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

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

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

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

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

on the dof region of the mimo gaussian two user interference channel with an instantaneous relay this paper studies inner and outer bounds on the degrees of freedom dof region of the multi antenna two user gaussian interference channel with an instantaneous relay ir or relay without delay it is assumed that the two transmitters and the two receivers have mathop mathrm mathsf m nolimits antennas while the ir receives through mathop mathrm mathsf n r nolimits antennas and transmits through mathop mathrm mathsf n t nolimits antennas in the proposed achievable scheme which generalizes a known one for the case mathop mathrm mathsf m nolimits mathop mathrm mathsf n r nolimits mathop mathrm mathsf n t nolimits to any mathop mathrm mathsf m nolimits mathop mathrm mathsf n r nolimits mathop mathrm mathsf n t nolimits the ir performs memoryless linear operations on its received signal so as to neutralize interference at the receivers and the beamforming matrices used by the ir and the transmitters are jointly designed this joint design strictly outperforms known achievable schemes two outer bounds are derived an information theoretic outer bound is obtained by giving the receivers or the ir genie side information so that the dof region of the resulting enhanced channel is known this converse is valid for any type of processing at the ir and shows the optimality of the proposed achievable scheme for some mathop mathrm mathsf m nolimits mathop mathrm mathsf n r nolimits mathop mathrm mathsf n t nolimits a linear processing outer bound is obtained when the ir is restricted to performs linear operations without any memoryless restriction on its received signal and shows the optimality of the proposed achievable scheme among all linear processing schemes at the ir as a result of independent interest the dof region of the classical multi antenna two user gaussian interference channel without relay when the channel matrices can have any structure is also derived which generalized available dof region results that were derived under certain assumptions on the structure of the channel matrices

duality gap in convex programming in this paper we consider general convex programming problems and give a sufficient condition for the equality of the infimum of the original problem and the supremum of its ordinary dual this condition may be seen as a continuity assumption on the constraint sets i e on the sublevel sets of the constraint function under linear perturbation it allows us to generalize as well as treat previous results in a unified framework our main result is in fact based on a quite general constraint qualification result for quasiconvex programs involving a convex objective function proven in the setting of a real normed vector space

some convex programs without a duality gap an important issue in convex programming concerns duality gap various conditions have been developed over the years that guarantee no duality gap including one developed by rockafellar network flows and monotropic programming wiley interscience new york 1984 involving separable objective function and affine constraints we show that this sufficient condition can be further relaxed to allow the constraint functions to be separable we also refine a sufficient condition involving weakly analytic functions by allowing them to be extended real valued

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

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

existence of optimal solutions and duality results under weak conditions in this paper we consider an ordinary convex program with no qualification conditions such as slater s condition and for which the optimal set is neither required to be compact nor to be equal to the sum of a compact set and a linear space it is supposed only that the infimum is finite a very wide class of convex functions is exhibited for which the optimum is always attained and is equal to the supremum of the ordinary dual program additional results concerning the existence of optimal solutions in the non convex case are also given

on conic qpccs conic qcqps and completely positive programs this paper studies several classes of nonconvex optimization problems defined over convex cones establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs the problems being studied include a conic quadratically constrained quadratic program qcqp a conic quadratic program with complementarity constraints qpcc and rank constrained semidefinite programs our results do not make any boundedness assumptions on the feasible regions of the various problems considered the first stage in the reformulation is to cast the problem as a conic qcqp with just one nonconvex constraint q x le 0 q x 0 where q x is nonnegative over the convex conic and linear constraints so the problem fails the slater constraint qualification a conic qpcc has such a structure we prove the converse namely that any conic qcqp satisfying a constraint qualification can be expressed as an equivalent conic qpcc the second stage of the reformulation lifts the problem to a completely positive program and exploits and generalizes a result of burer we also show that a frank wolfe type result holds for a subclass of this class of conic qcqps further we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper

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

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

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

analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets in this paper we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semialgebraic convex sets we establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semialgebraic convex sets and the dimension of the underlying space we achieve our results by exploiting the algebraic structure of the basic semialgebraic convex sets

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

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

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

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

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

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

ideal weakly efficient solutions for vector optimization problems we establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded the solutions to be referred are ideal strong weakly efficient weakly pareto also several necessary and or sufficient conditions for the solution set to be non empty and compact are established moreover some characterizations of the non emptiness boundedness of the convex solution set in case the solutions are searched in a subset of the real line are also given however the solution set fails to be convex in general in addition special attention is addressed when the underlying cone is the non negative orthant and when the semi strict quasiconvexity of each component of the vector valued function is assumed our approach is based on the asymptotic description of the functions and sets some examples illustrating such an approach are also exhibited

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

stable zero duality gaps in convex programming complete dual characterisations with applications to semidefinite programs the zero duality gap that underpins the duality theory is one of the central ingredients in optimisation in convex programming it means that the optimal values of a given convex program and its associated dual program are equal it allows in particular the development of efficient numerical schemes however the zero duality gap property does not always hold even for finite dimensional problems and it frequently fails for problems with non polyhedral constraints such as the ones in semidefinite programming problems over the years various criteria have been developed ensuring zero duality gaps for convex programming problems in the present work we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions for convex semidefinite programs we establish necessary and sufficient dual conditions for stable zero duality gaps as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right hand side and convex objective function zero duality gap results for second order cone programming problems are also given our approach makes use of elegant conjugate analysis and fenchel s duality

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

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

information networks with in block memory a class of channels is introduced for which there is memory inside blocks of a specified length and no memory across the blocks the multiuser model is called an information network with in block memory nibm it is shown that block fading channels channels with state known causally at the encoder and relay networks with delays are nibms a cut set bound is developed for nibms that unifies strengthens and generalizes existing cut bounds for discrete memoryless networks the bound gives new finite letter capacity expressions for several classes of networks including point to point channels and certain multiaccess broadcast and relay channels cardinality bounds on the random coding alphabets are developed that improve on existing bounds for channels with action dependent state available causally at the encoder and for relays without delay finally quantize forward network coding is shown to achieve rates within an additive gap of the new cut set bound for linear additive gaussian noise channels symmetric power constraints and a multicast session

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

causal relay networks in this paper we study causal discrete memoryless relay networks the network consists of multiple nodes each of which can be a source a relay and or a destination in the network there are two types of relays 1 relays with one sample delay strictly causal and 2 relays without delay causal whose transmit signals depend not only on the past received symbols but also on the current received symbols for this network we derive two new cut set bounds one when every node has a message and the other when only the strictly causal relays have messages using the examples of a causal vector gaussian two way relay channel and a causal vector gaussian relay channel we show that the new cut set bounds can be achieved by a simple amplify and forward type relaying our result for the causal relay channel strengthens the previously known capacity result for the same channel by el gamal hassanpour and mammen

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

duality gap of the conic convex constrained optimization problems in normed spaces in this paper motivated by a result due to champion math program 99 2004 we introduce a property for a conic quasi convex vector valued function in a general normed space we prove that this property characterizes the zero duality gap for a class of the conic convex constrained optimization problem in the sense that if this property is satisfied and the objective function f is continuous at some feasible point then the duality gap is zero and if this property is not satisfied then there exists a linear continuous function f such that the duality gap is positive we also present some sufficient conditions for the property

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

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

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

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

noncoercive optimization problems this paper modifies a recent sufficient condition for the existence of optimal solutions proposed by buttazzo and tomarelli to obtain a condition that is necessary and sufficient applications to mathematical programming are given and stability results established

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

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

a geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints we first establish a relaxed version of dines theorem associated to quadratic minimization problems with finitely many linear equality and a single nonconvex quadratic inequality constraints the case of unbounded optimal valued is also discussed then we characterize geometrically the strong duality and some relationships with the conditions employed in finsler theorem are established furthermore necessary and sufficient optimality conditions with or without the slater assumption are derived our results can be used to situations where none of the results appearing elsewhere are applicable in addition a revisited theorem due to frank and wolfe along with that due to eaves is established for asymptotically linear sets

set intersection theorems and existence of optimal solutions the question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics including the existence of optimal solutions the validity of the minimax inequality in zero sum games and the absence of a duality gap in constrained optimization we consider asymptotic directions of a sequence of closed sets and introduce associated notions of retractive horizon and critical directions based on which we provide new conditions that guarantee the nonemptiness of the corresponding intersection we show how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions and to derive some new results including a new extension of the frank wolfe theorem for nonconvex quadratic programming