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

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

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

fragile pt symmetry in a solvable model one of the simplest pseudo hermitian models with real spectrum viz square well on a real interval i of coordinates is re examined a pt symmetric complex deformation c of i is introduced and shown tractable via an innovated approach to matching conditions the result is surprising an arbitrarily small deformation i c implies a sudden collapse i e the spontaneous pt symmetry breaking of virtually all the spectrum i e up to its low energy part

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

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

nonuniqueness of the mathcal c operator in mathcal p mathcal t symmetric quantum mechanics the operator in symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations and these equations are difficult to solve exactly so perturbative methods have been used in the past to calculate the usual approach has been to express the hamiltonian as h h0 h1 and to seek a solution for in the form where q q q p is odd in the momentum p even in the coordinate q and has a perturbation expansion of the form q q1 3q3 5q5 in previous work it has always been assumed that the coefficients of even powers of in this expansion would be absent because their presence would violate the condition that q p q is odd in p in an earlier paper it was argued that the operator is not unique because the perturbation coefficient q1 is nonunique here the nonuniqueness of is demonstrated at a more fundamental level it is shown that the perturbation expansion for q actually has the more general form q q0 q1 2q2 in which all powers and not just odd powers of appear for the case in which h0 is the harmonic oscillator hamiltonian q0 is calculated exactly and in closed form and it is shown explicitly to be nonunique the results are verified by using powerful summation procedures based on analytic continuation it is also shown how to calculate the higher coefficients in the perturbation series for q

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

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

jump dynamics due to jump datum of compressible viscous navier stokes flows in a bounded plane domain abstract in this paper when the initial density has a jump across an interior curve in a bounded domain we show unique existence piecewise regularity and jump discontinuity dynamics for the density and the velocity vector governed by the navier stokes equations of compressible viscous barotropic flows a critical difficulty is in controlling the gradient of the pressure across the jump curve this is resolved by constructing a vector function associated with the pressure jump value on the convecting curve and extending it to the whole domain

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

inference for autocorrelations under weak assumptions abstract in this article we consider the large sample behavior of estimates of autocorrelations and autoregressive moving average arma coefficients as well as their distributions under weak conditions specifically the usual text book formulas for variances of these estimates are based on strong assumptions and should not be routinely applied without careful consideration such is the case when the time series follows an arma process with uncorrelated innovations that may not be assumed to be independent and identically distributed as a specific case it is well known that if the process is independent and identically distributed then the sample autocorrelation estimates scaled by the square root of the sample size are asymptotically standard normal this result is used extensively as a diagnostic check on the residuals of a fitted model or as an initial test on the observed time series to determine whether further model fitting is warranted in this article we show that this result can be quite

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

discrete cal pt symmetric models of scattering one dimensional scattering mediated by non hermitian hamiltonians is studied a schematic set of models is used which simulates two point interactions at a variable strength and distance the feasibility of the exact construction of the amplitudes is achieved via the discretization of the coordinate by direct construction it is shown that in all our models the probability is conserved this feature is tentatively attributed to the space and time reflection symmetry also known as symmetry of our specific hamiltonians

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

discontinuous solutions of steady state viscous compressible navier stokes equations abstract we study a steady state viscous compressible navier stokes flow in a rectangle 0 1 1 1 with the boundary condition u v 1 0 for the velocity field u v and the condition p 0 y p 0 y for the pressure p on 0 1 1 which is the part of the boundary where the stream lines emanate under the condition that p 0 y has a jump at y 0 we establish the existence and uniqueness of the solution having discontinuity along the stream line starting from the origin

critical parameters for non hermitian hamiltonians pt symmetric hamiltonians exhibit real eigenvalues real eigenvalues appear when pt symmetry is unbroken the range of real eigenvalues is determined by exceptional points or critical parameters the diagonalization method is suitable for the calculation of critical parameters degenerate eigenvalues require a more general definition of unbroken symmetry we calculate accurate critical parameters for a class of non hermitian hamiltonians by means of the diagonalization method we study three one dimensional models and two perturbed rigid rotors with pt symmetry one of the latter models illustrates the necessity of a more general condition for the appearance of real eigenvalues that we also discuss here

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

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

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

dynamics of singularity surfaces for compressible viscous flows in two space dimensions we prove the global existence of solutions of the navier stokes equations of compressible barotropic flow in two space dimensions which exhibit convecting singularity curves the fluid density and velocity gradient have jump discontinuities across these curves exactly as predicted by the rankine hugoniot conditions and these jump discontinuities decay exponentially in time more rapidly for smaller viscosities the singularity curves remain c1 despite the fact that the velocity fields which convect them are not continuously differentiable 2002 wiley periodicals inc

a fast estimation method for the vector autoregressive moving average model with exogenous variables abstract a very fast and simple algorithm for estimation of the parameters of large multivariate time series and distributed lag models is presented an analysis of the distribution of the estimates shows that they are asymptotically normal and unbiased and that they have a variance that decreases like 1 n n being the sample size the algorithm is especially applicable for estimation of large multivariate models where it is generally many times faster than maximalization algorithms

unit root tests in arma models with data dependent methods for the selection of the truncation lag abstract we analyze the choice of the truncation lag in the context of the said dickey test for the presence of a unit root in a general autoregressive moving average model it is shown that a deterministic relationship between the truncation lag and the sample size is dominated by data dependent rules that take sample information into account in particular we study data dependent rules that are not constrained to satisfy the lower bound condition imposed by said dickey akaike s information criterion falls into this category the analytical properties of the truncation lag selected according to a class of information criteria are compared to those based on sequential testing for the significance of coefficients on additional lags the asymptotic properties of the unit root test under various methods for selecting the truncation lag are analyzed and simulations are used to show their distinctive behavior in finite samples our results favor methods based on sequential tests over those based on informat

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

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

maximal couplings in pt symmetric chain models with the real spectrum of energies the domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non hermitian n n matrix hamiltonians h n with tridiagonal and symmetric structure at all dimensions n the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere for odd n 2m 1 or ellipsoid for even n 2k

a class of exactly solvable eigenvalue problems the class of differential equation eigenvalue problems y x x2n 2y x xn ey x n 1 0 1 2 3 on the interval x can be solved in closed form for all the eigenvalues e and the corresponding eigenfunctions y x the eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions the eigenfunctions can be rewritten as products of polynomials and functions that decay exponentially as x for odd n the polynomials that are obtained in this way are new and interesting classes of orthogonal polynomials for example when n 1 the eigenfunctions are orthogonal polynomials in x3 multiplying airy functions of x2 the properties of the polynomials for all n are described in detail

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

the potential iz m generates real eigenvalues only under symmetric rapid decay boundary conditions we consider the non hermitian eigenvalue problems u z iz mu z u z m 3 under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z plane we prove that the eigenvalues are all real and positive

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

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

small ϵ behavior of the non hermitian symmetric hamiltonian h p2 x2 ix ϵ the energy eigenvalues of the class of non hermitian pt symmetric hamiltonians h p 2 x 2 ix 0 are real positive and discrete the behavior of these eigenvalues has been studied perturbatively for small however until now no other features of h have been examined perturbatively in this paper the small expansion of the c operator and the equivalent isospectral dirac hermitian hamiltonian h are derived

finite dimensional pt symmetric hamiltonians this paper investigates finite dimensional pt symmetric hamiltonians it is shown here that there are two ways to extend real symmetric hamiltonians into the complex domain i the usual approach is to generalize such hamiltonians to include complex hermitian hamiltonians ii alternatively one can generalize real symmetric hamiltonians to include complex pt symmetric hamiltonians in the first approach the spectrum remains real while in the second approach the spectrum remains real if the pt symmetry is not broken both generalizations give a consistent theory of quantum mechanics but if d 2 a d dimensional hermitian matrix hamiltonian has more arbitrary parameters than a d dimensional pt symmetric matrix hamiltonian

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

on a class of non self adjoint periodic eigenproblems with boundary and interior singularities we prove that all the eigenvalues of a certain highly non self adjoint sturm liouville differential operator are real the results presented are motivated by and extend those recently found by various authors benilov et al 2003 3 davies 2007 7 and weir 2008 18 on the stability of a model describing small oscillations of a thin layer of fluid inside a rotating cylinder

regression procedures for arma estimation abstract regression procedures for parameter estimation in autoregression moving average arma models are discussed mainly for providing initial estimates for iterative maximization of a gaussian likelihood an iterative procedure of spliid 1983 is compared to a procedure of hannan and rissanen 1982 and a global convergence result is established for an iterative modification of spliid s procedure spliid s iteration does not always converge when it does it has the same asymptotic distribution as the second stage of the hannan rissanen procedure this second stage iteration gives the same problems as spliid s procedure so it is preferable to go immediately to the third stage which is asymptotically efficient in the gaussian case an example is provided by a first order arma model y t y t 1 e t e t 1 under the usual regularity conditions of stationarity and invertibility spliid s procedure fits an autoregression of predetermined order i e 2 to obtain estimates of the e t a

spiked and symmetrized decadic potentials supporting elementary n plets of bound states we show that the potential wells v x x10 ax8 bx6 cx4 dx2 f x2 with a central spike possess arbitrary finite multiplets of elementary exact bound states their strong asymptotic growth implies an ambiguity in their symmetrically generalized quantization via complex boundary conditions but the three eligible methods coincide at our exceptional solutions

diagnostic checking in arma models with uncorrelated errors we consider tests for lack of fit in arma models with nonindependent innovations in this framework the standard box pierce and ljung box portmanteau tests can perform poorly specifically the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration in this article we derive the asymptotic covariance matrix of a vector of autocorrelations for residuals of arma models under weak assumptions on the noise the asymptotic distribution of the portmanteau statistics follows a consistent estimator of and a modification of the portmanteau tests are proposed this allows us to construct valid asymptotic significance limits for the residual autocorrelations and asymptotically valid goodness of fit tests when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference a set of monte carlo experiments and an application to the standard poor 500 returns illustrate the p

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

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

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

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

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

regularity of solutions to the navier stokes equations for compressible barotropic flows on a polygon the navier stokes system for a steady state barotropic nonlinear compressible viscous flow with an inflow boundary condition is studied on a polygon d a unique existence for the solution of the system is established it is shown that the lowest order corner singularity of the nonlinear system is the same as that of the laplacian in suitable l q spaces let be the interior angle of a vertex p of d if and then the velocity u is split into singular and regular parts near the vertex p if 2 and or if 2 and 2 q it is shown that u h 2 q d 2

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

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

kurt symanzik a stable fixed point beyond triviality in 1970 kurt symanzik proposed a precarious 4 theory with a negative quartic coupling constant as a valid candidate for an asymptotically free theory of strong interactions symanzik s deep insight into the non trivial properties of this theory has been overruled since then by the hermitian intuition of generations of scientists who considered or consider this actually non hermitian highly important theory to be unstable this short certainly controversial communication tries to shed some light on the historical and formalistic context of symanzik s theory in order to sharpen our quantum intuition about non perturbative theoretical physics between non triviality and asymptotic freedom

estimation and forecasting in vector autoregressive moving average models for rich datasets we address the issue of modelling and forecasting macroeconomic variables using rich datasets by adopting the class of vector autoregressive moving average varma models we overcome the estimation issue that arises with this class of models by implementing an iterative ordinary least squares iols estimator we establish the consistency and asymptotic distribution of the estimator for weak and strong varma p q models monte carlo results show that iols is consistent and feasible for large systems outperforming the mle and other linear regression based efficient estimators under alternative scenarios our empirical application shows that varma models are feasible alternatives when forecasting with many predictors we show that varma models outperform the ar 1 arma 1 1 bayesian var and factor models considering different model dimensions

var analysis nonfundamental representations blaschke matrices abstract macroeconomic models may produce arma structures where the determinant of the ma matrix polynomial has some roots inside the unit circle this implies that the impulse response functions are no longer identified and may vary in an infinite dimensional space this paper deals with this problem in the var or structural var framework we provide a method to strongly limit the research for economically interesting nonfundamental impulse response functions and show how to construct such representations from estimated var coefficients we also give two empirical applications gnp unemployment usa data and interest rate inflation french data

an evolution compressible stokes system in a polygon abstract an evolution compressible stokes system is studied in a polygon a lowest order corner singularity by the laplacian operator is sorted out and an increased regularity is established in a suitable banach space in the lowest level for each time the order of the corner singularity of the system is the same as that of the laplacian in addition a singular function for pressure is defined and shows that the corner singularities originated from the corners of polygon may propagate along the directed streamlines

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

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

unbounded mathcal c symmetries and their nonuniqueness it is shown that if the operator for a symmetric hamiltonian with simple eigenvalues is not unique then it is unbounded the fact that the operator is unbounded is significant because while there is a formal equivalence between a symmetric hamiltonian and a conventionally hermitian hamiltonian in the sense that the two hamiltonians are isospectral the hilbert spaces are inequivalent this is so because the mapping from one hilbert space to the other is unbounded this shows that symmetric quantum theories are mathematically distinct from conventional hermitian quantum theories this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

recursive estimation of mixed autoregressive moving average order summary the order p q of an autoregressive moving average sequence y t may be estimated by minimizing a criterion log 2 p q log t t with respect to p and q where 2 is the maximum likelihood estimate of the variance of the innovations t it is suggested that instead a2 be estimated from a series of regressions of y t on y t 1 y t 2 y t p 5 t 1 t q where the t are obtained by fitting a long autoregression to the data it is shown how the sequence of regressions may for p q be economically recursively calculated by embedding them in a sequence of bivariate autoregressions asymptotic properties of the procedure are established under very general conditions

interior jump and piecewise regularity for compressible stokes equations in the t shaped cavity domain abstract in the t shaped cavity domain the fluid flows consist of three essential phenomena that are the flows in the top region with inflow and outflow the ones in the bottom cavity region and the interface fluctuation between two flows we consider a simple modeling problem of compressible stokes flows to illustrate them the domain is split by the interior curve connecting the upstream and downstream corners the transport equation is directed by the vector field having jump discontinuity with respect to the curve so the continuity equation shows the pressure jump solution across the curve and the velocity vector by the momentum equations has the singular behaviors due to the pressure gradient plus the corner singularities at the non convex corners we establish a piecewise maximal regularity by constructing a vector function corresponding to the pressure jump on the interior curve without subtracting the corner singularities

pseudospectra in non hermitian quantum mechanics we propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non hermitian operators we relate pseudospectral properties to quasi hermiticity similarity to self adjoint operators and basis properties of eigenfunctions the abstract results are illustrated by unexpected wild properties of operators familiar from pt symmetric quantum mechanics

large bayesian varmas vector autoregressive moving average varma models have many theoretical properties which should make them popular among empirical macroeconomists however they are rarely used in practice due to over parameterization concerns difficulties in ensuring identification and computational challenges with the growing interest in multivariate time series models of high dimension these problems with varmas become even more acute accounting for the dominance of vars in this field in this paper we develop a bayesian approach for inference in varmas which surmounts these problems it jointly ensures identification and parsimony in the context of an efficient markov chain monte carlo mcmc algorithm we use this approach in a macroeconomic application involving up to twelve dependent variables we find our algorithm to work successfully and provide insights beyond those provided by vars

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

closed formula for the metric in the hilbert space of a symmetric model we introduce a very simple exactly solvable symmetric non hermitian model with a real spectrum and derive a closed formula for the metric operator which relates the problem to a hermitian one

perturbation theory of symmetric hamiltonians in the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric hamiltonians is proved using stability techniques we apply this method to symmetric unperturbed hamiltonians perturbed by symmetric additional interactions

calculation of the hidden symmetry operator in symmetric quantum mechanics in a recent paper it was shown that if a hamiltonian h has an unbroken symmetry then it also possesses a hidden symmetry represented by the linear operator the operator commutes with both h and the inner product with respect to is associated with a positive norm and the quantum theory built on the associated hilbert space is unitary in this paper it is shown how to construct the operator for the non hermitian symmetric hamiltonian h 1 2p2 1 2x2 ix3 using perturbative techniques it is also shown how to construct the operator for h 1 2p2 1 2x2 x4 using nonperturbative methods

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

symmetric quantum theory defined in a krein space we provide a mathematical framework for symmetric quantum theory which is applicable irrespective of whether a system is defined on or a complex contour whether symmetry is unbroken and so on the linear space in which symmetric quantum theory is naturally defined is a krein space constructed by introducing an indefinite metric into a hilbert space composed of square integrable complex functions in a complex contour we show that in this krein space every symmetric operator is hermitian if and only if it has transposition symmetry as well from which the characteristic properties of the symmetric hamiltonians found in the literature follow some possible ways to construct physical theories are discussed within the restriction to the class k h