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

superconvergence of solution derivatives for the shortley weller difference approximation of poisson s equation part i smoothness problems the finite difference method fdm using the shortley weller approximation can be viewed as a special kind of the finite element methods fems using the piecewise bilinear and linear functions and involving some integration approximation when u c3 s i e u c3 0 s and f c2 s the superconvergence rate o h2 of solution derivatives in discrete h1 norms by the fdm is derived for rectangular difference grids where h is the maximal mesh length of difference grids used and the difference grids are not confined to be quasiuniform comparisons are made on the analysis by the maximum principle and the fem analysis conversions between the fdm and the linear and bilinear fems are discussed and numerical experiments are provided to support superconvergence analysis made

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

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

on solution uniqueness of elliptic boundary value problems in this paper we consider the problem of solution uniqueness for the second order elliptic boundary value problem by looking at its finite element or finite difference approximations we derive several equivalent conditions which are simpler and easier than the boundedness of the entries of the inverse matrix given in yamamoto et al t yamamoto s oishi q fang discretization principles for linear two point boundary value problems ii numer funct anal optim 29 2008 213 224 the numerical experiments are provided to support the analysis made strictly speaking the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem and this condition should be checked by solving the corresponding eigenvalue problems an application of the equivalent conditions is that we may discover the uniqueness simultaneously while seeking the approximate solutions of elliptic boundary equations

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

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

path integral solutions for klein gordon particle in vector plus scalar generalized hulthen and woods saxon potentials the green s function for a klein gordon particle under the action of vector plus scalar deformed hulthen and woods saxon potentials is evaluated by exact path integration explicit path integration leads to the green s function for different shapes of the potentials from the singularities of the latter green s function the bound states are extracted for q 1 and 1 ln q r the analytic expression of the energy spectrum and the normalized wave functions for the l states are obtained within the framework of an approximation to the centrifugal term when the deformation parameter q is 0 q 1 or q 0 it is found that the quantization conditions are transcendental equations involving the hypergeometric function that require a numerical solution for the s state energy levels particular cases of these potentials are also discussed briefly

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

fourth order eigenvalue approximation by extrapolation on domains with reentrant corners the eigenvalue problem of the laplace operator is considered on a non convex domain composed of rectangles this model problem may be solved by the finite element method with bilinear elements on a rectangular mesh it is known thato h 2 1 convergence can be obtained for the eigenvalues if the mesh haso h 2 points a simple extrapolation scheme is presented which on appropriately graded meshes increases the rate of convergence too h 4

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

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

bound state solutions of the dirac equation for the trigonometric and hyperbolic scarf grosche potentials using the nikiforov uvarov method the relativistic dirac equation with equal trigonometric and hyperbolic scarf grosche scalar and vector potentials in 3 1 and 1 1 space time dimension are solved by using the nikiforov uvarov method the exact energy equations and the spinor wave functions are obtained for these sets of potentials the bound states energy are calculated numerically also with appropriate transformations solutions of the fourth potential v4 can be deduced from the solutions of the third potential v3

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

exact solution of the schrodinger and klein gordon equations for generalised hulthen potentials for the class of generalised hulthen s wave potentials vgh p r sigma i 1p giai exp gamma ir 1 sigma j 1p aj exp gamma jr p or 1 the exact solution is presented for both the schrodinger and klein gordon equations this simultaneous solubility represents a new non perturbative treatment of the kinematical relativistic corrections and is based on a non standard generalisation of the hypergeometric function

exact solutions of the schrodinger equation with non central potential by the nikiforov uvarov method the general solutions of the schrodinger equation for a non central potential are obtained by using the nikiforov uvarov method the schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions are derived analytically by making special selections the non central potential is reduced to coulomb and hartmann ring shaped potentials and the obtained results are compared with the solutions of coulomb and hartmann potentials given in the literature

teitelbaum s exceptional zero conjecture in the anticyclotomic setting teitelbaum formulated a conjecture relating first derivatives of the mazur swinnerton dyer adic l functions attached to modular forms of even weight k 2 to certain invariants arising from shimura curve parametrizations this article formulates an analogue of teitelbaum s conjecture in which the cyclotomic zp extension of q is replaced by the anticyclotomic zp extension of an imaginary quadratic field this analogue is then proved by using the cerednik drinfeld theory of 7 adic uniformisation of shimura curves introduction let 0 y an 2 on to n the classical l function l cf s admits an analytic continuation to the entire complex plane and a functional equation which relates its values at s and k s of special arithmetic interest for the present work is the central value l j k 2 for example when k 2 the birch and swinnerton dyer conjecture relates the behavior of l j s at s 1 to the arithmetic of the abelian variety a associated to j by the eichler shimura construction in mtt a adic variant of the birch and swinnerton dyer conjecture is formulated with l s replaced by a 7 adic analogue lp f s attached to the cyclotomic zp extension of q when p divides n exactly and ap 1 which implies that a has split multiplicative reduction at p the function lp f s vanishes at s 1 in this case the conjectures of mtt imply the following relationship between the first

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

uniform asymptotic and jwkb expansions for anharmonic oscillators we show explicitly the relation between the uniform asymptotic and the jeffreys wentzel kramers brillouin jwkb wavefunctions and between the matching of uniform asymptotic expansions and the complete jwkb connection formulae written in terms of stokes multipliers and loop integrals as an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and via dispersion relations the corresponding asymptotic behaviour of the rayleigh schrodinger perturbation theory coefficients

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

position dependent effective mass dirac equations with pt symmetric and non pt symmetric potentials we present a new procedure to construct the one dimensional non hermitian imaginary potential with a real energy spectrum in the context of the position dependent effective mass dirac equation with the vector coupling scheme in 1 1 dimensions in the first example we consider a case for which the mass distribution combines linear and inversely linear forms the dirac problem with a pt symmetric potential is mapped into the exactly solvable schrodinger like equation problem with the isotonic oscillator by using the local scaling of the wavefunction in the second example we take a mass distribution with smooth step shape the dirac problem with a non pt symmetric imaginary potential is mapped into the exactly solvable schrodinger like equation problem with the rosen morse potential the real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method

on the reality of the eigenvalues for a class of symmetric oscillators we study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays where is a real polynomial and we prove that if for some we have for all then the eigenvalues are all positive real we then sharpen this to a larger class of polynomial potentials

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

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

generalized heegner cycles and p adic rankin l series this article studies a distinguished collection of so called generalized heegner cycles in the product of a kuga sato variety with a power of a cm elliptic curve its main result is a p adic analogue of the gross zagier formula which relates the images of generalized heegner cycles under the p adic abel jacobi map to the special values of certain p adic rankin l series at critical points that lie outside their range of classical interpolation

new expansions of numerical eigenvalues for δ by nonconforming elements the paper explores new expansions of the eigenvalues for u pu in s with dirichlet boundary conditions by the bilinear element denoted q 1 and three nonconforming elements the rotated bilinear element denoted q 1 rot the extension of q 1 rot denoted eq 1 rot and wilson s elements the expansions indicate that q 1 and q 1 rot provide upper bounds of the eigenvalues and that eq 1 rot and wilson s elements provide lower bounds of the eigenvalues by extrapolation the o h 4 convergence rate can be obtained where h is the maximal boundary length of uniform rectangles numerical experiments are carried out to verify the theoretical analysis made

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

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

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

error analysis of the trefftz method for solving laplace s eigenvalue problems for solving laplace s eigenvalue problems we propose new algorithms using the trefftz method tm i e the boundary approximation method bam by means of degeneracy of numerical helmholtz equations since piecewise particular solutions can be fully adopted the new algorithms benefit high accuracy of eigenvalues and eigenfunctions low cost in cpu time and computer storage also the algorithms can be applied to solve the problems with multiple interfaces and singularities in this paper error estimates are derived for the approximate eigenvalues and eigenfunctions obtained numerical experiments for smooth and singular solutions are reported in this paper to show significance of the algorithms proposed and to verify the theoretical results made

on the p adic height of heegner cycles 0 1 in gr za gross and zagier proved a remarkable formula which relates the first derivative of the l function of a modular form f of weight 2 on 0 n over a suitable imaginary quadratic field and the neron tate height of a heegner point over the same quadratic field on the f part of the jacobian of the modular curve x0 n later perrin riou pr 2 proved a p adic version of this formula kolyvagin s method of euler systems ko combined with the formula of gross zagier proves the conjecture of birch and swinnerton dyer up to a controlled rational factor for all modular elliptic curves over q with analytic rank 1

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

an improved calculation for screened coulomb potentials in rayleigh schrodinger perturbation theory earlier works on screened coulomb potentials using rayleigh schrodinger perturbation theory have been re examined instead of working with the usual hulthen potential as the unperturbed hamiltonian the authors propose that a scaled hulthen potential with modified strength and screening coefficient represents the lowest order approximation for the static screened coulomb and exponential cosine screened coulomb potentials the scale parameter appearing in the new hulthen potential is then determined from the notion of the virial theorem and intuitive physical arguments it is found that the accuracy of the predicted energy eigenvalues for the bound s states improves significantly even when the screening parameter is large and quite close to its critical values for which the quantum state becomes just bound in spite of the simplicity of the approach the numerical results compare fairly well with those obtained from rigorous analytic approximation methods

on the eigenproblems of pt symmetric oscillators we consider the non hermitian hamiltonian h d2 dx2 p x2 ix 2n 1 on the real line where p x is a polynomial of degree at most n 1 with all non negative real coefficients possibly p 0 it is proved that the eigenvalues must be in the sector arg 2n 3 also for the cubic case h d2 dx2 ix 3 we establish a zero free region of the eigenfunction u and its derivative u and we find some other interesting properties of eigenfunctions

integral structures in automorphic line bundles on the p adic upper half plane given an automorphic line bundle ox k of weight k on the drinfel d upper half plane x over a local field k we construct a gl2 k equivariant integral lattice o x k in ox k k k as a coherent sheaf on the formal model x underlying x k k here k k is ramified of degree 2 this generalizes a construction of teitelbaum from the case of even weight k to arbitrary integer weight k we compute h x o x k and obtain applications to the de rham cohomology h 1 dr x sym k k st with coefficients in the k th symmetric power of the standard representation of sl2 k where k 0 of projective curves x uniformized by x namely we prove the degeneration of a certain reduced hodge spectral sequence computing h 1 dr x sym k k st we re prove the hodge decomposition of h 1 dr x sym k k st and show that the monodromy operator on h 1 dr x sym k k st respects integral de rham struc tures and is induced by a universal monodromy operator defined on x i e before passing to the quotient

dirac operators in tensor categories and the motive of quaternionic modular forms we define a motive whose realizations afford modular forms of arbitrary weight on an indefinite division quaternion algebra this generalizes work of iovita spiess to odd weights in the spirit of jordan livne it also generalizes a construction of scholl to indefinite division quaternion algebras and provides the first motivic construction of new subspaces of modular forms

the teitelbaum conjecture in the indefinite setting let f be a new cusp form on gamma 0 n of even weight k 2 geq 2 suppose that there is a prime p parallel n and that we may write n pn n where n is the squarefree product of an even number of primes there is a darmon style cal l invariant cal l n f attached to this factorization which is the orton cal l invariant when n 1 we prove that cal l n f does not depend on the chosen factorization of n and it is equal to the other known cal l invariants we also give a formula for the computation of the logarithmic p adic abel jacobi image of the darmon cycles this formula is crucial for the computations of the derivatives of the p adic l functions of the weight variable attached to a real quadratic field k bbb q such that the primes dividing n are split and the primes dividing pn are inert