quasi exactly solvable quartic elementary integrals and asymptotics we study elementary eigenfunctions y peh of operators l y y py where p h and p are polynomials in one variable for the case when h is an odd cubic polynomial we investigate the real level crossing points and asymptotics of eigenvalues this study leads to an interesting identity with elementary integrals littlewood when he makes use of an algebraic identity always saves himself the trouble of proving it he maintains that an identity if true can be verified in few lines by anybody obtuse enough to feel the need of verification freeman dyson 1949 some guesses in theory of partitions eureka 8 10 15

a unified treatment of schrodinger s equation for anharmonic and double well potentials the authors exploit some exact soluble model potentials vn alpha beta v a b lambda x6 in order to extrapolate as lambda to 0 reliable eigenvalues of the one dimensional schrodinger equations with either anharmonic or symmetric double well potentials of the form v a b 1 2ax2 bx2 b 0 their procedure which corresponds to low order rayleigh schrodinger perturbation theory is found to be competitive with both high order pade summation of conventional rspt and large scale variational calculations using harmonic oscillators basis functions

reversible joint hilbert and linear canonical transform without distortion generalized analytic signal associated with the linear canonical transform lct was proposed recently by fu and li generalized analytic signal associated with linear canonical transform opt commun vol 281 pp 1468 1472 2008 however most real signals especially for baseband real signals cannot be perfectly recovered from their generalized analytic signals therefore in this paper the conventional hilbert transform ht and analytic signal associated with the lct are concerned to transform a real signal into the lct of its ht two integral transforms i e the ht and lct are required the goal of this paper is to simplify cascades of multiple integral transforms which may be the ht analytic signal lct or inverse lct the proposed transforms can reduce the complexity when realizing the relationships among the following six kinds of signals a real signal its ht and analytic signal and the lct of these three signals most importantly all the proposed transforms are reversible and undistorted using the proposed transforms several signal processing applications are discussed and show the advantages and flexibility over simply using the analytic signal or the lct

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

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

on the two gap locus for the elliptic calogero moser model we give an analytical description of the locus of the two gap elliptic potentials associated with the corresponding flow of the calogero moser system we start with the description of treibich verdier two gap elliptic potentials the explicit formulae for the covers wavefunctions and lame polynomials are derived together with a new lax representation for the particle dynamics on the locus we then consider more general potentials within the weierstrass reduction theory of theta functions to lower genera the reduction conditions in the moduli space of the genus 2 algebraic curves are given this is a subvariety of the humbert surface which can be singled out by the condition of the vanishing of some theta constants

estimating group delay and phase delay via discrete time analytic cross correlation starting with two finite n point real valued discrete time signals frequency domain algorithm procedures are provided for estimating the group delay and phase delay between the signals these delays are extracted from the complex valued discrete time analytic cross correlation sequence computed using the two signals special adjustments of the cross correlation transform end points are shown to be necessary in order to generate a proper analytic cross correlation sequence two cases are considered for delay estimation 1 to sample interval resolution and 2 to subsample interval resolution by interpolation

momentum space orthogonal polynomial projection quantization the orthogonal polynomial projection quantization oppq is an algebraic method for solving schrodinger s equation by representing the wave function as an expansion in terms of polynomials orthogonal with respect to a suitable reference function r x which decays asymptotically not faster than the bound state wave function the expansion coefficients are obtained as linear combinations of power moments in turn the s are generated by a linear recursion relation derived from schrodinger s equation from an initial set of low order moments it can be readily argued that for square integrable wave functions representing physical states rapidly converging discrete energies are obtained by setting coefficients to zero at arbitrarily high order this paper introduces an extention of oppq in momentum space by using the representation where q n k are polynomials orthogonal with respect to a suitable reference function t k the advantage of this new representation is that it can help solving problems for which there is no coordinate space moment equation this is because the power moments in momentum space are the taylor expansion coefficients which are recursively calculated via schrodinger s equation we show the convergence of this new method for the sextic anharmonic oscillator and an algebraic treatment of gross pitaevskii nonlinear equation

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

simple analytical approximations to the integrals of the bessel functions jν application to the transmittance of a circular aperture two accurate yet simple analytic approximations to the integral of the bessel function j0 are presented these first and second order approximations are obtained by improving on the recently developed method known as two point quasi rational approximations the accuracy of the first order approximant is better than 0 05 the second order approximant is practically indistinguishable from the true integral even for very large values of the argument overall accuracy is better than 0 002 05 our approximants are in addition analytic and therefore replace with significant advantages both the well known power series and the asymptotic formulae of the integral approximants to the transmittance function of a plane wave through a circular aperture are derived a problem which arises in diffraction theory and particle scattering the second order approximant to the transmittance is analytic too and can be evaluated for small and large values of the argument just with a hand calculator its accuracy is better than 0 0011 as an extension two first order approximations to the integrals of the bessel functions j of fractional order are derived

on the variation of interaural time differences with frequency interaural time difference itd is a major cue to sound localization in humans and animals for a given subject and position in space itd depends on frequency this variation is analyzed here using a head related transfer functions hrtfs database collected from the literature and comprising human hrtfs from 130 subjects and animal hrtfs from six specimens of different species for humans the itd is found to vary with frequency in a way that shows consistent differences with respect to a spherical head model maximal itd values were found to be about 800 s in low frequencies and 600 s in high frequencies the itd variation with frequency up to 200 s for some positions occurs within the frequency range where itd is used to judge the lateral position of a sound source in addition itd varies substantially within the bandwidth of a single auditory filter leading to systematic differences between envelope and fine structure itds because the frequency dependent pattern of itd does not display spherical symmetries it potentially provides cues to elevation and resolves front back confusion the fact that the relation between position and itds strongly depends on the sound s spectrum in turn suggests that humans and animals make use of this relationship for the localization of sounds

on stieltjes relations painleve iv hierarchy and complex monodromy a generalization of the stieltjes relations for the fourth painleve piv transcendents and their higher analogues determined by dressing chains is proposed it is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher piv equation the approach is based on the interpretation of the stieltjes relations as local trivial monodromy conditions for certain schrodinger equations in the complex domain as a corollary a new class of schrodinger operators with trivial monodromy is constructed in terms of the piv transcendents

failure of the hill determinant method for the sextic anharmonic oscillator an extended analysis of eigenvalues and wavefunctions resulting from the hill determinant method is carried out for the one dimensional sextic anharmonic oscillator it is shown that in spite of a seemingly natural ansatz of the method irrelevant wavefunctions may appear leading to incorrect eigenvalues the domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

on the rational monodromy free potentials with sextic growth we study the rational potentials v x with sextic growth at infinity such that the corresponding one dimensional schrodinger equation has no monodromy in the complex domain for all values of the spectral parameter we investigate in detail the subclass of such potentials which can be constructed by the darboux transformations from the well known class of quasiexactly solvable potentials v x6 x2 l l 1 x2 we show that in contrast with the case of quadratic growth there are monodromy free potentials which have quasirational eigenfunctions but which cannot be given by this construction we discuss the relations between the corresponding algebraic varieties and present some elementary solutions of the calogero moser problem in the external field with sextic potential

the energy levels of the sextic double well potential an infinite set of exact solutions of the sextic double well potential is derived these solutions that are obtained for special configurations of the well are shown to provide an excellent description of the entire negative energy spectrum for arbitrary geometry of the well a universal feature of this class of deep double well potentials is deduced

computing the discrete time analytic signal via fft starting with a real valued n point discrete time signal frequency domain algorithms are provided for computing 1 the complex valued standard n point discrete time analytic signal of the same sample rate 2 the complex valued decimated n 2 point discrete time analytic signal of half the original sample rate and 3 the complex valued interpolated nm point discrete time analytic signal of m times the original sample rate special adjustment of the transform end points are shown to be necessary in order to generate proper discrete time analytic signals

energy spectrum of the potential v ax2 x4 suitable sequences of quasi exactly solvable hamiltonians are shown to provide stringent upper bounds to the energy eigenvalues of the bound state potential v ax2 x4 procedures to convert these bounds into even further improved energy estimates are developed for the quartic anharmonic oscillator a 0 case a simple argument is provided to indicate that the conventional small parameter energy expansion does not converge as a taylor series an accurate closed form parameterization of the entire quartic a 0 spectrum is noted the energy difference between the lowest lying levels of a quartic double well a 0 is satisfactorily recovered and for deep wells a useful expression is deduced for it empirically

two point quasifractional approximant in physics truncation error the quasifractional approximation method is developed in a systematic manner this method uses simultaneously the power series and at a second point the asymptotic expansion the usual form of the approximants is two or more rational fractions in terms of a suitable variable combined with auxiliary nonfractional functions coincidence in the singularities in the region of interest is pursued equal denominators in the rational fractions is required so that the solution of only linear algebraic equations is needed to determine the parameters of the approximant an upper bound is obtained for the truncation error for a certain class of functions which contains most of the functions for which this method has been applied so far it is shown that quasifractional approximants can be derived as a mixed german and latin polynomial problem in the context of hermite pade approximation theory

a note on the application of the hilbert transform to time delay estimation it is shown that the hilbert transform may be employed to convert the maximum finding task of a delay estimator to one of finding a zero crossing the basic theory of the technique is shown and a few applications are described its primary application is in servo tracking and servo velocity measurement systems which employ correlation techniques a review of prior work in this field is given

zeros of wronskians of hermite polynomials and young diagrams abstract for a certain class of partitions a simple qualitative relation is observed between the shape of the young diagram and the pattern of zeros of the wronskian of the corresponding hermite polynomials in the case of the two term wronskian w h n h n k we give an explicit formula for the asymptotic shape of the zero set as n some empirical asymptotic formulas are given for the zero sets of three term and four term wronskians

polynomial families and schrodinger equation one example for nonhypergeometric type of correspondence we discuss the explicit construction of the schrodinger equations admitting representation through some family of general nonorthogonal polynomials the specific choice of the third order polynomial coefficient functions that lead to quasi solvable families of schrodinger potentials is considered in detail

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

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

a moments analysis of quasi exactly solvable systems a new perspective on the sextic potential gx 6 bx 4 mx 2 beta over x 2 a quantum system is quasi exactly solvable qes when a subset of its eigenvalues can be obtained in closed form or they are the roots of closed form expressions in one dimension qes states assume the form involving a known positive reference function and a polynomial p x so that the taylor expansion of truncates at a finite order for non qes states this truncation procedure for suitable reference functions can provide approximate values for the eigenvalues it corresponds to the hill determinant method which is known to be unstable when assumes the controlling asymptotic form of the physical state for this reason it cannot be used to simultaneously generate exact qes states while approximating non qes states to address these limitations the orthogonal polynomial projection quantization oppq method was developed handy and vrinceanu 2013 j phys a math theor 46 135202 2013 j phys b atom mol opt phys 46 115002 it is demonstrated in this paper that oppq can directly and transparently provide at the same time both algebraic solutions for qes states and convergent numerically stable approximations for non qes states within the oppq analysis for the wavefunction representation the bender dunne energy orthogonal polynomials correspond exactly up to a numerical factor to the energy dependent power moments p dx xp x within this perspective the existence of qes states is associated with an anomalous kink behavior in the order of the finite difference moment equation corresponding to the s suggesting a change in the number of degrees of freedom this was first noted through the implementation of the eigenvalue moment method the first application of semidefinite programming analysis to quantum operators handy and bessis 1985 phys rev lett 55 931 this moments perspective also reveals additional properties for the non qes states of the same symmetry as the qes states their lower order moments must be zero we demonstrate our results in the context of the two sextic potentials vsa x x6 mx2 bx4 and vss x x6 mx2 b x2

whittaker hill equation and semifinite gap schrodinger operators a periodic one dimensional schrodinger operator is called semifinite gap if every second gap in its spectrum is eventually closed we construct explicit examples of semifinite gap schrodinger operators in trigonometric functions by applying darboux transformations to the whittaker hill equation we give a criterion for the regularity of the corresponding potentials and investigate the spectral properties of the new operators