a note on controlled diffusions on line with time averaged cost existence of stable optimal markov controls is established for a class of cost functions for controlled one dimensional diffusions with time averaged cost

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

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

general aspects of pt symmetric and p self adjoint quantum theory in a krein space in our previous work we proposed a mathematical framework for symmetric quantum theory and in particular constructed a krein space in which symmetric operators would naturally act in this work we explore and discuss various general consequences and aspects of the theory defined in the krein space not only spectral properties and symmetry breaking but also several issues crucial for the theory to be physically acceptable such as time evolution of state vectors probability interpretation uncertainty relation classical quantum correspondence completeness existence of a basis and so on in particular we show that for a given real classical system we can always construct the corresponding symmetric quantum system which indicates that symmetric theory in the krein space is another quantization scheme rather than a generalization of the traditional hermitian one in the hilbert space we propose a postulate for an operator to be a physical observable in this framework

szego polynomials associated with wiener levinson filters abstract szego polynomials are studied in connection with wiener levinson filters formed from discrete signals xn xn k n 1k 0 our main interest is in the frequency analysis problem of finding the unknown frequencies j when the signal is a trigonometric polynomial x n k j i i j e i j k associated with this signal is the sequence of monic szego polynomials n n z n 0 orthogonal on the unit circle with respect to a distribution function n explicit expressions for the weight function n and associated szego function dn z are given in terms of the z transform xn z of the signal xn several theorems are given to support the following conjecture which was suggested by numerical experiments as n and n increase the 2i 1 zeros of n n z of largest modulus approach the points ei j we conclude by showing that the reciprocal polynomials n n z z n n n 1 z are pade numerators for pade approximants of fixed denominator degree to a meromorphic function related to dn z

relaxation labeling with learning automata relaxation labeling processes are a class of mechanisms that solve the problem of assigning labels to objects in a manner that is consistent with respect to some domain specific constraints we reformulate this using the model of a team of learning automata interacting with an environment or a high level critic that gives noisy responses as to the consistency of a tentative labeling selected by the automata this results in an iterative linear algorithm that is itself probabilistic using an explicit definition of consistency we give a complete analysis of this probabilistic relaxation process using weak convergence results for stochastic algorithms our model can accommodate a range of uncertainties in the compatibility functions we prove a local convergence result and show that the point of convergence depends both on the initial labeling and the constraints the algorithm is implementable in a highly parallel fashion

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

j self adjoint operators with mathcal c symmetries an extension theory approach a well known tool in conventional von neumann quantum mechanics is the self adjoint extension technique for symmetric operators it is used e g for the construction of dirac hermitian hamiltonians with point interaction potentials here we reshape this technique to allow for the construction of pseudo hermitian j self adjoint hamiltonians with complex point interactions we demonstrate that the resulting hamiltonians are bijectively related to the so called hypermaximal neutral subspaces of the defect krein space of the symmetric operator this symmetric operator is allowed to have arbitrary but equal deficiency indices n n general properties of the operators for these hamiltonians are derived a detailed study of operator parametrizations and krein type resolvent formulae is provided for j self adjoint extensions of symmetric operators with deficiency indices 2 2 the technique is exemplified on 1d pseudo hermitian schr dinger and dirac hamiltonians with complex point interaction potentials

scattering theory for quantum fields with indefinite metric in this work we discuss the scattering theory of local relativistic quantum fields with indefinite metric since the results of haag ruelle theory do not carry over to the case of indefinite metric 4 we propose an axiomatic framework for the construction of in and out states such that the lsz asymptotic condition can be derived from the assumptions the central mathematical object for this construction is the collection of mixed vacuum expectation values of local in and out fields called the form factor functional which is required to fulfill a hilbert space structure condition given a scattering matrix with polynomial transfer functions we then construct interpolating local relativistic quantum fields with indefinite metric which fit into the given scattering framework

polynomial zerofinders based on szego polynomials abstract the computation of zeros of polynomials is a classical computational problem this paper presents two new zerofinders that are based on the observation that after a suitable change of variable any polynomial can be considered a member of a family of szeg polynomials numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial

some remarks on quasi hermitian operators a quasi hermitian operator is an operator that is similar to its adjoint in some sense via a metric operator i e a strictly positive self adjoint operator whereas those metric operators are in general assumed to be bounded we analyze the structure generated by unbounded metric operators in a hilbert space following our previous work we introduce several generalizations of the notion of similarity between operators then we explore systematically the various types of quasi hermitian operators bounded or not finally we discuss their application in the so called pseudo hermitian quantum mechanics

parameter estimation for nonincreasing exponential sums by prony like methods abstract let z j e f j with f j 0 i be distinct nodes for j 1 m with complex coefficients c j 0 we consider a nonincreasing exponential sum h x c 1 e f 1 x c m e f m x x 0 many applications in electrical engineering signal processing and mathematical physics lead to the following problem determine all parameters of h if 2 n sampled values h k k 0 2 n 1 n m are given this parameter estimation problem is a nonlinear inverse problem for noiseless sampled data we describe the close connections between prony like methods namely the classical prony method the matrix pencil method and the esprit method further we present a new efficient algorithm of matrix pencil factorization based on qr decomposition of a rectangular hankel matrix the algorithms of parameter estimation are also applied to sparse fourier approximation and nonlinear approximation

szego polynomials applied to frequency analysis abstract this paper is concerned with the problem of determining unknown frequencies 1 i using an observed discrete time signal xn xn m arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well defined frequencies j periods p j 2 n j n z j 1 i we investigate the wiener levinson method formulated here in terms of szeg polynomials n n z orthogonal on the unit circle with respect to a distribution function n defined by the n observed values of the signal we prove that if n0 denotes the number of critical points eiwj then for every n n0 and n 1 the zeros z j n n of n n z can be arranged so that limn z j n n eiwj for each of the frequencies wj this result confirms one of the main parts of the conjecture given by jones njastad and saff this journal 1990 on the convergence of zeros of the szeg polynomials a related result on the convergence of corresponding two point pade approximants is also given

one dimensional schrodinger operators with symmetric zero range potentials non hermitian hamiltonians appearing as operator realizations of symmetric symmetric in physical literature zero range singular perturbations of one dimensional schrodinger operators are studied in particular hamiltonians with a real spectrum are described in terms of parameters of singular perturbations and moreover it is shown that only part of them are similar to hermitian ones in this case they can be used as exactly solvable models of symmetric quantum mechanics

on the boundary conditions characterizing j selfadjoint extensions of j symmetric operators the study of boundary value problems involving linear differentiai equations with real valued coefficients is by now a well established area of analysis on the other hand much less is known about the solvability of such problems when the coefficients or boundary conditions are known to be complex examples of this latter type arise naturally for example in nuclear physics e g the so called optical mode1 for low energy scattering 1 p ijo electromagnetic field theory dielectric waveguides with heat loss c f 19 i or the propagation of radio waves through inhomogeneous media s and elsewhere in cases like these the relevant differential expressions are no longer formally symmetric and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available to facilitate the study of such problems glazman introduced in 4 the concept of a j symmetric operator in a complex hilbert space r let j be a given conjugation operator on 2 i e j is a conjugate linear involution with jx 3 u x for all x and y in 2 a closed densely defined linear operator t in 3 is said to be j symmetric if

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

pt symmetric models in curved manifolds we consider the laplace beltrami operator in tubular neighborhoods of curves on two dimensional riemannian manifolds subject to non hermitian parity and time preserving boundary conditions we are interested in the interplay between the geometry and spectrum after introducing a suitable hilbert space framework in the general situation which enables us to realize the laplace beltrami operator as an m sectorial operator we focus on solvable models defined on manifolds of constant curvature in some situations notably for non hermitian robin type boundary conditions we are able to prove either the reality of the spectrum or the existence of complex conjugate pairs of eigenvalues and establish similarity of the non hermitian m sectorial operators to normal or self adjoint operators the study is illustrated by numerical computations

convergence of learning algorithms with constant learning rates the behavior of neural network learning algorithms with a small constant learning rate epsilon in stationary random input environments is investigated it is rigorously established that the sequence of weight estimates can be approximated by a certain ordinary differential equation in the sense of weak convergence of random processes as epsilon tends to zero as applications backpropagation in feedforward architectures and some feature extraction algorithms are studied in more detail

continuation methods for the computation of zeros of szego polynomials abstract let j j 0 be a family of monic polynomials that are orthogonal with respect to an inner product on the unit circle the polynomials j arise in time series analysis and are often referred to as szego polynomials or levinson polynomials knowledge about the location of their zeros is important for frequency analysis of time series and for filter implementation we present fast algorithms for computing the zeros of the polynomials n based on the observation that the zeros are eigenvalues of a rank one modification of a unitary upper hessenberg matrix h n 0 of order n the algorithms first determine the spectrum of h n 0 by one of several available schemes that require only o n 2 arithmetic operations the eigenvalues of the rank one perturbation are then determined from the eigenvalues of h n 0 by a continuation method the computation of the n zeros of n in this manner typically requires only o n 2 arithmetic operations the algorithms have a structure that lends itself well to parallel computation the latter is of significance in real time applications

moyal products a new perspective on quasi hermitian quantum mechanics the rationale for introducing non hermitian hamiltonians and other observables is reviewed and open issues identified we present a new approach based on moyal products to compute the metric for quasi hermitian systems this approach is not only an efficient method of computation but also suggests a new perspective on quasi hermitian quantum mechanics which invites further exploration in particular we present some first results which link the berry connection and curvature to non perturbative properties and the metric

ergodic control of multidimensional diffusions 1 the existence results the existence of optimal stable markov relaxed controls for the ergodic control of multidimensional diffusions is established by direct probabilistic methods based on a characterization of a s limit sets of empirical measures the optimality of the above is established in the strong i e almost sure sense among all admissible controls under very general conditions

complex symmetric operators and applications ii a bounded linear operator t on a complex hilbert space h is called complex symmetric if t ct c where c is a conjugation an isometric antilinear involution of h we prove that t cj t where j is an auxiliary conjugation commuting with t t t we consider numerous examples including the poincare neumann singular integral bounded operator and the jordan model operator compressed shift the decomposition t cj t also extends to the class of unbounded c selfadjoint operators originally introduced by glazman in this context it provides a method for estimating the norms of the resolvents of certain unbounded operators

pt symmetry cartan decompositions lie triple systems and krein space related clifford algebras gauged quantum mechanics ptqm and corresponding krein space setups are studied for models with constant non abelian gauge potentials and extended parity inversions compact and noncompact lie group components are analyzed via cartan decompositions a lie triple structure is found and an interpretation as symmetrically generalized jaynes cummings model is possible with close relation to recently studied cavity qed setups with transmon states in multilevel artificial atoms for models with abelian gauge potentials a hidden clifford algebra structure is found and used to obtain the fundamental symmetry of krein space related j self adjoint extensions for ptqm setups with ultra localized potentials

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

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 a class of j self adjoint operators with empty resolvent set in the present paper we investigate the set j of all j self adjoint extensions of an operator s which is symmetric in a hilbert space h with deficiency indices 2 2 and which commutes with a non trivial fundamental symmetry j of a krein space h sj js our aim is to describe different types of j self adjoint extensions of s which in general are non self adjoint operators in the hilbert space h one of our main results is the equivalence between the presence of j self adjoint extensions of s with empty resolvent set and the commutation of s with a clifford algebra cl2 j r where r is an additional fundamental symmetry with jr rj this enables one to parameterize in terms of cl2 j r the set of all j self adjoint extensions of s with stable c symmetry here an extension has stable c symmetry if it commutes with a fundamental symmetry and in turn this fundamental symmetry commutes with s such a situation occurs naturally in many applications here we discuss the case of indefinite sturm liouville operators and the case of a one dimensional dirac operator with point interaction

signal recovery by discrete approximation and a prony like method abstract we introduce an algorithm which combines ideas of prony s approach to recover signals from given samples with approximation methods we solve two overdetermined systems of linear equations with linear programming methods and calculate the zeros of a suitable prony like polynomial we get the bandwidth m the frequencies as well as the amplitudes and some other characteristics of the signal especially it is reconstructed if sufficient many at least 2 m samples are given if we have too few samples or if they are too much noised we get an approximation of the original noiseless signal even if we have sufficient many but possible erroneous samples the obtained signal interpolates at least m of them usually the low noised or noiseless ones and of course all samples if the signal is recovered the described method behaves well to moderate sampling errors and is resistant to outliers in the samples which can be detected filtered off or corrected during the calculations to improve the quality of the computed signal

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

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

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

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

on domains of cal p cal t symmetric operators related to y x 1 nx2ny x in recent years a generalization of hermiticity has been investigated using a complex deformation h p2 x2 ix of the harmonic oscillator hamiltonian where is a real parameter these complex hamiltonians possessing symmetry the product of parity and time reversal can have a real spectrum we will consider the most simple case even in this paper we describe all self adjoint hermitian and at the same time symmetric operators associated with h p2 x2 ix surprisingly it turns out that there is a large class of self adjoint operators associated with h p2 x2 ix which are not symmetric

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

metrics and isospectral partners for the most generic cubic pt symmetric non hermitian hamiltonian we investigate properties of the most general symmetric non hermitian hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family for various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and moyal products we elaborate on the subtleties of this approach for special choices of the ten parameters the hamiltonian reduces to various models previously studied such as to the complex cubic potential the so called swanson hamiltonian or the transformed version of the from below unbounded quartic x4 potential in addition it also reduces to various models not considered in the present context namely the single site lattice reggeon model and a transformed version of the massive sextic x6 potential which plays an important role as a toy model to identify theories with vanishing cosmological constant