deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator classes of coherent states are presented by replacing the labelling parameter z of klauder perelomov type coherent states by confluent hypergeometric functions with specific parameters temporally stable coherent states for the isotonic oscillator hamiltonian are presented and these states are identified as a particular case of the so called mittag leffler coherent states

algebraic isomonodromic deformations of logarithmic connections on the riemann sphere and finite braid group orbits on character varieties we study algebraic isomonodromic deformations of flat logarithmic connections on the riemann sphere with n ge 4 poles for arbitrary rank we introduce a natural property of algebraizability for the germ of universal deformation of such a connection we relate this property to a peculiarity of the corresponding monodromy representation to yield a finite braid group orbit on the appropriate character variety under reasonable assumptions on the deformed connection we may actually establish an equivalence between both properties we apply this result in the rank two case to relate finite branching and algebraicity for solutions of garnier systems for general rank a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension

survey paper research on probabilistic methods for control system design a novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty the main objective of this paper is to provide a broad perspective on this area of research known as probabilistic robust control and to address in a systematic manner recent advances the focal point is on design methods based on the interplay between uncertainty randomization and convex optimization and on the illustration of specific control applications

the generalized quantum isotonic oscillator recently it has been proved that a nonlinear quantum oscillator generalization of the isotonic one is exactly solvable for certain values of its parameters here we show that the schrodinger equation for such an oscillator can be transformed into a confluent heun equation we give a very simple and efficient algorithm to solve it numerically no matter what the values of the parameters are algebraic quasi polynomial solutions for particular values of the parameters are found

construction of solvable quantum mechanical potentials a method of generating some quantum mechanical exactly solvable potentials esps using the properties of classical orthogonal polynomials cops is presented it is illustrated using hypergeometric polynomial utilization of other cops to get more esps is indicated

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

exact solutions of the harmonic oscillator plus non polynomial interaction the exact solutions to a one dimensional harmonic oscillator plus a non polynomial interaction ax2bx2 1cx2 a 0 c 0 are given by the confluent heun functions hc z the minimum value o

the stability of the grey linear system this paper discusses the stability problem of linear systems whose state matrices are interval matrices and presents some sufficient conditions whereby the stability of the systems can be guaranteed examples are given to illustrate the points

the q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

special linear combinations of orthogonal polynomials abstract the paper lists a number of problems that motivate consideration of special linear combinations of polynomials orthogonal with the weight p x on the interval a b we study properties of the polynomials as well as the necessary and sufficient conditions for their orthogonality the special linear combinations of chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations

remarks on the solution of the position dependent mass schrodinger equation an approximate method is proposed to solve the position dependent mass pdm schrodinger equation the procedure suggested here leads to the solution of the pdm schrodinger equation without transforming the potential function to the mass space or vice versa the method based on the asymptotic taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the pdm schrodinger equation the results show that the pdm and constant mass schrodinger equations are not isospectral the calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

an observation of quadratic algebra dual family of nonlinear coherent states and their non classical properties in the generalized isotonic oscillator in this paper we construct nonlinear coherent states for the generalized isotonic oscillator and study their non classical properties in detail by transforming the deformed ladder operators suitably which generate the quadratic algebra we obtain heisenberg algebra from the algebra we define two non unitary and one unitary displacement type operators while the action of one of the non unitary type operators reproduces the original nonlinear coherent states the other one fails to produce a new set of nonlinear coherent states dual pair we show that these dual states are not normalizable for the nonlinear coherent states we evaluate the parameter a3 and examine the non classical nature of the states through quadratic and amplitude squared squeezing effect further we derive analytical formula for the p function q function and the wigner function for the nonlinear coherent states all of them confirm the non classicality of the nonlinear coherent states in addition to the above we obtain the

extending romanovski polynomials in quantum mechanics some extensions of the third class romanovski polynomials also called romanovski pseudo jacobi polynomials which appear in bound state wavefunctions of rationally extended scarf ii and rosen morse i potentials are considered for the former potentials the generalized polynomials satisfy a finite orthogonality relation while for the latter an infinite set of relations among polynomials with degree dependent parameters is obtained both types of relations are counterparts of those known for conventional polynomials in the absence of any direct information on the zeros of the romanovski polynomials present in denominators the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second order differential equations of schrodinger type it is also shown that on going from scarf i to scarf ii or from rosen morse ii to rosen morse i potentials the variety of rational extensions is narrowed down from types i ii and iii to type iii only

spectral properties of a confined nonlinear quantum oscillator in one and three dimensions we analyze the spectral behaviour of a nonlinear quantum oscillator model under confinement the underlying potential is given by a harmonic oscillator interaction plus a nonlinear term that can be weakened or strengthened through a parameter numerical eigenvalues of the model in one and three dimensions are presented the asymptotic behaviour of the eigenvalues for confinement relaxation and for vanishing nonlinear term in the potential is investigated our findings are compared with existing results

quantization of the lienard ii equation and jacobi s last multiplier in this paper the role of jacobi s last multiplier in mechanical systems with a position dependent mass is unveiled in particular we map the lienard ii equation to a position dependent mass system the quantization of the lienard ii equation is then carried out using the point canonical transformation method together with the von roos ordering technique finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated laguerre and exceptional laguerre functions

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

on a novel supersymmetric connections between harmonic and isotonic oscillators abstract beginning from the harmonic oscillator we obtain a one parameter family of potentials whose spectra coincide with that of the isotonic oscillator diverse arguments concerning supersymmetric transformations factorization procedures and riccati equations are used in a systematic way in particular we take advantage of the non standard factorization of the schrodinger equation which appears when considering the energy of the first excited level as reference to be more specific supersymmetry is neither exact nor spontaneously broken but realizes itself acting on wave functions vanishing at x 0

a report on the nonlinear squeezed states and their non classical properties of a generalized isotonic oscillator we construct nonlinear squeezed states of a generalized isotonic oscillator potential we demonstrate the non existence of a dual counterpart of nonlinear squeezed states in this system we investigate the statistical properties exhibited by the squeezed states in particular mandel s parameter second order correlation function photon number distributions and parameter a3 in detail we also examine the quadrature and amplitude squared squeezing effects finally we derive the expression for the s parameterized quasi probability distribution function of these states all this information about the system is new to the literature

new solvable singular potentials we obtain three new solvable real shape invariant potentials starting from the harmonic oscillator poschl teller i and poschl teller ii potentials on the half axis and extending their domain to the full line while taking special care to regularize the inverse square singularity at the origin the regularization procedure gives rise to a delta function behaviour at the origin our new systems possess underlying nonlinear potential algebras which can also be used to determine their spectra analytically

a generalized cole hopf transformation for nonlinear odes we introduce a hybrid cole hopf darboux transformation to relate the solutions of nonlinear and linear second order differential equations and derive classification and sufficient condition for this correspondence we explore physical applications of this correspondence to nonlinear oscillations the duffing equation and a nonlinear form of the schr dinger equation for the nonrelativistic hydrogen atom in addition we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation these nonlinear equations can be viewed as the class of special nonlinear equations which correspond to the linear differential equations which define the special functions of mathematical physics

disconjugacy regularity of multi indexed rationally extended potentials and laguerre exceptional polynomials the power of the disconjugacy properties of second order differential equations of schrodinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re examining the extensions of the isotonic oscillator or radial oscillator potential derived in kth order supersymmetric quantum mechanics or multistep darboux backlund transformation method the function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term whose value is calculated by induction over k the sign of this term being the same as that of the already known highest degree term the potential denominator has the same sign at both extremities of the definition interval a property that is shared by the seed eigenfunction used in the potential construction by virtue of disconjugacy such a property implies the nodeless character of both the eigenfunction and the resulting potential

the sobolev orthogonality and spectral analysis of the laguerre polynomials l n k for positive integers k for k n we consider the analysis of the classical laguerre differential expression l k y x 1 x ke x x k 1e xy x rx ke xy x x 0 where r 0 is fixed in several nonisomorphic hilbert and hilbert sobolev spaces in one of these spaces specifically the hilbert space l2 0 x ke x it is well known that the glazman krein naimark theory produces a self adjoint operator a k generated by l k that is bounded below by rl where i is the identity operator on l2 0 x ke x consequently as a result of a general theory developed by littlejohn and wellman there is a continuum of left definite hilbert spaces h s k v s k s k s 0 and left definite self adjoint operators b s k s 0 associated with the pair l2 0 x ke x a k for a k and each of the operators b s k it is the case that the tail end sequence l n k n k of laguerre polynomials form a complete set of eigenfunctions in the corresponding hilbert spaces in 1995 kwon and littlejohn introduced a hilbert sobolev space w k 0 in which the entire sequence of laguerre polynomials is orthonormal in this paper we construct a self adjoint operator in this space generated by the second order laguerre differential expression l k having l n k n 0 as a complete set of eigenfunctions the key to this construction is in identifying a certain closed subspace of w k 0 with the kth left definite vector space v k k

algebraic and geometric isomonodromic deformations using the gauss manin connection picard fuchs differential equation and a result of malgrange a special class of algebraic solutions to isomonodromic deformation equations the geometric isomonodromic deformations i s defined from families of families of algebraic varieties geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over p1 the complete list of geometric

new ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two dimensional systems new ladder operators are constructed for a rational extension of the harmonic oscillator associated with type iii hermite exceptional orthogonal polynomials and characterized by an even integer m the eigenstates of the hamiltonian separate into m 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebra these ladder operators are used to construct a higher order integral of motion for two superintegrable two dimensional systems separable in cartesian coordinates the polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite dimensional unitary irreducible representations

orthogonality of the jacobi polynomials with negative integer parameters it is well known that the jacobi polynomials pn x are orthogonal with respect to a quasi definite linear functional whenever and 1 are not negative integer numbers recently sobolev orthogonality for these polynomials has been obtained for a negative integer and not a negative integer and also for the case negative integer numbers in this paper we give a sobolev orthogonality for the jacobi polynomials in the remainder cases

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

integrability of the one dimensional schrodinger equation we present a definition of integrability for the one dimensional schrodinger equation which encompasses all known integrable systems i e systems for which the spectrum can be explicitly computed for this we introduce the class of rigid functions built as liouvillian functions but containing all solutions of rigid differential operators in the sense of katz and a notion of natural of boundary conditions we then make a complete classification of rational integrable potentials many new integrable cases are found some of them physically interesting

probabilistic enhancement of classical robustness margins the unirectangularity concept abstract the focal point of this paper is a control system subjected to parametric uncertainty motivated by the newly emerging theory of probabilistic robustness the risk of performance violation is assessed with uncertainty bounds which exceed classical deterministic margins for a wide class of problems the uniformity principle up developed by barmish and lagoa math control signals systems 10 1997 203 222 makes it possible to estimate the probability of performance satisfaction with almost no a priori statistical information about the uncertainty the application of the up is however limited to problems satisfying certain convexity and symmetricity conditions since such conditions are violated in many practical problems the objective in this paper is to extend the application of the up to this end by working with a so called unirectangularity condition a procedure is implemented for computing probabilities of performance and the associated improvements of deterministic robustness margins that is given any robustness radius r 0 which is computable via deterministic methods a probabilistic enhancement of this margin r 0 e r 0 with pre specified level of risk e 0 is provided the radius r 0 e is called a risk adjusted robustness margin

unified derivation of exact solutions for a class of quasi exactly solvable models we present a unified treatment of exact solutions for a class of four quantum mechanical models namely the anharmonic singular potential the generalized quantum isotonic oscillator the soft core coulomb potential and the non polynomially modified oscillator we show that all four cases are reducible to the same basic ordinary differential equation which is quasi exactly solvable a systematic and closed form solution to the basic equation is obtained via the bethe ansatz method using the result general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations a hidden sl 2 algebraic structure is also discovered in these models

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

monotonic relaxations for robust control new characterizations parameterized linear matrix inequalities plmis that is lmis depending on a parameter confined to a compact set frequently arise in both analysis and synthesis problems of robust control as a major difficulty plmis are equivalent to an infinite family of lmi constraints and consequently are very hard to solve numerically known approaches to find solutions exploit relaxations inferred from convexity arguments these relaxations involve a finite family of lmis the number of which grows exponentially with the number of scalar parameters in this note we propose a novel approach based on monotonicity concept which allows us to solve plmis via a finite and of polynomial order family of lmis the effectiveness and viability of our approach are demonstrated by numerical examples such as robust stability analysis and linear parameter varying lpv synthesis for which we clearly show that no additional conservatism is entailed as compared to earlier techniques

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

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

stochastic robustness of linear time invariant control systems a simple numerical procedure for estimating the stochastic robustness of a linear time invariant system is described monte carlo evaluation of the system s eigenvalues allows the probability of instability and the related stochastic root locus to be estimated this analysis approach treats not only gaussian parameter uncertainties but also nongaussian cases including uncertain but bound variations confidence intervals for the scalar probability of instability address computational issues inherent in monte carlo simulation trivial extensions of the procedure admit consideration of alternate discriminants thus the probabilities that stipulated degrees of instability will be exceeded or that closed loop roots will leave desirable regions can also be estimated results are particularly amenable to graphical presentation

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

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

time dependent potentials associated with exceptional orthogonal polynomials we introduce a method for generating rational extensions of time dependent potentials such that the associated schrodinger equation admits solutions in terms of exceptional orthogonal polynomials our method is applicable to position dependent schrodinger equations as well as to their conventional counterparts for constant mass

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

on avoiding vertexization of robustness problems the approximate feasibility concept for a large class of robustness problems with uncertain parameter vector q confined to a box q there are many papers providing results along the following lines the desired performance specification is robustly satisfied for all q spl isin q if and only if it is satisfied at each vertex q sup i of q since the number of vertices of q explodes combinatorically with the dimension of q the computation associated with the implementation of such results is often intractable the main point of this paper is to introduce a new approach to such problems aimed at alleviation of this computational complexity problem to this end the notion of approximate feasibility is introduced and the theory which follows from this definition is vertex free

quantization of quadratic lienard type equations by preserving noether symmetries abstract the classical quantization of a family of a quadratic lienard type equation lienard ii equation is achieved by a quantization scheme nucci 2011 28 that preserves the noether point symmetries of the underlying lagrangian in order to construct the schrodinger equation this method straightforwardly yields the schrodinger equation as given in choudhury and guha 2013 6

a class of exactly solvable rationally extended non central potentials in two and three dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions

factorization solution of a family of quantum nonlinear oscillators in a recent paper carinena j f perelomov a m ranada m f and santander m 2008 j phys a math gen 41 085301 analyzed a non polynomial one dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators in particular they proved that it is schrodinger soluble and explicitly obtained the wavefunctions and energies of the bound states in this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator we then use this observation to generate an infinite set of potentials which can exactly be solved in a similar manner

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

solvable rational extensions of the morse and kepler coulomb potentials we show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential this is made by using darboux backlund transformations based on unphysical regular riccati schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials

a generalization of a theorem of bochner we describe all polynomial solutions to the general second order operator equation in the askey wilson operators thus extending a theorem of bochner from differential operators to askey wilson divided difference operators

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

distributionally robust control of constrained stochastic systems we investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process i e when only the first two moments of the disturbance distribution are known we consider two types of distributionally robust constraints in the first case we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments these constraints are commonly referred to as distributionally robust chance constraints in the second case we impose conditional value at risk cvar constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information such constraints are referred to as distributionally robust cvar constraints with second order moment specifications we propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems we prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust cvar and chance constrained problems the proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives

invariance properties of wronskian type determinants of classical and classical discrete orthogonal polynomials abstract given a finite set f f 1 f k of nonnegative integers written in increasing order of magnitude and a classical discrete family p n n of orthogonal polynomials charlier meixner krawtchouk or hahn we consider the casorati determinant det p f i x j 1 i j 1 k in this paper we prove an invariance property of this kind of casorati determinants when the set f is substituted by the set i f 0 1 2 max f max f f f f our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators krall discrete polynomials these polynomials are orthogonal with respect to certain christoffel transforms of the classical discrete measures by passing to the limit this invariance property is extended to wronskian type determinants whose entries are hermite laguerre and jacobi polynomials

probabilistic enhancement of classical robustness margins a class of nonsymmetric distributions in this note we address the problem of risk assessment when the robustness margin is exceeded without a priori knowledge of the distribution of the uncertainty the only assumption is that the distribution belongs to a given class in contrast to previous work this class contains both symmetric and nonsymmetric distributions we prove that the assessment of risk can be done using only a subset of the admissible distributions also if the set of uncertainties that verify the specifications is convex it is proven that risk assessment can be done using only a finite subset of the class finally a way of estimating risk is provided for the nonconvex case

a new potential with the spectrum of an isotonic oscillator with the method of factorisation a new potential with free parameters is generated this potential possesses exactly the same energy spectrum as the isotonic oscillator harmonic oscillator with centripetal barrier the wavefunctions and shift operators of the new system are also presented

new quasi exactly solvable sextic polynomial potentials a hamiltonian is said to be quasi exactly solvable qes if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in a closed form an entirely new class of qes hamiltonians having sextic polynomial potentials is constructed these new hamiltonians are different from the sextic qes hamiltonians in the literature because their eigenfunctions obey symmetry rather than hermitian boundary conditions these new hamiltonians present a novel problem that is not encountered when the hamiltonian is hermitian it is necessary to distinguish between the parametric region of unbroken symmetry in which all of the eigenvalues are real and the region of broken symmetry in which some of the eigenvalues are complex the precise location of the boundary between these two regions is determined numerically using the extrapolation techniques and analytically using the wkb analysis

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

bi hamiltonian structure of the bi dimensional superintegrable nonlinear isotonic oscillator the higher order superintegrability of the two dimensional isotonic oscillator noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator with rational ratio of frequencies is directly related with the existence of some complex functions with interesting poisson bracket properties first the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi hamiltonian complex structure in the second part several real symplectic structures are obtained and the properties of the recursion operators are studied

on the solutions of the position dependent effective mass schrodinger equation of a nonlinear oscillator related with the isotonic oscillator we study exact solutions of the position dependent effective mass schrodinger equation by considering the new solvable nonlinear oscillator that relates to the isotonic oscillator through the method of point canonical transformations we construct exactly solvable potentials by choosing physically important position dependent mass distributions we also provide the energy spectrum of the bound states and the wavefunctions of the solvable potentials

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them