krein adler transformations for shape invariant potentials and pseudo virtual states for 11 examples of one dimensional quantum mechanics with shape invariant potentials the darboux crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to krein adler transformations deleting multiple eigenstates with shifted parameters these are based upon infinitely many polynomial wronskian identities of classical orthogonal polynomials i e the hermite laguerre and jacobi polynomials which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape invariant potentials

uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

density classification performance and ergodicity of the gacs kurdyumov levin cellular automaton model iv almost four decades ago gacs kurdyumov and levin introduced three different cellular automata to investigate whether one dimensional nonequilibrium interacting particle systems are capable of displaying phase transitions and as a byproduct they introduced the density classification problem the ability to classify arrays of symbols according to their initial density in the cellular automata literature their model ii became a well known model in theoretical computer science and statistical mechanics the other two models however did not receive much attention here we characterize the density classification performance of gacs kurdyumov and levin s model iv a four state cellular automaton with three absorbing states only two of which are attractive by numerical simulations we show that model iv compares well with its sibling model ii in the density classification task the additional states slow down the convergence to the majority state but confer a slight advantage in classification performance we also show that unexpectedly initial states diluted in one of the nonclassifiable states are more easily classified the performance of model iv under the influence of noise was also investigated and we found signs of an ergodic nonergodic phase transition at some small finite positive level of noise although the evidence is not entirely conclusive we set an upper bound on the critical point for the transition if any

supersymmetry and algebraic darboux transformations we describe a class of algebraically solvable susy models by considering the deformation of invariant polynomial flags by means of the darboux transformation the algebraic deformations corresponding to the addition of a bound state to a shape invariant potential are particularly interesting the polynomial flags in question are indexed by a deformation parameter m 1 2 and lead to new algebraically solvable models we illustrate these ideas by considering deformations of the hyperbolic p schl teller potential

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

uniqueness of positive solutions to some nonlinear neumann problems abstract using the moving plane method we obtain a liouville type theorem for nonnegative solutions of the neumann problem div y a u x y 0 x r n y 0 lim y 0 y a u y x y f u x 0 x r n under general nonlinearity assumptions on the function f r r for any constant a 1 1

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

a direct method of moving spheres on fractional order equations abstract in this paper we introduce a direct method of moving spheres for the fractional laplacian 2 with 0 2 in which a key ingredient is the narrow region maximum principle as immediate applications we classify non negative solutions for semilinear equations involving the fractional laplacian in r n we prove a non existence result for the prescribing q curvature equation on s n then by combining the direct method of moving planes and moving spheres we establish a liouville type theorem on a half euclidean space we expect to see more applications of this method to many other nonlinear equations involving non local operators

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

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

extensions of solvable potentials with finitely many discrete eigenstates we address the problem of rational extensions of six examples of shape invariant potentials having finitely many discrete eigenstates the overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions which are an essential ingredient for multi indexed and iso spectral extensions of these potentials they have exactly the same form as the eigenfunctions but their degrees are much higher than nmax so that their energies are lower than the groundstate energy

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

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

the cauchy process and the steklov problem let xt be a cauchy process in rd d 1 we investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain d we establish a connection between the semigroup of this process and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem using this we derive a variational characterization for the eigenvalues of the cauchy process in d this characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the cauchy process inspired by those for brownian motion our results are new even in the simplest geometric setting of the interval 1 1 where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions such results although trivial for the laplacian take considerable work to prove for the cauchy processes and remain open for general symmetric stable processes along the way we present other general properties of the eigenfunctions such as real analyticity which even though well known in the case of the laplacian are not available for more general symmetric stable processes

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

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

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

two novel classes of solvable many body problems of goldfish type with constraints two novel classes of many body models with nonlinear interactions of goldfish type are introduced they are solvable provided the initial data satisfy a single constraint in one case in the other two constraints i e for such initial data the solution of their initial value problem can be achieved via algebraic operations such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials entirely isochronous versions of some of these models are also exhibited i e versions of these models whose nonsingular solutions are all completely periodic with the same period

information entropy of conditionally exactly solvable potentials we evaluate shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials the bialynicki birula mycielski inequality has also been tested for a number of states

new algebraic quantum many body problems we develop a systematic procedure for constructing quantum many body problems whose spectrum can be partially or totally computed by purely algebraic means the exactly solvable models include rational and hyperbolic potentials related to root systems in some cases with an additional external field the quasi exactly solvable models can be considered as deformations of the previous ones which share their algebraic character

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

two step rational extensions of the harmonic oscillator exceptional orthogonal polynomials and ladder operators the type iii hermite xm exceptional orthogonal polynomial family is generalized to a double indexed one with m1 even and m2 odd such that m2 m1 and the corresponding rational extensions of the harmonic oscillator are constructed by using second order supersymmetric quantum mechanics the new polynomials are proved to be expressible in terms of mixed products of hermite and pseudo hermite ones while some of the associated potentials are linked with rational solutions of the painlev iv equation a novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial heisenberg algebra of order m2 m1 1 which may alternatively be interpreted in terms of a special type of m2 m1 2 th order shape invariance property

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

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

extended krein adler theorem for the translationally shape invariant potentials considering successive extensions of primary translationally shape invariant potentials we enlarge the krein adler theorem to mixed chains of state adding and state deleting darboux backlund transformations it allows us to establish novel bi linear wronskian and determinantal identities for classical orthogonal polynomials

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

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

supersymmetric transformations of real potentials on the line a systematic study of supersymmetric or darboux factorizations on the line is performed all possible pairs of supersymmetric transformations with the same factorization energy are reviewed for such pairs there is no condition on this energy iterations of single transformations and of pairs both allow arbitrary modifications of the bound spectrum different iterative methods lead to compact analytical equations depending on the initial potential and its solutions iterations of single transformations are able to transform an even potential into an even potential particular cases of a method based on iteration of pairs conserve either the reflection coefficient at all energies or the norming constants of bound states these results are compared with previous methods established in other contexts the most general supersymmetric transformation of a given potential is finally described both with and without modifications of the bound spectrum

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

the darboux transformation and algebraic deformations of shape invariant potentials we investigate the backward darboux transformations addition of the lowest bound state of shape invariant potentials on the line and classify the subclass of algebraic deformations those for which the potential and the bound states are simple elementary functions a countable family m 0 1 2 of deformations exists for each family of shape invariant potentials we prove that the m th deformation is exactly solvable by polynomials meaning that it leaves invariant an infinite flag of polynomial modules p m m p m 1 m where p n m is a codimension m subspace of in particular we prove that the first m 1 algebraic deformation of the shape invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules p n 1 by construction these algebraically deformed hamiltonians do not have an sl 2 hidden symmetry algebra structure

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

from pca s to equilibrium systems and back stationary measures for probabilistic cellular automata pca s ind dimensions give rise to space time histories whose statistics may naturally be described by gibbs states ind 1 dimensions for an interaction energy obtained from the pca in this note we study the converse question do all gibbs states for this correspond to statistical space time histories for the pca our main result states that the answer is yes at least for translation invariant or periodic gibbs states thus ergodicity questions for pca s can at least partially be formulated as questions of uniqueness of gibbs states

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

dissipation anisotropy and the stabilization of computationally complex states of homogeneous media abstract the unbounded complexity exhibited by homogeneous media such as cellular automata is ordinarily quite unstable with respect to noise with an arbitrarily small generic probabilistic perturbation causing the medium instead to relax to a unique finitely complicated state independent of the initial conditions we survey recent results on how irreversibility and spatial anisotropy allow unboundedly complex behavior to persist in spite of such noise

reply to comment on the darboux transformation and algebraic deformations of shape invariant potentials we reply to the comment on our recent paper made by drs sinha and roy we agree that the backward darboux transformation method used in our paper is equivalent to the approach based on ces potentials but we stress that the emphasis and the results of our paper are different