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

behavior of various complexity functions for a nonempty closed set email protected a n with email protected a we consider 3 complexity functions of k 0 1 2 1 block complexity p w b l k w 0 1 k 1 2 maximal pattern complexity p w k sup s n s k w s 3 minimal pattern complexity p w k inf s n s k w s where denotes the number of elements in a set and w s is the restriction of w to email protected n if p w k p w k k 0 1 2 holds then the above 3 complexities coincide and are called uniform complexity denoted by p w k behaviors of these 4 complexity functions are discussed

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

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

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

empirical null and false discovery rate analysis in neuroimaging current strategies for thresholding statistical parametric maps in neuroimaging include control of the family wise error rate control of the false discovery rate fdr and thresholding of the posterior probability of a voxel being active given the data the latter derived from a mixture model of active and inactive voxels correct inference using any of these criteria depends crucially on the specification of the null distribution of the test statistics in this article we show examples from fmri and dti data where the theoretical null distribution does not match well the observed distribution of the test statistics as a solution we introduce the use of an empirical null a null distribution empirically estimated from the data itself allowing for global corrections of theoretical null assumptions the theoretical null distributions considered are normal t chi 2 and f all commonly encountered in neuroimaging the empirical null estimate is accompanied by an estimate of the proportion of non active voxels in the data based on the two class mixture model we present the equivalence between the strategies of controlling fdr and thresholding posterior probabilities in the context of neuroimaging and show that the fdr estimates derived from the empirical null can be seen as empirical bayes estimates

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

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

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

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

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

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

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

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

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

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

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