the statistics of the absolute minimum which dominates at zero temperature and for the configurations with nearly degenerate minima which govern the thermal fluctuations at very low temperature for the dynamics we compute the distribution over samples of the equilibration time or equivalently the distribution of the largest barrier in the system we also study the properties of the rare configurations presenting an anomalously large equilibration time which govern the long time dynamics we compute the disorder averaged diffusion front which interpolates between the kesten distribution of the sinai model at short rescaled time and the reaching of equilibrium at long rescaled time finally the method allows to describe the full coarsening i e many domain walls of the 1d rfim in a field gradient as well as its equilibrium

on solvable potentials related to so 2 2 scattering problems in one dimension associated with the so 2 2 group are studied the s matrices for the systems under consideration are calculated by using the theory of intertwining operators for semisimple lie groups the wave functions are expressed in terms of matrix elements of principal series representation of the so 2 1 group

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

conjecture on the interlacing of zeros in complex sturm liouville problems the zeros of the eigenfunctions of self adjoint sturm liouville eigenvalue problems interlace for these problems interlacing is crucial for completeness for the complex sturm liouville problem associated with the schrodinger equation for a non hermitian pt symmetric hamiltonian completeness and interlacing of zeros have never been examined this paper reports a numerical study of the sturm liouville problems for three complex potentials the large n limit of a ix n potential a quasiexactly solvable x4 potential and an ix3 potential in all cases the complex zeros of the eigenfunctions exhibit a similar pattern of interlacing and it is conjectured that this pattern is universal understanding this pattern could provide insight into whether the eigenfunctions of complex sturm liouville problems form a complete set

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

density and correlation functions of vortex and saddle points in open billiard systems we present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random wave model rwm in particular due to an improvement of the experimental setup we determine experimentally the spatial correlation of saddle points of the current field an asymptotic expression for the vortex saddle and saddle saddle correlation functions based on the rwm is derived with experiment and theory agreeing well we also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the rwm and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying dirichlet conditions

moyal products a new perspective on quasi hermitian quantum mechanics the rationale for introducing non hermitian hamiltonians and other observables is reviewed and open issues identified we present a new approach based on moyal products to compute the metric for quasi hermitian systems this approach is not only an efficient method of computation but also suggests a new perspective on quasi hermitian quantum mechanics which invites further exploration in particular we present some first results which link the berry connection and curvature to non perturbative properties and the metric

poschl teller paradoxes in the well known poschl teller trigonometric potential well a symmetric regularization x x ie of the impenetrable end point barriers is performed this leads to the four different solvable generalizations of the model as a byproduct the scheme clarifies certain apparent paradoxes encountered in the classically forbidden coupling regime

the chaotic analytic function in contrast to a stationary gaussian random function of a real variable which is free to have any correlation function the closest analogous analytic random function in the complex plane has no true freedom it is statistically unique since it has arisen only recently as an apparently universal feature in the physical context of quantum chaos i refer to it here as the chaotic analytic function i note that it is implied by the assumption that a quantum chaotic wavefunction has gaussian randomness and has a constant value for the average of its wigner function in phase space interpreted literally this shows that the chaotic analytic function is the bargmann function of a pure white noise wavefunction more physically if constant is replaced by smooth on the scale of a planck area these assumptions are the semiclassical ones made by berry for chaotic eigenstates the analysis shows that the chaotic analytic function is still obtained semiclassically

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

on scattering systems related to the group scattering systems related to the noncompact groups g in the sense that the hamiltonian of the system can be written as a function of the casimir operator of g are considered the s matrix for such systems are defined in terms of an intertwining operator of underling symmetry group g the s matrices for one dimensional scattering systems with symmetry group are classified

algebraic approach to non central potentials an algebraic approach to class of separable non central hamiltonians is presented we show that the bound states of quantum systems under consideration are described by unitary representations of the algebra

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

models with sl 2 c symmetry and their s matrices we study quantum systems whose scattering modes are governed by unitary representation of the algebra the s matrices of the systems under consideration are defined from intertwining relations for the weyl equivalent representations of the group sl 2 c or its lie algebra

energy distribution of maxima and minima in a one dimensional random system we study the energy distribution of maxima and minima of a simple one dimensional disordered hamiltonian we find that in systems with short range correlated disorder there is energy separation between maxima and minima such that at fixed energy only one kind of stationary points is dominant in number over the other on the other hand in the case of systems with long range correlated disorder maxima and minima are completely mixed

quasi exact solvability and entropies of the one dimensional regularised calogero model the calogero model can be regularised through the introduction of a cutoff parameter which removes the divergence in the interaction term in this work we show that the one dimensional two particle regularised calogero model is quasi exactly solvable and that for certain values of the hamiltonian parameters the eigenfunctions can be written in terms of heun s confluent polynomials these eigenfunctions are such that the reduced density matrix of the two particle density operator can be obtained exactly as well as its entanglement spectrum we found that the number of non zero eigenvalues of the reduced density matrix is finite in these cases the limits for the cutoff distance going to zero calogero and infinity are analysed and all the previously obtained results for the calogero model are reproduced once the exact eigenfunctions are obtained the exact von neumann and renyi entanglement entropies are studied to characterise the physical traits of the model the quasi exactly solvable character of the model is assessed studying the numerically calculated renyi entropy and entanglement spectrum for the whole parameter space

the klein gordon equation of generalized hulthen potential in complex quantum mechanics we have investigated the reality of exact bound states of complex and or pt symmetric non hermitian exponential type generalized hulthen potential the klein gordon equation has been solved by using the nikiforov uvarov method which is based on solving the second order linear differential equations by reduction to a generalized equation of hypergeometric type in many cases of interest negative and positive energy states have been discussed for different types of complex potentials

gradual spontaneous breakdown of mathcal pt symmetry in a solvable potential a new solvable potential is introduced by transforming the schrodinger equation into the differential equation of the jacobi polynomials the z x variable transformation function depends on two parameters c and and can be obtained implicitly as x z the remaining two parameters set the strength of the even and odd potential components systematic analysis is given for c 0 0 when the potential contains the scarf ii and rosen morse i potentials as special cases the normalized bound state solutions are obtained in closed form both in the real and symmetric cases the energy eigenvalues are determined from the roots of a quartic algebraic equation despite this implicitly defined spectrum the spectral structure of the potential can be determined easily in the symmetric case the spontaneous breakdown of symmetry can also be studied in a controllable way and it is found that with increasing non hermiticity the complexification of the energy eigenvalues is a gradual process that starts with the ground state the general form of the potential contains the scarf i rosen morse ii generalized poschl teller eckart symmetric ginocchio and dutt khare varshni dkv potentials too and can be identified as a subclass of the general six parameter natanzon class this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

matching method and exact solvability of discrete symmetric square wells discrete symmetric square wells are studied their wavefunctions are found to be proportional to classical tshebyshev polynomials of complex argument the compact secular equations for energies are derived giving the real spectra in certain intervals of non hermiticity strengths z it is amusing to note that although the known square well re emerges in the usual continuum limit a twice as rich upside down symmetric spectrum is exhibited by all its present discretized predecessors

the band spectrum of periodic potentials with script pscript t symmetry a real band condition is shown to exist for one dimensional periodic complex non hermitian potentials exhibiting symmetry we use an exactly solvable ultralocal periodic potential to obtain the band structure and discuss some spectral features of the model specially those concerning the role of the imaginary parameters of the couplings analytical results as well as some numerical examples are provided

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

on the pseudo norm and admissible solutions of the symmetric scarf i potential the physically admissible solutions of the symmetric scarf i potential are identified in the domain of real and complex energies it is found that generally there are no admissible complex energy solutions and there is one with real energy in a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi parity q however the two sets are not orthogonal in the domain of real energies the sign of the pseudo norm of states with real energy is found to oscillate as 1 n in accordance with the example of other symmetric potentials possessing an infinite number of discrete levels it is argued that the spontaneous breakdown of symmetry cannot be defined for the scarf i potential a comparison with some symmetric extensions of the infinite square well is presented

signed zeros of gaussian vector fields density correlation functions and curvature we calculate correlation functions of the signed density of zeros of gaussian distributed vector fields we are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract riemann cartan or riemannian manifold as an application we discuss one and two point functions the zeros of a two dimensional gaussian vector field model the distribution of topological defects in the high temperature phase of two dimensional systems with orientational degrees of freedom such as superfluid films thin superconductors and liquid crystals

statistics of nodal points of in plane random waves in elastic media we consider the nodal points nps u 0 and v 0 of the in plane vectorial displacements u u v which obey the navier cauchy equation similar to the berry conjecture of quantum chaos we present the in plane eigenstates of chaotic billiards as the real part of the superposition of longitudinal and transverse plane waves with random phases by an average over random phases we derive the mean density and correlation function of nps consequently we consider the distribution of the nearest distances between nps

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

path integral quantization of certain noncentral systems with dynamical symmetries path integral quantization is done for the five classes of potentials appearing in the systematic search for nonrelativistic systems with dynamical symmetries done by makarov smorodinsky valiev and winternitz nuovo cimento a 52 1061 1967 by an iterated application of bateman s series formula to the polar coordinate path integral an expansion is obtained of the feynman kernel or the green s function whichever is possible in terms of hypergeometric functions of the polar and azimuthal parts and a radial path integral is obtained whose evaluation yields the energy eigenvalues and the normalized wave functions special cases include the hartmann potential and the ring shaped oscillator

coupled channel version of the pt symmetric square well a coupled pair of pt symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non hermitian commuting observables via the diagonalization of our hamiltonian h h and spin like observable we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues en n 0 1 energies and 1 spin projections remain real in such a measurable regime the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction dependent scalar products

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

shape invariant potentials with pt symmetry suitable complexification of the well known hyperbolic rosen morse oscillator in one dimension is shown to give the fourth and last exactly solvable model which combines the shape and invariance

some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant comparison between the exact value of the spectral zeta function zh 1 5 6 5 3 2cos 5 2 1 5 3 5 and the results of numeric and wkb calculations supports the conjecture by daniel bessis 1995 private communication that all the eigenvalues of this pt invariant hamiltonian are real for one dimensional schrodinger operators with complex potentials having a monotonic imaginary part the eigenfunctions and the imaginary parts of their logarithmic derivatives have no real zeros

antilinear deformations of coxeter groups an application to calogero models we construct complex root spaces remaining invariant under antilinear involutions related to all coxeter groups we provide two alternative constructions one is based on deformations of factors of the coxeter element and the other based on the deformation of the longest element of the coxeter group motivated by the fact that non hermitian hamiltonians admitting an antilinear symmetry may be used to define consistent quantum mechanical systems with real discrete energy spectra we subsequently employ our constructions to formulate deformations of coxeter models remaining invariant under these extended coxeter groups we provide explicit and generic solutions to the schrodinger equation of these models for the eigenenergies and corresponding wavefunctions a new feature of these novel models is that when compared with the undeformed case their solutions are usually no longer singular for an exchange of an amount of particles less than the dimension of the representation space of the roots the simultaneous scattering of all particles in the model leads to anyonic exchange factors for processes which have no analogue in the undeformed case

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

correlations and screening of topological charges in gaussian random fields two point topological charge correlation functions of several types of geometric singularity in gaussian random fields are calculated explicitly using a general scheme zeros of n dimensional random vectors signed by the sign of their jacobian determinant critical points gradient zeros of real scalars in two dimensions signed by the hessian and umbilic points of real scalars in two dimensions signed by their index the functions in each case depend on the underlying spatial correlation function of the field these topological charge correlation functions are found to obey the first stillinger lovett sum rule for ionic fluids

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

symmetrically regularized eckart poschl teller and hulthen potentials parity times time reversal symmetry of complex hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for hermiticity perhaps an equally important role is played by the related strengthened analyticity assumptions in a constructive illustration we complexify a few potentials solvable only in s wave then we continue their domain from the semi axis to the whole axis and obtain new exactly solvable models their energies turn out to be real as expected the new one dimensional spectra themselves differ quite significantly from their s wave predecessors

construction of the operator for a symmetric model we obtain a closed form expression of the operator for the symmetric scarf i potential it is also shown that the eigenfunctions form a complete set

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

pt symmetry and its spontaneous breakdown in three dimensions the mechanism of introducing non hermiticity to non central symmetric potentials through both the azimuth and polar angles is discussed generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials such as the scarf i or rosen morse i potentials are used in the eigenvalue equation of the polar component it is shown that the spontaneous breakdown of symmetry can also occur in these non central potentials several potentials are proposed in the azimuthal eigenvalue equation too where the use of periodic boundary conditions is essential possible generalizations of the results are outlined

a lie algebraic approach to non hermitian hamiltonians with real spectra an algebraic technique useful in studying non hermitian hamiltonians with real spectra is presented the method is illustrated by explicit application to a family of one dimensional potentials

on the existence of compressive solitary waves in compacting media necessary conditions for the existence of compressive solitary wave solutions of a partial differential equation derived by scott and stevenson 1986 which describes the two phase fluid flow in a medium compacting under gravity are derived it is shown that for compressive solitary wave solutions to exist which satisfy certain boundary conditions it is necessary that n m 1 where n and m are the exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix respectively to the voidage the effect of the value of the exponents n and m on the shape of the solitary wave is investigated by using existing analytical solutions and new numerical solutions

a family of complex potentials with real spectrum we consider a two parameter non hermitian quantum mechanical hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions in contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane our model is integrable along a line parallel to the real axis

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

remarks on the pseudo norm in symmetric quantum mechanics this paper presents an underlying analytical relationship between the pseudo norm associated with the symmetric hamiltonian h p2 v q and the stokes multiplier of the differential equation corresponding to this hamiltonian we show that the sign alternation of the pseudo norm which has been observed as a generic feature of the inner product is essentially controlled by the derivative of a stokes multiplier with respect to the eigenparameter

calculation of the hidden symmetry operator for a symmetric square well it has been shown that a hamiltonian with an unbroken symmetry also possesses a hidden symmetry that is represented by the linear operator this symmetry operator guarantees that the hamiltonian acts on a hilbert space with an inner product that is both positive definite and conserved in time thereby ensuring that the hamiltonian can be used to define a unitary theory of quantum mechanics in this paper it is shown how to construct the operator for the symmetric square well using perturbative techniques

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

non hermitian multi particle systems from complex root spaces we provide a general construction procedure for antilinearly invariant complex root spaces the proposed method is generic and may be applied to any weyl group allowing us to take any element of the group as a starting point for the construction worked out examples for several specific weyl groups are presented focusing especially on those cases for which no solutions were found previously when applied to the defining relations of models based on root systems this usually leads to non hermitian models which are nonetheless physically viable in a self consistent sense as they are antilinearly invariant by construction we discuss new types of calogero models based on these complex roots in addition we propose an alternative construction leading to q deformed roots we employ the latter type of roots to formulate a new version of affine toda field theories based on non simply laced root systems these models exhibit on the classical level a strong weak duality in the coupling constant equivalent to a lie algebraic duality which is known for the quantum version of the undeformed case

periodic potentials and supersymmetry we review the current status of one dimensional periodic potentials and also present several new results it is shown that using the formalism of supersymmetric quantum mechanics one can considerably enlarge the limited class of analytically solvable one dimensional periodic potentials further using the landen transformations as well as cyclic identities for jacobi elliptic functions discovered by us recently it is shown that a linear superposition of lame as well as associated lame potentials is also analytically solvable finally using anti isospectral transformations we also obtain a class of analytically solvable complex pt invariant periodic potentials having real band spectra

nonlocal symmetric potentials the factorization approach for complex hamiltonians has been used to obtain the exactly solvable nonlocal variant of symmetric local potentials the formalism is used to obtain exact eigenvalues and eigenfunctions of the nonlocal symmetric scarf potential

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

orientational relaxation in a dispersive dynamic medium generalization of the kubo ivanov anderson jump diffusion model to include fractional environmental dynamics the ivanov anderson model and an earlier treatment by kubo envisages a decay of the orientational correlation by random but large amplitude molecular jumps as opposed to infinitesimal small jumps assumed in brownian diffusion recent computer simulation studies on water and viscous liquids have shown that large amplitude motions may indeed be more of a rule than exception existing theoretical studies on jump diffusion mostly assume an exponential poissonian waiting time distribution for jumps thereby again leading to an exponential decay here we extend the existing formalism of ivanov and anderson to include an algebraic waiting time distribution between two jumps as a result the first l 1 and second l 2 rank orientational time correlation functions show the same long time power law but their short time decay behavior is quite different the predicted cole cole plot of dielectric relaxation reproduces various features of non debye behavior observed experimentally we also developed a theory where both unrestricted small jumps and large angular jumps coexist simultaneously the small jumps are shown to have a large effect on the long time decay particularly in mitigating the effects of algebraic waiting time distribution and in giving rise to an exponential like decay with a time constant surprisingly less than the time constant that arises from small amplitude decay alone

vortex free complex landscapes and umbilic free real landscapes although complex scalar functions varying over the infinite plane r x y commonly possess vortices points of phase singularity it is possible to devise exceptional functions with no vortices similarly although real scalar functions representing surfaces as small departures from the plane commonly possess umbilic locally spherical points it is possible to devise exceptional functions that do not possess them examples of both are given

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

the morphology of nodal lines random waves versus percolation in this paper we investigate the properties of nodal structures in random wave fields and in particular we scrutinize their recently proposed connection with short range percolation models we propose a measure which shows the difference between monochromatic random waves which are characterized by long range correlations and gaussian fields with short range correlations which are naturally assumed to be better modelled by percolation theory we also study the relevance of the quantities which we compute to the probability that nodal lines are in the vicinity of a given reference line

complex square well a new exactly solvable quantum mechanical model recently a class ofpt invariant quantum mechanical models described by the non hermitian hamiltonianhdp2 x2 ix was studied it was found that the energy levels for this theory are real for all 0 here the limit as 1is examined it is shown that in this limit the theory becomes exactly solvable a generalization of this hamiltonian h d p 2 x 2m ix md 1 2 3 is also studied and thispt symmetric hamiltonian becomes exactly solvable in the large limit as well in effect what is obtained in each case is a complex analogue of the hamiltonian for the square well potential expansions about the large limit are obtained

metric operator in pseudo hermitian quantum mechanics and the imaginary cubic potential we present a systematic perturbative construction of the most general metric operator and positive definite inner product for quasi hermitian hamiltonians of the standard form h 1 2 p 2 v x in one dimension we show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in 1 1 dimensions for the case that v x is purely imaginary the latter have the form of a nonhomogeneous wave equation which admits an exact solution we apply our general method to obtain the most general metric operator for the imaginary cubic potential v x i x 3 this reveals an infinite class of previously unknown cpt as well as non cpt inner products we compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system our results for the imaginary cubic potential show that unlike the quantum system the corresponding classical system is not sensitive to the choice of the metric operator as another application of our method we give a complete characterization of the pseudo hermitian canonical quantization of a free particle moving in r that is consistent with the usual choice for the quantum hamiltonian finally we discuss subtleties involved with higher dimensions and systems having a fixed length scale

four easy pieces this paper is a mini summary of four new advances in the theory of pt symmetric quantum mechanics it describes some new calculations that were completed in 2005 the first advance concerns the classical coordinate space trajectories in some pt symmetric theories depending on the initial conditions one can find arbitrarily long periodic pt symmetric classical trajectories the longer trajectories originate from smaller regions of initial conditions there is an interesting resemblance to the so called period doubling route to chaos the second advance concerns the perturbative construction of the c operator for a pt symmetric square well the result has notable similarities to and differences from that for the pt symmetric cubic oscillator the third advance is a detailed comparison of a pt symmetric hamiltonian and the corresponding hermitian hamiltonian it is argued that a perturbative construction of the hermitian hamiltonian leads to an almost intractable theory the fourth advance concerns reflectionless potentials and pt symmetry and a possible connection with problems in cosmology

bender wu branch points in the cubic oscillator the analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods bender wu branch points at which level crossing occurs are calculated and labelled by a process of analytic continuation the different resonances are the values that a single analytic function takes on different sheets of a riemann surface whose topology is described

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

classical trajectories of 1d complex non hermitian hamiltonian systems classical motion of complex 1d non hermitian hamiltonian systems is investigated analytically to identify periodic unbounded and chaotic trajectories expressions for the lyapunov exponent for 1d complex hamiltonians are derived complex potentials and v2 x x3 are studied in detail and their lyapunov exponents are obtained analytically it was found that when is complex all the trajectories of v1 are chaotic with lyapunov exponent im and most of the trajectories of v2 are periodic when is pure imaginary but for other complex values of trajectories of v2 are non periodic and show infinite oscillations unbounded neighbouring trajectories of v2 show power law divergence rather than exponential divergence as in the case of v1

distribution of nearest distances between nodal points for the berry function in two dimensions according to berry a wave chaotic state may be viewed as a superposition of monochromatic plane waves with random phases and amplitudes here we consider the distribution of nodal points associated with this state using the property that both the real and imaginary parts of the wave function are random gaussian fields we analyze the correlation function and densities of the nodal points using two approaches the poisson and bernoulli we derive the distribution of nearest neighbor separations furthermore the distribution functions for nodal points with specific chirality are found comparison is made with results from numerical calculations for the berry wave function

statistics of nodal lines and points in chaotic quantum billiards perimeter corrections fluctuations curvature for real time reversal symmetric quantum billiards the mean length l of nodal line is calculated for the nth mode n 1 with wavenumber k using a gaussian random wave model adapted locally to satisfy dirichlet or neumann boundary conditions the leading term is of order k i e n and the first perimeter correction dominated by an unanticipated long range boundary effect is of order log k i e log n with the same sign negative for both boundary conditions the leading order state to state fluctuations l are of order log k for the curvature of nodal lines and 2 are of order k but 3 and higher moments diverge for complex e g aharonov bohm billiards the mean number n of nodal points phase singularities in the mode has a leading term of order k2 i e n the perimeter correction again a long range effect is of order klog k i e nlog n and positive notwithstanding nodal depletion near the boundary and the fluctuations n are of order k log k generalizations of the results for mixed robin boundary conditions are stated

solvability and pt symmetry in a double well model with point interactions the concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non hermitian so called symmetric quantum mechanical systems this is illustrated with a double well model the easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous symmetry breaking at a certain strong non hermiticity boundary below this limit the model is shown to be suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in symmetric quantum mechanics

integrable models from mathcal pt symmetric deformations we address the question of whether integrable models allow for symmetric deformations which preserve their integrability for this purpose we carry out the painleve test for symmetric deformations of burgers and the korteweg de vries equations we find that the former equation allows for infinitely many deformations which pass the painleve test for a specific deformation we prove the convergence of the painleve expansion and thus establish the painleve property for these models which are therefore thought to be integrable the korteweg de vries equation does not allow for deformations which pass the painleve test in complete generality but we are able to construct a defective painleve expansion

extrema statistics in the dynamics of a non gaussian random field when the equations that govern the dynamics of a random field are nonlinear the field can develop with time non gaussian statistics even if its initial condition is gaussian here we provide a general framework for calculating the effect of the underlying nonlinear dynamics on the relative densities of maxima and minima of a two dimensional field using this simple geometrical probe we can identify the size of the non gaussian contributions in the random field or alternatively the magnitude of the nonlinear terms in the underlying equations of motion we demonstrate our approach by applying it to an initially gaussian field that evolves according to the deterministic kpz equation which models surface growth and shock dynamics

numerical evidence that the perturbation expansion for a non hermitian pt symmetric hamiltonian is stieltjes recently several studies of non hermitian hamiltonians having pt symmetry have been conducted most striking about these complex hamiltonians is how closely their properties resemble those of conventional hermitian hamiltonians this paper presents further evidence of the similarity of these hamiltonians to hermitian hamiltonians by examining the summation of the divergent weak coupling perturbation series for the ground state energy of the pt symmetric hamiltonian h p2 14x2 i x3 recently studied by bender and dunne for this purpose the first 193 nonzero coefficients of the rayleigh schrodinger perturbation series in powers of 2 for the ground state energy were calculated pade summation and pade prediction techniques recently described by weniger are applied to this perturbation series the qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator which is known to be a stieltjes se

algorithmic complexity in the minority game in this paper we present our approach for the study of the complexity of minority game using tools from thermodynamics and statistical physics previous attempts were based on the behavior of volatility an observable of the financial markets our approach focuses on some properties of the binary stream of outcomes of the game physical complexity a magnitude rooted in kolmogorov chaitin theory allows us to explain some properties of collective behavior of the agents mutual information function a measure related to shannon s information entropy was useful to observe a kind of phase transition when applied to the binary string of the whole history of the game

pt symmetric potentials and the so 2 2 algebra starting from a differential realization of the generators of the so 2 2 algebra we connect the eigenvalue equation of the casimir invariant either with the hypergeometric equation or the schr odinger equation in the latter case we consider problems for whichso 2 2 appears as a potentia la lgebra connecting states with the same energy in different potentials we analyse the role of the two so 2 1 subalgebras and point out their importance for pt symmetric problems where the doubling of bound states is known to occur we present two mechanisms for this and illustrate them with the example of the scarf and the p oschl teller ii potentials we also analyse scattering states transmission

dynamical symmetry group and quantum splittings for a free particle on the group manifold ads3 it is shown that the set of all quantum states corresponding to the motion of a free particle on the group manifold ads3 as the bases with two different labels constitute a hilbert space the second label is bounded by the first one however the first label is semibounded the casimir operator corresponding to the simultaneous and agreeable shifting generators of both labels along with the cartan subalgebra generator describe the hamiltonian of a free particle on ads3 with dynamical symmetry group u 1 1 and infinite fold degeneracy for the energy spectrum the hilbert space for the lie algebra of the dynamical symmetry group is a reducible representation space but the hilbert subspaces constructed by all the bases which have a given constant value for the difference of two their labels constitute an irreducible representation for it it is also shown that the irreducible representation subspaces of the lie algebras u 1 1 and u 2 are separately spanned by the bases which have the same value for the s