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

the channel capacity of a certain noisy timing channel the effect of noise on a simple covert timing channel is investigated shannon s information theory is used to quantify the resulting information flow across the channel in particular how a probabilistic response time to a query by the receiver affects the mutual information and channel capacity is studied the channel capacity is expressed in terms of the critical probability for the mutual information function which is given in closed form in terms of wright s hypergeometric function

m dimensional lattice sums of generalized hypergeometric functions representations and convergence criteria for infinite m dimensional lattice sums of generalized hypergeometric functions pfp 1 are deduced by appealing to the principle of mathematical induction in particular we show that such a lattice sum may be expressed essentially as a finite sum of mellin transforms of products of bessel functions of order m 2 and the functions pfp 1 in the lattice sum in addition a direct derivation for the three dimensional case is provided moreover we construct a countably infinite class of null functions on increasingly larger open intervals which are parametrically independent of the functions pfp 1 generating them

spectral differentiation matrices for the numerical solution of schrodinger s equation schrodinger s equation is solved numerically using the spectral collocation pseudospectral method based on hermite weighted polynomial interpolants several sets of numerical results from the literature are reproduced using this approach matlab codes that can serve as templates for further exploration are provided both in the paper and online a new proposal is made for solving schrodinger s equation in stokes wedges

on the pseudo hermitian nondiagonalizable hamiltonians we consider a class of possibly nondiagonalizable pseudo hermitian operators with discrete spectrum and we establish for such a class the equivalence between the pseudo hermiticity property and the existence of an antilinear involutory symmetry moreover we prove that this class actually coincides with the one of possibly nondiagonalizable weak pseudo hermitian operators and that in no case unless they are diagonalizable and have a real spectrum they are hermitian with respect to a definite inner product finally we show that a typical degeneracy of the real eigenvalues which reduces to the well known kramers degeneracy in the hermitian case occurs whenever a fermionic possibly nondiagonalizable pseudo hermitian hamiltonian admits an antilinear symmetry like the time reversal operator t some consequences and applications are briefly discussed

universal properties of voronoi tessellations of hard discs the authors describe a 2d mosaic obtained by the voronoi tessellation of a monosize assembly of discs at different packing fractions the experimental device hard discs moving on an air table produces for every packing fraction a succession of mosaics in statistical equilibrium which constitutes a statistical ensemble this ensemble is large enough for fluctuations from the most probable distributions to be negligible they use the maximum entropy principle to get the distribution of the polygons in 2d mosaics generated from an assembly of hard discs steric exclusion yields an extra conservation law which is sufficient to give a good agreement with the experimental data a similar behaviour in the six fold parameter seems to hold for other mosaics

nonnuclear subalgebras of af algebras we show that any non type i separable unital af algebra b can be modeled from inside by a nonnuclear c algebra and from outside by a nonexact c algebra more precisely there exist unital separable quasidiagonal c algebras a b c of real rank zero stable rank one such that a is nonnuclear c is nonexact and both a and c are asymptotically homotopy equivalent to b in particular a b and c have the same ordered k theory groups hence isomorphic ideal lattices and a and b have affinely homeomorphic trace spaces

airy function exact wkb results for potentials of odd degree an exact wkb wentzel kramers brillouin treatment of 1d homogeneous schrodinger operators with the confining potentials n even is extended to odd degrees n the resulting formalism is first illustrated theoretically and numerically upon the spectrum of the cubic oscillator potential concerning the linear potential n 1 the theory exhibits a duality in which the airy functions ai become paired with the spectral determinants of the quartic oscillator n 4 classic identities for the airy function as well as some less familiar ones appear in this new perspective as special cases in a general setting

generating converging bounds to the complex discrete states of the p2 ix3 iαx hamiltonian the eigenvalue moment method emm is applied to the h p2 ix3 i x hamiltonian enabling the algebraic numerical generation of converging bounds to the complex energies of the l2 states as argued through asymptotic methods by delabaere and trinh the robustness of the formalism and its computational implementation suggest that the present non negativity formulation implicitly contains the key algebraic relations by which to prove bessis conjecture that the eigenenergies of the h0 hamiltonian are real the required algebraic analysis of the emm procedure pertaining to this problem will be presented in a forthcoming paper

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

analytic continuation of eigenvalues of a quartic oscillator we consider the schrodinger operator on the real line with even quartic potential x4 x2 and study analytic continuation of eigenvalues as functions of parameter we prove several properties of this analytic continuation conjectured by bender wu loeffel and martin 1 all eigenvalues are given by branches of two multi valued analytic functions one for even eigenfunctions and one for odd ones 2 the only singularities of these multi valued functions in the complex plane are algebraic ramification points and there are only finitely many singularities over each compact subset of the plane

optical lattices with pt symmetry are not transparent in diffraction of a plane wave by a non hermitian pt symmetric optical lattice the sum of the bragg beam intensities need not be conserved even though the gain and loss are equally distributed the evolution is not unitary instead different sums are conserved in which the intensities are weighted with real numbers positive or negative several such sum rules are derived two beam diffraction from a refractive index of the form constant a cos x ib sin x is studied in detail the sum rule depends on the balance between the real hermitian parameter a and the real anti hermitian parameter b

lattice sums of generalized hypergeometric functions representations and convergence criteria for one dimensional 1d and two dimensional 2d lattice sums of generalized hypergeometric functions pfp 1 are derived by using apparently new integral representations for the latter functions and 1d and 2d forms of the poisson summation formula these lattice sum representations concisely unify and generalize various specializations given previously by several authors and should prove useful in analysing finite size effects in systems subjected to non periodic boundary conditions

extension of a spectral bounding method to the pt invariant states of the ix n non hermitian potential the eigenvalue moment method emm developed by handy j phys a math gen 2001 34 l271 j phys a math gen 2001 34 5065 handy et al j phys a math gen 2001 34 5593 and handy and xiao qian wang j phys a math gen 2001 34 8297 which generates converging lower and upper bounds to the complex discrete state energies is extended to the case of discrete states with non real support in particular bender and boettcher phys rev lett 1998 80 5243 have argued on the reality of the discrete state spectrum for the ix n potential for n integer 4 such pt invariant solutions can only exist on appropriate complex contours we develop and apply the necessary emm formalism to such cases in particular the restriction of emm to the anti stokes angles significantly increases the convergence rate of the bounds

decomposition rank and z stability we show that separable simple nonelementary unital c algebras with finite decomposition rank absorb the jiang su algebra mathcal z tensorially this has a number of consequences for elliott s program to classify nuclear c algebras by their k theory data in particular it completes the classification of c algebras associated to uniquely ergodic smooth minimal dynamical systems by their ordered k groups

two parametric pt symmetric quartic family we describe a parametrization of the real spectral locus of the two parametric family of pt symmetric quartic oscillators for this family we find a parameter region where all eigenvalues are real extending the results of dorey et al 2007 j phys a math theor 40 r205 83 and shin 2005 j phys a math gen 38 6147 66 2002 commun math phys 229 543 64

revisiting the fradkin vilkovisky theorem the status of the usual statement of the fradkin vilkovisky theorem claiming complete independence of the batalin fradkin vilkovisky path integral on the gauge fixing fermion even within a nonperturbative context is critically reassessed basic but subtle reasons why this statement cannot apply as such in a nonperturbative quantisation of gauge invariant theories are clearly identified a criterion for admissibility within a general class of gauge fixing conditions is provided for a large ensemble of simple gauge invariant systems this criterion confirms the conclusions of previous counter examples to the usual statement of the fradkin vilkovisky theorem comment 21 pages no figures to appear in jnl phys a

on the conjectures of henkel and weston recent conjectures of henkel and weston 1992 on certain two dimensional sums are examined and some significant generalizations of these sums are developed the results so obtained should be useful in analysing finite size effects in systems subjected to non periodic boundary conditions

geometry of random sequential adsorption by sequentially adding line segments to a line or disks to a surface at random positions without overlaps we obtain configurations of the one and two dimensional random sequential adsorption rsa problem we have simulated the one and two dimensional problem with periodic boundary condition the one dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation in two dimensions the geometrical properties of the rsa configuration are discussed and in addition known results of the rsa process are reproduced various statistical distributions of the voronoi dirichlet vd network corresponding to the rsa disk configuration are analyzed in order to characterize pores in the rsa configuration we introduce circular holes there is a direct correspondence between vertices of the vd network and these holes and also between direct indirect geometrical neighbors and these holes the hole size distribution is found to be a parabola we also find general relations that connect the asymptotic behavior of the surface coverage the correlation function and the hole size distribution

time dependent hamiltonians with 100 evolution speed efficiency the evolution speed in projective hilbert space is considered for hermitian hamiltonians and for non hermitian nh ones based on the hilbert schmidt norm and the spectral norm of a hamiltonian resource related upper bounds on the evolution speed are constructed these bounds are valid also for nh hamiltonians and they are illustrated for an optical nh hamiltonian and for a nh pt symmetric matrix hamiltonian furthermore the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective hilbert space a recipe for the construction of time dependent hamiltonians which ensure 100 speed efficiency is given generally these efficient hamiltonians are nh but there is a hermitian efficient hamiltonian as well finally the extremal case of a nh non diagonalizable hamiltonian with vanishing energy difference is shown to produce a 100 efficient evolution with minimal resources consumption

on certain schlomilch type series abstract by using a form of the poisson summation formula together with a generalization due to srivastava and exton of the discontinuous integral of weber and schafheitlin and the heretofore not readily available cosine transform of the hypergeometric function 1 f 2 b 2 t 2 b 0 several new schlomilch and fourier type series are evaluated by specialization of the latter series numerous results appearing in the literature are obtained in a unified way

pade summability of the cubic oscillator we prove the pade stieltjes summability of the perturbation series of any energy level en 1 of the cubic anharmonic oscillator as suggested by the numerical studies of bender and weniger at the same time we give a simple proof of the positivity of every level of the symmetric hamiltonian h1 for positive bessis zinn justin conjecture the n zeros of a state n 1 stable at 0 are confined for on the cut complex plane and are related to the level en 1 by the bohr sommerfeld quantization rule semiclassical phase integral condition we also prove the absence of non perturbative eigenvalues and the simplicity of the spectrum of our hamiltonians

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

generalized pt symmetry and real spectra the fact that eigenvalues of pt symmetric hamiltonians h can be real for some values of ap arameter and complex for others is explained by showing that the matrix elements ofh a ndhence the secular equation are real not only for pt but also for an ya ntiunitary operator a satisfying a 2k 1w ith k odd the argument is illustrated by a 2 2m atrix hamiltonian and two examples of the generalization are given

an analytic family of cal p cal t symmetric hamiltonians with real eigenvalues we study nonlinear perturbative expansions for symmetric local schrodinger operators the schrodinger operator is a sum of the harmonic oscillator hamiltonian and a local symmetric potential depending in general nonlinearly on the perturbation parameter a specific class of models having real spectrum for any value of the parameter is proposed

on the correlation between sizes and shapes of cells in epithelial mosaics it is shown that lewis s empirical linear relationship between the average area of a cell and the number of its sides in two dimensional mosaics corresponds to maximal arbitrariness in the cellular distribution an expression for the distribution is given in the general case

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

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

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

on the mellin transform of products of bessel and generalized hypergeometric functions we deduce in an elementary way representations for the mellin transform of a product of bessel functions 0f1 a2x2 and generalized hypergeometric functions pfp 1 b2x2 for a b 0 as a corollary we obtain a transformation formula for p 1fp 1 which was discovered by wimp in 1987 by using bailey s method for the specialization 3f2 1

beyond the wkb approximation in pt symmetric quantum mechanics the mergings of energy levels associated with the breaking of symmetry in the model of bender and boettcher and in its generalization to incorporate a centrifugal term are analysed in detail even though conventional wkb techniques fail it is shown how the ode im correspondence can be used to obtain a systematic approximation scheme which captures all previously observed features nonperturbative effects turn out to play a crucial role governing the behaviour of almost all levels once the symmetry breaking transition has been passed in addition a novel treatment of the radial schrodinger equation is used to recover the values of local and non local conserved charges in the related integrable quantum field theories without any need for resummation even when the angular momentum is nonzero

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

extension of a spectral bounding method to complex rotated hamiltonians with application to p2 i x3 we show that a recently developed method for generating bounds for the discrete energy states of the non hermitian ix3 potential handy c r 2001 j phys a math gen 34 l271 is applicable to complex rotated versions of the hamiltonian this has important implications for extension of the method in the analysis of resonant states regge poles and general bound states in the complex plane bender and boettcher 1998 phys rev lett 80 5243

the spectrum of the cubic oscillator we prove the simplicity and analyticity of the eigenvalues of the cubic oscillator hamiltonian begin array ll h beta frac d 2 dx 2 x 2 i sqrt beta x 3 end array for in the cut plane mathcal c c mathcal c backslash mathcal r moreover we prove that the spectrum consists of the perturbative eigenvalues en n 0 labeled by the constant number n of nodes of the corresponding eigenfunctions in addition for all beta in mathcal c c e n beta can be computed as the stieltjes pade sum of its perturbation series at 0 this also gives an alternative proof of the fact that the spectrum of h is real when is a positive number this way the main results on the repulsive pt symmetric and on the attractive quartic oscillators are extended to the cubic case

singular bohr sommerfeld rules the object of this paper is to present a geometrical formulation of the singular bohr sommerfeld rules given in 8 in a more general context including characteristic manifolds with several saddle points several examples are detailed and numerical checkings of the results are provided

generalized inductive limits of quasidiagonal c algebras if a is a unital quasidiagonal c algebra we construct a generalized inductive limit ba which is simple unital and inherits many structural properties from a if a is the unitization of a non simple purely infinite algebra e g the cone over a cuntz algebra then ba is tracially af which among other things lends support to a conjecture of toms

on certain two dimensional schlomilch series by using representations for one dimensional 1d schlomilch series lommel s expansion for bessel functions of the first kind and elementary series manipulations several algebraic formulae for two dimensional 2d schlomilch series are derived one of these formulae contains as a special case an identity involving a certain trigonometric lattice sum conjectured by henkel and weston 1993 which has recently been derived by boersma and de doelder 1993 using the 2d poisson summation formula the latter method is used to give an alternative derivation for 2d schlomilch series further it is shown that representation by 2d schlomilch series of a null function is not unique in addition representations for several other 1d type series are given

on some trigonometric and exponential lattice sums certain two dimensional trigonometric lattice series the ordinary convergence of which involves unresolved questions of a deep and delicate number theoretic nature are shown to be summable by a special abelian method and their abelian sums are obtained this is done by first evaluating an absolutely convergent exponential lattice series and then analytically extending its sum

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

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

an alternative construction of the positive inner product for pseudo hermitian hamiltonians examples in this paper we build on our earlier proposal for the construction of a positive inner product for pseudo hermitian hamiltonians and present examples to clarify the procedure we focus on two detailed calculations where the method is used namely a simple generalized 2 2 matrix pseudo hermitian hamiltonian which can be diagonalized and a second system where the hamiltonian cannot be diagonalized but can be described as a perturbation of the harmonic oscillator when the quantum mechanical system cannot be diagonalized exactly our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically for real eigenvalues in this paper we build on our earlier proposal for the construction of a positive inner product for pseudo hermitian hamiltonians and present examples to clarify the procedure we focus on two detailed calculations where the method is used namely a simple generalized 2 2 matrix pseudo hermitian hamiltonian which can be diagonalized and a second system where the hamiltonian cannot be diagonalized but can be described as a perturbation of the harmonic oscillator when the quantum mechanical system cannot be diagonalized exactly our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically for real eigenvalues

canonical expansion of symmetric operators and perturbation theory let h be any symmetric schrodinger operator of the type 2 x21 x2d igw x1 xd on where w is any odd homogeneous polynomial and it is proved that is self adjoint and that its eigenvalues coincide up to a sign with the singular values of h i e the eigenvalues of moreover we explicitly construct the canonical expansion of h and determine the singular values j of h through the borel summability of their divergent perturbation theory the singular values yield estimates of the location of the eigenvalues j of h by weyl s inequalities

lower bounds for critical values of a cancellative model we consider the following one dimensional discrete time cancellative model whose evolution is given by n 1 x n x 1 n x 1 mod 2 with probability p and n 1 0 with probability 1 p concerning critical probabilities pc and pc on a survival probability it is known that 0 706 pc pc 1 under a condition in this paper we give improved lower bounds of 0 771 and 0 781 on pc and pc respectively by finding suitable supermartingales for the model

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

on the simpleness of zeros of stokes multipliers abstract the aim of this paper is to discuss the simpleness of zeros of stokes multipliers associated with the differential equation x w x x 0 where w x x m a 1 x m 1 a m is a real monic polynomial we show that under a suitable hypothesis on the coefficients a k all the zeros of the stokes multipliers are simple

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

on some harris fkg type correlation inequalities for a non attractive model we consider the following non attractive one dimensional model whose evolution is given by n 1 x n x 1 n x 1 mod 2 with probability p 0 with probability 1 p in the present letter we prove that some harris fkg type correlation inequalities hold for this model moreover it is shown that a correlation inequality is also correct for the more general non attractive class

spacing distributions for real symmetric 2 2 generalized gaussian ensembles for ensembles of 2 2 real symmetric matrices the normalized spacing distributions p s form a family whose parameter space is the unit cube with coordinates determined by the means and variances of the diagonal and off diagonal elements the cube contains a variety of spacing distributions that are calculated analytically they include the wigner poisson transition distributions with singularities and gaussians unfolding is superfluous for 2 2 matrices but it can be implemented giving rise to a further variety of spacing distributions some surprising

distributional borel summability of odd anharmonic oscillators it is proved that the divergent rayleigh schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are borel summable in the distributional sense to the resonances naturally associated with the system

eigenvalues of pt symmetric oscillators with polynomial potentials we study the eigenvalue problem u z iz m pm 1 iz u z u z with the boundary condition that u z decays to zero as z tends to infinity along the rays in the complex plane where pm 1 z a1zm 1 a2zm 2 am 1z is a polynomial and integers m 3 we provide an asymptotic expansion of the eigenvalues n as n and prove that for each real polynomial pm 1 the eigenvalues are all real and positive with only finitely many exceptions

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

spectral analysis of the complex cubic oscillator using the exact semiclassical analysis we study the spectrum of a one parameter family of complex cubic oscillators the invariance property of the complex hamiltonians and the reality property of the spectrum are discussed analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated the global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by bender and wu for the quartic oscillator

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

complex wkb analysis of energy level degeneracies of non hermitian hamiltonians the hamiltonian h p 2 x 4 iax where a is a real parameter is investigated the spectrum of h is discrete and entirely real and positive for a 3 169 as a increases past this point adjacent pairs of energy levels coalesce and then become complex starting with the lowest lying energy levels for large energies the values of a at which this merging occurs scale as the three quarters power of the energy that is as a and e at the points of coalescence the ratio a a e 3 4 approaches a constant whose numerical value is acrit 1 1838363072914 conventional wkb theory determines the high lying energy levels but cannot be used to calculate acrit this critical value is predicted exactly by complex wkb theory

non self adjoint hamiltonians with complex eigenvalues motivated by what one observes dealing with pt symmetric quantum mechanics we discuss what happens if a physical system is driven by a diagonalizable hamiltonian with not all real eigenvalues in particular we consider the functional structure related to systems living in finite dimensional hilbert spaces and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators we also analyze a simple model computing the transition probabilities in the broken and in the unbroken regime

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

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

an alternative derivation of closed form representations for multidimensional lattice sums of generalized hypergeometric functions some representations for infinite m dimensional lattice sums of generalized hypergeometric functions which were deduced previously by the first named author are rederived here by appealing to an alternative more direct method in particular it is shown that these lattice sums are proportional to a finite sum of meijer s g functions

comment on a recent paper by mezincescu it has been conjectured that for 0 the entire spectrum of the non hermitian symmetric hamiltonian hn p2 x2 ix where n 2 is real strong evidence for this conjecture for the special case n 3 was provided in a recent paper by mezincescu mezincescu g a 2000 j phys a math gen 33 4911 in which the spectral zeta function z3 1 for the hamiltonian h3 p2 ix3 was calculated exactly here the calculation of mezincescu is generalized from the special case n 3 to the region of all n 2 0 and the exact spectral zeta function zn 1 for hn is obtained using zn 1 it is shown that to extremely high precision about three parts in 1018 the spectrum of hn for other values of n such as n 4 is entirely real

an algorithmic test for diagonalizability of finite dimensional pt invariant systems a non hermitian operator does not necessarily have a complete set of eigenstates contrary to a hermitian one an algorithm is presented which allows one to decide whether the eigenstates of a given pt invariant operator on a finite dimensional space are complete or not in other words the algorithm checks whether a given pt symmetric matrix is diagonalizable the procedure neither requires to calculate any single eigenvalue nor any numerical approximation

harmonic oscillator well with a screened coulombic core is quasi exactly solvable in the quantization scheme which weakens the hermiticity of a hamiltonian to its mere t invariance the superposition v x x2 ze2 x of the harmonic and coulomb potentials is defined at the purely imaginary effective charges ze2 if and regularized by a purely imaginary shift of x this model is quasi exactly solvable we show that at each excited n 1 th harmonic oscillator energy e 2n 3 there exists not only the well known harmonic oscillator bound state at the vanishing charge f 0 but also a normalizable n 1 plet of the further elementary sturmian eigenstates n x at eigencharges f f n 0 n 0 1 n beyond the smallest multiplicities n we recommend perturbative methods for their construction

a method for finding bounds on critical values for non attractive interacting particle systems the method is to use a computer to find a function for the process that always has negative expectation by considering all relevant configurations of 0 s and 1 s at the boundary of a finite process in one dimension it is shown that a branching annihilating random walk will die out if the diffusion parameter is greater than 0 176 the method may also be applied to attractive processes and is used for the contact process in one dimension obtaining the same values as ziezold and grillenberger 1988 for up to 10 places in from the boundaries

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

exact semiclassical expansions for one dimensional quantum oscillators a set of rules is given for dealing with wkb expansions in the one dimensional analytic case whereby such expansions are not considered as approximations but as exact encodings of wave functions thus allowing for analytic continuation with respect to whichever parameters the potential function depends on with an exact control of small exponential effects these rules which include also the case when there are double turning points are illustrated on various examples and applied to the study of bound state or resonance spectra in the case of simple oscillators it is thus shown that the rayleigh schrodinger series is borel resummable yielding the exact energy levels in the case of the symmetrical anharmonic oscillator one gets a simple and rigorous justification of the zinn justin quantization condition and of its solution in terms of multi instanton expansions

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

effect of packing method on the randomness of disc packings the randomness of disc packings generated by random sequential adsorption rsa random packing under gravity rpg and mason packing mp which gives a packing density close to that of the rsa packing has been analysed based on the delaunay tessellation and is evaluated at two levels i e the randomness at individual subunit level which relates to the construction of a triangle from a given edge length distribution and the randomness at network level which relates to the connection between triangles from a given triangle frequency distribution the delaunay tessellation itself is also analysed and its almost perfect randomness at the two levels is demonstrated which verifies the proposed approach and provides a random reference system for the present analysis it is found that i the construction of a triangle subunit is not random for the rsa mp and rpg packings with the degree of randomness decreasing from the rsa to mp and then to rpg packing ii the connection of triangular subunits in the network is almost perfectly random for the rsa packing acceptable for the mp packing and not good for the rpg packing packing method is an important factor governing the randomness of disc packings

metric operators for non hermitian quadratic su 2 hamiltonians a class of non hermitian quadratic su 2 hamiltonians having an anti linear symmetry is constructed this is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra in fact different realizations for this type of symmetry are obtained including the natural occurrence of charge conjugation together with parity and time reversal once specified the underlying anti linear symmetry of the hamiltonian the former if unbroken leads to a purely real spectrum and the latter can be mapped to a hermitian counterpart by amongst other possibilities a similarity transformation here lie algebraic methods which were used to investigate the generalized swanson hamiltonian assis and fring 2009 j phys a math theor 42 015203 are employed to identify the class of quadratic hamiltonians that allow for such a mapping to the hermitian counterpart whereas for the linear su 2 system every hamiltonian of this type can be mapped to a hermitian counterpart by a transformation which is itself an exponential of a linear combination of su 2 generators the situation is more complicated for quadratic hamiltonians therefore the possibility of more elaborate similarity transformations including quadratic exponents is also explored in detail the existence of finite dimensional representations for the su 2 hamiltonian as opposed to the su 1 1 studied before allows for comparison with explicit diagonalization results for finite matrices finally the similarity transformations constructed are compared with the analogue of swanson s method for exact diagonalization of the problem establishing a simple relation between both approaches

a class of generalized hypergeometric summations abstract summations over the positive integers n of the generalized hypergeometric expressions 1 n p f p 1 n 2 x 2 x 0 are derived in closed form the specialization p 0 for example reduces to known results for schlomilch series in addition we record the apparently not readily available sine and cosine transforms of p f p 1 b 2 x 2 b 0 the latter of which is used together with a form of the poisson summation formula to deduce the aforementioned results

exact resolution method for general 1d polynomial schrodinger equation the stationary 1d schrodinger equation with a polynomial potential v q of degree n is reduced to a system of complex exact quantization conditions of bohr sommerfeld form they arise from bilinear wronskian functional relations pairing spectral determinants of n 2 generically distinct operators all the transforms of one quantum hamiltonian under a cyclic group of complex scalings the determinants zeros define n 2 semi infinite chains of points in the complex spectral plane and these encode the original quantum problem now each chain can be assigned an exact quantization condition which constrains it relative to its neighbours resulting in closed equilibrium conditions for the global chain system these are supplemented by the standard bohr sommerfeld quantization conditions which bind the infinite tail of each chain asymptotically this reduced problem is then probed numerically for effective solvability upon test cases mostly symmetric quartic oscillators we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues eigenfunctions we conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable contractive and thus reconstruct the exact quantum data as their fixed point to date this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials v q

wkb analysis of mathcal p mathcal t symmetric sturm liouville problems most studies of symmetric quantum mechanical hamiltonians have considered the schr dinger eigenvalue problem on an infinite domain this paper examines the consequences of imposing the boundary conditions on a finite domain as is the case with regular hermitian sturm liouville problems the eigenvalues of the symmetric sturm liouville problem grow like n2 for large n however the novelty is that a eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another for the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane wkb analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points the method turns out to be surprisingly accurate even at low energies this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

a pseudo hermitian β hermite family of matrices abstract a family of pseudo hermitian random matrices is constructed whose eigenvalues are the same of the hermite ensemble of random matrix theory it is shown that matrices of this family are unstable in such a way that small perturbations cause the eigenvalues to move into the complex plane in conjugate pairs