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

confluent hypergeometric equations and related solvable potentials in quantum mechanics the connection between the schrodinger and confluent hypergeometric equations is discussed it is shown that the factorization of the confluent hypergeometric equation gives a unifying powerful algebraic tool in order to study some quantum mechanical eigenvalue problems that description includes the linear and n dimensional harmonic oscillators as well as the coulomb and morse potentials

on the schrodinger radial ladder operator using the radial ladder operators a l and a l defined in solving the hydrogen atom radial equation with the factorisation method the author develops a new device by which the calculations on the average value of rs the inner product of the radial wavefunctions and the matrix element of r are simplified and by which some new recurrence relations are derived

so 3 algebraic approach to the morse potential we construct so 3 algebra associated with the morse potential and show that these operators obey so 3 commutation relations a so 3 algebraic method is proposed in order to obtain the eigenvalues and eigenfunctions of the morse potential this method exhibits that cartan operator jz the lowering operator j and the raising operator j determine successfully energy eigenvalues the lowest energy eigenfunction and excited energy eigenfunctions respectively

a simple algebraic approach to coherent states for the morse oscillator minimum uncertainty coherent states and annihilation operator coherent states for the morse oscillator are derived and shown to be equivalent they reduce in the limit of small anharmonicity constant or equivalently in the limit of large well depth to the approximate coherent states derived previously from the use of generalized displacement operator

on constraint preservation in numerical simulations of yang mills equations the problem of constraint preservation for discretizations of nonlinear pdes is addressed in the example of the hyperbolic yang mills equations in temporal gauge these equations preserve a nonlinear divergence field analogous to the electric charge for maxwell s equations we introduce and discuss several discretizations of these equations on finite element spaces of lie algebra valued differential forms numerical experiments indicate that simply restricting the variational formulation to the galerkin spaces yields substantial drift in the charge contrary to the linear maxwell case we then propose a fully discrete method constrained with lagrange multipliers for which we prove discrete charge conservation and observe excellent energy conservation

limit laws of erdos renyi shepp type on obtient la forme complete des theoremes d erdos renyi 1970 et shepp 1964 les conditions etendent celles de deheuvels devroye et lynch 1986 pour couvrir une classe de distributions plus grande

isospectral problem interplay between liouville equations darboux transforms and mckean trubowitz flows systems of liouville equations are generated by isospectral transforms of the schrodinger operator and describe the isospectral evolution of both a potential and eigenfunctions relations of these equations with the mckean trubowitz isospectral flows are studied the role of these flows among other isospectral flows is discussed

exact convergence rate in the limit theorems of erdos renyi and shepp on considere le comportement asymptotique de u n sup o i n k s i k s i et on montre que le terme o k 1 2 peut etre remplace par o k 1 log k

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

an integrated approach to ladder and shift operators for the morse oscillator radial coulomb and radial oscillator potentials the morse oscillator radial coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods these problems correspond to different realizations of the so 2 1 algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems the resultant transition operators act as ladder or energy changing operators in the cases of the coulomb and harmonic oscillator potentials whereas they act as shift operators acting at constant energy in the case of the morse potential this is a consequence of the so 2 1 dynamical symmetry whereby the morse hamiltonian is expressible solely in terms of the casimir operator of the algebra an alternative algebraic approach the use of the method of supersymmetric quantum mechanics or factorization produces in each case a set of shift operators relations between the various ladder and shift operators may be identified by means of the appropriate mappings and these results can be generalized so as to relate the one dimensional morse oscillator to the radial coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions

unified treatment of the coulomb and harmonic oscillator potentials in d dimensions quantum mechanical models and practical calculations often rely on some exactly solvable models like the coulomb and the harmonic oscillator potentials the d dimensional generalized coulomb potential contains these potentials as limiting cases thus it establishes a continuous link between the coulomb and harmonic oscillator potentials in various dimensions we present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations we define a hilbert space basis the generalized coulomb sturmian basis and calculate the green s operator on this basis and also present an su 1 1 algebra associated with it we formulate the problem for the one dimensional case too and point out that the complications arising due to the singularity of the one dimensional coulomb problem can be avoided with the use of the generalized coulomb potential

simplicial gauge theory on spacetime we define a discrete gauge invariant yang mills higgs action on spacetime cylindrical meshes with simplicial spatial base the formulation is a generalization of classical lattice gauge theory and we prove consistency of the action in the finite element sense in addition we perform numerical tests of convergence towards exact continuum results for several choices of gauge fields in pure gauge theory

the su 1 1 dynamical algebra from the schrodinger ladder operators for n dimensional systems hydrogen atom mie type potential harmonic oscillator and pseudo harmonic oscillator we apply the schrodinger factorization to construct the ladder operators for the hydrogen atom mie type potential harmonic oscillator and pseudo harmonic oscillator in arbitrary dimensions by generalizing these operators we show that the dynamical algebra for these problems is the su 1 1 lie algebra

algebraic evaluation of matrix elements in the laguerre function basis the laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron structure theory with applications in hadronic and nuclear theory as well while similar in form to the coulomb bound state eigenfunctions from the schrodinger eigenproblem or the coulomb sturmian functions from a related sturm liouville problem the laguerre functions unlike these former functions constitute a complete discrete orthonormal set for square integrable functions in three dimensions we construct the su 1 1 so 3 dynamical algebra for the laguerre functions and apply the ideas of factorization or supersymmetric quantum mechanics to derive shift operators for these functions we use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators involving powers of the radial coordinate and radial derivative in the laguerre function basis we illustrate how matrix elements for more general spherical tensor operators in three dimensional space such as the gradient may then be constructed from these radial matrix elements

algebraic treatment of super integrable potentials the so 2 1 lie algebra is applied to three classes of two and three dimensional smorodinsky winternitz super integrable potentials for which the path integral discussion has been recently presented in the literature we have constructed the green s functions for two important super integrable potentials in r2 among the super integrable potentials in r3 we have considered two examples one is maximally super integrable and another one minimally super integrable the discussion is made in various coordinate systems the energy spectrum and the suitably normalized wave functions of bound and continuous states are then deduced

q deformed ladder and shift operators for three exactly solvable potentials obeying so 2 1 symmetry the quantum deformation algebra soq 2 1 is studied and applied to derive the q analogues of the ladder and shift operators for the radial coulomb radial harmonic oscillator and morse oscillator potentials the q deformed operators in all three cases are found to act like shift operators called q shift operators their possible similarity with the quasi shift operators arising in supersymmetric quantum mechanics or factorization of the radial harmonic oscillator is also pointed out

ladder operators for subtle hidden shape invariant potentials ladder operators can be constructed for all potentials that present the integrability condition known as shape invariance satisfied by most of the exactly solvable potentials using the superalgebra of supersymmetric quantum mechanics we construct the ladder operators for two exactly solvable potentials that present a subtle hidden shape invariance

an su 1 1 dynamical algebra for the poschl teller potential an su 1 1 dynamical algebra to describe the continuum part of the spectrum for the modified poschl teller potential is proposed the space associated with this algebra is given in terms of a family of orthonormal functions n characterized by the parameter this set is constructed from polynomials which are orthogonal with respect to a weighting function corresponding to a poschl teller ground state an analysis of the associated algebra is investigated in detail the functions are identified with poschl teller like functions associated with different potential depths we prove that by choosing appropriately the parameter it is possible to decouple the discrete states of a given parity from the continuum part of the spectrum a discussion of the matrix elements of the dipole function operator is also included

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

exact convergence rates in strong approximation laws for large increments of partial sums consider partial sumss n of an i i d sequencex1x2 of centered random variables having a finite moment generating function in a neighborhood of zero the asymptotic behaviour of u n mathop max limits 0 leqq k leqq n b n s k b n s k is investigated where 1 b n n denotes an integer sequence such thatb n logn asn in particular ifb n o log p n asn for somep 1 the exact convergence rate ofu n b n n 1 0 1 is determined where n depends uponb n and the distribution ofx1 in addition a weak limit law foru n is derived finally it is shown how strong invariance takes over if mathop lim limits n to infty b n loglogn 2 log3n

coherent states for isospectral oscillator hamiltonians coherent states for a family of isospectral oscillator hamiltonians are derived from a suitable choice of annihilation and creation operators the fock bargmann representation is also considered

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

improved erdos renyi and strong approximation laws for increments of partial sums let x 1 x 2 cdots be an i i d sequence with ex 1 0 ex 2 1 1 ee tx 1 infty t t 0 and partial sums s n x 1 cdots x n starting from some analogous results for the wiener process this paper studies the almost sure limiting behaviour of max 0 leq n leq n a n a 1 2 n s n a n s n as n rightarrow infty under various conditions on the integer sequence a n improvements of the erdos renyi law of large numbers for partial sums are obtained as well as strong invariance principle type versions via the komlos major tusnady approximation an appearing gap between these two results is also going to be closed

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

an su 1 1 dynamical algebra for the morse potential an su 1 1 dynamical algebra to describe both the discrete and the continuum parts of the spectrum for the morse potential is proposed the space associated with this algebra is given in terms of a family of orthonormal functions n characterized by the parameter this set is constructed from polynomials which are orthogonal with respect to a weighting function related to a morse ground state an analysis of the associated algebra is investigated in detail the functions are identified with morse like functions associated with different potential depths we prove that for a particular choice of the discrete and the continuum parts of the spectrum decouple the connection of this treatment with the supersymmetric quantum mechanics approach is established a closed expression for the mecke dipole moment function is obtained

single and coupled channel radial inverse scattering with supersymmetric transformations the present status of the three dimensional inverse scattering method with supersymmetric transformations is reviewed for the coupled channel case we first revisit in a pedagogical way the single channel case where the supersymmetric approach is shown to provide a complete efficient and elegant solution to the inverse scattering problem for the radial schrodinger equation with short range interactions a special emphasis is put on the differences between conservative and non conservative transformations i e transformations that do or do not conserve the behaviour of solutions of the radial schrodinger equation at the origin in particular we show that for the zero initial potential a non conservative transformation is always equivalent to a pair of conservative transformations these single channel results are illustrated on the inversion of the neutron proton triplet eigenphase shifts for the s and d waves we then summarize and extend our previous works on the coupled channel case i e on systems of coupled radial schrodinger equations and stress remaining difficulties and open questions of this problem by putting it in perspective with the single channel case we mostly concentrate on two channel examples to illustrate general principles while keeping mathematics as simple as possible in particular we discuss the important difference between the equal threshold and different threshold problems for equal thresholds conservative transformations can provide non diagonal jost and scattering matrices iterations of such transformations in the two channel case are studied and shown to lead to practical algorithms for inversion a convenient particular technique where the mixing parameter can be fitted without modifying the eigenphases is developed with iterations of pairs of conjugate transformations this technique is applied to the neutron proton triplet s d scattering matrix for which exactly solvable matrix potential models are constructed for different thresholds conservative transformations do not seem to be able to provide a non trivial coupling between channels in contrast a single non conservative transformation can generate coupled channel potentials starting from the zero potential and is a promising first step towards a full solution to the coupled channel inverse problem with threshold differences

matrix elements of u and p for the modified poschl teller potential closed analytical expressions in terms of a single sum are obtained for the matrix elements of the momentum and the natural variable u tanh x in the basis of the modified poschl teller mpt bound eigenstates these matrix elements are first expressed in terms of franck condon factors which thereafter are substituted for analytic expressions expansions of the variables p and u in terms of creation and annihilation operators associated with the mpt bound eigenfunctions are also presented

vector coherent states with matrix moment problems canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer m the requirement that these states realize a resolution of the identity typically results in a moment problem where the moments form the positive sequence of real numbers m m 0 in this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments m of the canonical coherent states by n n matrices associated oscillator algebras are discussed with the aid of a generalized matrix factorial two physical examples are discussed in the first example coherent states are obtained for the jaynes cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states in the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator

dynamical potential algebras for gendenshtein and morse potentials a differential realization of so 2 1 is shown to be the potential algebra for the one dimensional systems with the morse or gendenshtein potentials this shows that two classes of gendenshtein potentials will support the same eigenvalues as the morse potential and that the three sets of eigenfunctions may be derived in a common formalism the potential algebra is then extended to a dynamical potential algebra with operators connecting states both in different potentials and with different energies giving new dynamical algebras for the gendenshtein problem the matrix elements of certain corresponding operators in the three types of system may then be given by a single formula involving so 2 1 wigner coefficients the authors also give ladder operators connecting the gendenshtein potential eigenstates

a gauge invariant discretization on simplicial grids of the schrodinger eigenvalue problem in an electromagnetic field we propose a method to compute approximate eigenpairs of the schrodinger operator on a bounded domain in the presence of an electromagnetic field formulated for simplicial meshes the method combines techniques from lattice gauge theory and finite elements retaining the discrete gauge invariance of the former but allowing for noncongruent space elements as in the latter the error of the method is studied in the framework of strang s variational crimes comparing with a standard galerkin approach when the meshes are quasi uniform and satisfy the discrete maximum principle we prove that for a smooth electromagnetic field the crime is of the order of the mesh width h for a coulomb potential it is of order h log h and for a general finite energy electromagnetic field it is of order h 1 2

a certain double well potential related to su 2 symmetry in the present paper we analyse a class of partially exactly solvable one dimensional double well potentials we show how the underlying su 2 dynamical symmetry makes it possible analytically to find several energy levels

erdős renyi laws for gibbs measures can one detect a phase transition from a single large sample of a gibbs measure what information does one get on the other gibbs distributions with the same potential we approach these questions via erd s renyi laws in particular we prove almost sure limit theorems for sets of empirical distributions of sub samples of the given one for suitable sub samples size this set converges to the set of stationary gibbs measures moreover we formulate erd s renyi laws for general families of random variables with suitable large deviation principles

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related