parametric symmetries in exactly solvable real and pt symmetric complex potentials in this paper we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex pt symmetric potentials we focus our attention on the conventional potentials such as the generalized poschl teller gpt scarf i and pt symmetric scarf ii which are invariant under certain parametric transformations the resulting set of potentials is shown to yield a completely different behavior of the bound state solutions further the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and pt symmetric complex potentials these potentials are also observed to be shape invariant si in nature we subsequently take up a study of the newly discovered rationally extended si potentials corresponding to the above mentioned conventional potentials whose bound state solutions are associated with the exceptional orthogonal polynomials eops we discuss the transformations of the corresponding

eigenstate structure in graphs and disordered lattices we study wave function structure for quantum graphs in the chaotic and disordered regime using measures such as the wave function intensity distribution and the inverse participation ratio the result is much less ergodicity than expected from random matrix theory even though the spectral statistics are in agreement with random matrix predictions instead analytical calculations based on short time semiclassical behavior correctly describe the eigenstate structure

quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 as observables we take the l 2 functions on the interval the proof is based on the periodic orbit expansion of a majorant of the quantum variance specifically given a one dimensional lebesgue measure preserving map of an interval we consider an increasingly refined sequence of partitions of the interval to this sequence we associate a sequence of graphs whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

infinitely many shape invariant potentials and cubic identities of the laguerre and jacobi polynomials we provide analytic proofs for the shape invariance of the recently discovered odake and sasaki phys lett b 679 414 2009 two families of infinitely many exactly solvable one dimensional quantum mechanical potentials these potentials are obtained by deforming the well known radial oscillator potential or the darboux poschl teller potential by a degree l l 1 2 eigenpolynomial the shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the laguerre or jacobi polynomials these identities are proved elementarily by combining simple identities

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

the local time distribution of a particle diffusing on a graph we study the local time distribution of a brownian particle diffusing along the links on a graph in particular we derive an analytic expression of its laplace transform in terms of the green function on the graph we show that the asymptotic behaviour of this distribution has non gaussian tails characterized by a nontrivial large deviation function

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

series with general exponents abstract we define the artinian and noetherian algebra which consist of formal series involving exponents which are not necessarily integers all of the usual operations are defined here and characterized as an application we compute the algebra of symmetric functions with nonnegative real exponents the applications to logarithmic series and the umbral calculus are deferred to another paper 9

localized eigenfunctions in seba billiards we describe some new families of quasimodes for the laplacian perturbed by the addition of a potential formally described by a dirac delta function as an application we find under some additional hypotheses on the spectrum subsequences of eigenfunctions of seba billiards that localize around a pair of unperturbed eigenfunctions

the classical umbral calculus and the flow of a drinfeld module the notion of a flow appears in many areas of mathematics and physics for example it is one of the fundamental notions in studying ordinary differential equations classically a flow on a set x is a group action of the additive group of the set of real numbers r on the set x one of the classical theorems in analysis the classical taylor theorem can be expressed in terms of a flow more precisely let d dt be the formal derivative acting on the polynomial ring r t for each x r let e r t r t be the linear operator defined by

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

scattering theory on graphs we consider the scattering theory for the schrodinger operator dx2 v x on the graphs made of one dimensional wires connected to external leads we derive two expressions for the scattering matrix on arbitrary graphs one involves matrices that couple arcs oriented bonds the other involves matrices that couple vertices we discuss a simple way to tune the coupling between the graph and the leads the efficiency of the formalism is demonstrated with a few known examples

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

visibility of quantum graph spectrum from the vertices we investigate the relation between the eigenvalues of the laplacian with kirchhoff vertex conditions on a finite metric graph and a corresponding titchmarsh weyl function a parameter dependent ne

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

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

occupation times distribution for brownian motion on graphs considering a brownian motion on a general graph we study the joint law for the occupation times on all the bonds in particular we show that the laplace transform of this distribution can be expressed as the ratio of two determinants we give two formulations with arc or vertex matrices for this result and discuss a simple example

umbral calculus and special functions developpement du calcul symbolique des puissances et de la methode des coalgebres et des comodules associee application a l etude des fonctions speciales classiques hypergeometriques de laguerre sonine d hermite weber de legendre et de bessel

higher order susy linearized nonlinear heisenberg algebras and coherent states using an iterative construction of the first order intertwining technique we find k parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below e0 it is seen that the natural ladder operators for these systems give place to polynomial nonlinear algebras and it is shown that these algebras can be linearized the coherent states construction is performed in the nonlinear and linearized cases

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

algebraic treatment of second poschl teller morse rosen and eckart equations the method of an earlier paper see ibid vol 20 p 4075 1987 is applied to the non compact case to solve a family of second poschl teller morse rosen and eckart equations with quantised coupling constants both discrete and continuous spectra bound state and scattering wavefunctions transmission coefficients are found from the matrix elements of group representations

quantum ergodicity on regular graphs we give three different proofs of the main result of anantharaman and le masson duke math j 164 4 723 765 2015 establishing quantum ergodicity a form of delocalization for eigenfunctions of the laplacian on large regular graphs of fixed degree these three proofs are much shorter than the original one quite different from one another and we feel that each of the four proofs sheds a different light on the problem the goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics such as the anderson model on large regular graphs regular graphs with weighted edges or possibly certain models of non regular graphs a source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs

value distribution of the eigenfunctions and spectral determinants of quantum star graphs we compute the value distributions of the eigenfunctions and spectral determinant of the schrodinger operator on families of star graphs the values of the spectral determinant are shown to have a cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related this is in contrast to the spectral determinants of random matrices for which the logarithm is known to satisfy a gaussian limit distribution the value distribution of the eigenfunctions also differs from the corresponding random matrix result we argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of seba type billiards

continuous sheffer families ii abstract continuous sheffer families have been recently introduced by the authors these are continuous versions of the sheffer sequences arising in the umbral calculus we show here that quite a number of classical special functions are examples of such families

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

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

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

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

the potential group approach and hypergeometric differential equations this paper proposes a generalized realization of the potential groups so 2 1 and so 2 2 to describe the confluent hypergeometric and the hypergeometric equations respectively it implies that the classes of schrodinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure

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

universal spectral statistics in quantum graphs we prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations as predicted by random matrix theory the stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

potentials generated by su 1 1 a systematic procedure for deriving a class of potentials with the underlying symmetry group su 1 1 starting the commutation relations for the generators of su 1 1 is presented

a new realisation of dynamical groups and factorisation method a new method of algebraisation of quantum mechanical eigenvalue equations is presented in this method the dynamical algebra is represented on the space of group matrix elements the ladder operators of the dynamical algebra are obtained from infeld hull miller factorisations the method is used to study the first poschl teller equation even in the non symmetric case the energy spectrum and the exact normalised solutions are obtained in agreement with the results of non algebraic methods

the interated logarithmic algebra abstract we generalize the umbral calculus of g c rota adv in math 27 1978 95 188 by studying not only sequences of polynomials and inverse power series or even the logarithms studied by d loeb and g c rota adv in math 75 1989 1 118 but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power using a theory of formal power series with real exponents and a more general definition of factorial binomial coefficient and stirling numbers to all the real numbers we define the iterated logarithmic algebra j its elements are the formal representations of the asymptotic expansions of a large class of real functions and we define the harmonic logarithm basis of j which will be interpreted as a generalization of the powers xn since it behaves nicely with respect to the derivative we classify all operators over j which commute with the derivative classically there are known as shift invariant operators and formulate several equivalent definitions of a sequence of binomial type we then derive many formulas useful towards the calculation of these sequences including the recurrence formula the transfer formula and the lagrange inversion formula in the sequel 22 we study sheffer sequences and give many examples

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

orbit bifurcations and the scarring of wave functions we extend the semiclassical theory of scarring of quantum eigenfunctions n q by classical periodic orbits to include situations where these orbits undergo generic bifurcations it is shown that n q 2 averaged locally with respect to position q and the energy spectrum en has structure around bifurcating periodic orbits with an amplitude and length scale whose dependence is determined by the bifurcation in question specifically the amplitude scales as and the length scale as and values of the scar exponents and are computed for a variety of generic bifurcations in each case the scars are semiclassically wider than those associated with isolated and unstable periodic orbits moreover their amplitude is at least as large and in most cases larger in this sense bifurcations may be said to give rise to superscars the competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed we argue that there is a resulting universal scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase space dynamics finally a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps

families of line graphs and their quantization any directed graph g with n vertices and j edges has an associated line graph l g where the j edges form the vertices of l g we show that the non zero eigenvalues of the adjacency matrices are the same for all graphs of such a family ln g we give necessary and sufficient conditions for a line graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions line graphs may therefore serve as models to study the semiclassical limit of large matrix size of a quantum dynamics on graphs with fixed classical behaviour

polynomial heisenberg algebras polynomial deformations of the heisenberg algebra are studied in detail some of their natural realizations are given by the higher order susy partners and not only by those of first order as is already known of the harmonic oscillator for even order polynomials here it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd moreover it will be proved that the general systems ruled by such kinds of algebras in the quadratic and cubic cases involve painleve transcendents of types iv and v respectively

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

an exact periodic orbit formula for the energy levels of the three pronged star graph we derive an exact periodic orbit expansion formula for the energy levels of the three pronged star graph in addition we present a proof of the one root per root cell property and derive exact periodic orbit sum rules a spin off of our periodic orbit formulae is novel arithmetic series for 2

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