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

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

gradual spontaneous breakdown of mathcal pt symmetry in a solvable potential a new solvable potential is introduced by transforming the schrodinger equation into the differential equation of the jacobi polynomials the z x variable transformation function depends on two parameters c and and can be obtained implicitly as x z the remaining two parameters set the strength of the even and odd potential components systematic analysis is given for c 0 0 when the potential contains the scarf ii and rosen morse i potentials as special cases the normalized bound state solutions are obtained in closed form both in the real and symmetric cases the energy eigenvalues are determined from the roots of a quartic algebraic equation despite this implicitly defined spectrum the spectral structure of the potential can be determined easily in the symmetric case the spontaneous breakdown of symmetry can also be studied in a controllable way and it is found that with increasing non hermiticity the complexification of the energy eigenvalues is a gradual process that starts with the ground state the general form of the potential contains the scarf i rosen morse ii generalized poschl teller eckart symmetric ginocchio and dutt khare varshni dkv potentials too and can be identified as a subclass of the general six parameter natanzon class this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

construction of exact solutions to eigenvalue problems by the asymptotic iteration method we apply the asymptotic iteration method aim ciftci hall and saad 2003 j phys a math gen 36 11807 to solve new classes of second order homogeneous linear differential equation in particular solutions are found for a general class of eigenvalue problems which includes schrodinger problems with coulomb harmonic oscillator or poschl teller potentials as well as the special eigenproblems studied recently by bender et al 2001 j phys a math gen 34 9835 and generalized in the present paper to arbitrary dimension

on solvable potentials related to so 2 2 ii natanzon potentials general natanzon potentials related to the so 2 2 group are studied the s matrices for systems under consideration are related to intertwining operators between weyl equivalent most degenerate principal series representations of so 2 2

symmetrically regularized eckart poschl teller and hulthen potentials parity times time reversal symmetry of complex hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for hermiticity perhaps an equally important role is played by the related strengthened analyticity assumptions in a constructive illustration we complexify a few potentials solvable only in s wave then we continue their domain from the semi axis to the whole axis and obtain new exactly solvable models their energies turn out to be real as expected the new one dimensional spectra themselves differ quite significantly from their s wave predecessors

shape invariance raising and lowering operators in hypergeometric type equations the schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self adjoint operators defined in the theory of hypergeometric type equations the fundamental formulae occurring in a supersymmetric approach to these hamiltonians are consequences of some formulae concerning the general theory of associated special functions we use this connection in order to obtain a general theory of schrodinger equations exactly solvable in terms of associated special functions and to extend certain results known in the case of some particular potentials

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

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

an exactly solvable schrodinger equation with finite positive position dependent effective mass the solution of the one dimensional schrodinger equation is discussed in the case of position dependent mass the general formalism is specified for potentials that are solvable in terms of generalized laguerre polynomials and mass functions that are positive and bounded on the whole real x axis the resulting four parameter potential is shown to belong to the class of implicit potentials closed expressions are obtained for the bound state energies and the corresponding wave functions including their normalization constants the constant mass case is obtained by a specific choice of the parameters it is shown that this potential contains both the harmonic oscillator and the morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials possible generalizations of the method are outlined

the functional analytic versus the functional integral approach to quantum hamiltonians the one dimensional hydrogen atoma the capabilities of the functional analytic and of the functional integral approach for the construction of the hamiltonian as a self adjoint operator on hilbert space are compared in the context of non relativistic quantum mechanics differences are worked out by taking the one dimensional hydrogen atom as an example that is a point mass on the euclidean line subjected to the inverse distance potential this particular choice is made with the intent to clarify a long lasting discussion about its spectral properties in fact for the four parameter family of possible hamiltonians the corresponding energy dependent green functions are derived in closed form the multiplicity of hamiltonians should be kept in mind when modeling certain experimental situations as for instance in quantum wires

on the coulomb sturmian matrix elements of relativistic coulomb green s operators the hamiltonian of the radial coulomb klein gordon and second order dirac equations are shown to possess an infinite symmetric tridiagonal matrix structure on the relativistic coulomb sturmian basis this allows us to give an analytic representation for the corresponding coulomb green s operators in terms of continued fractions the poles of the green s matrix reproduce the exact relativistic hydrogen spectrum

the cal pt symmetric rosen morse ii potential effects of the asymptotically non vanishing imaginary potential component bound and scattering solutions of the symmetric rosen morse ii potential are investigated the energy eigenvalues and the corresponding wavefunctions are written in a closed analytic form and it is shown that this potential always supports at least one bound state it is found that with increasing non hermiticity the real bound state energy spectrum does not turn into complex conjugate pairs i e the spontaneous breakdown of symmetry does not occur rather the energy eigenvalues remain real and shift to positive values closed expression is found for the pseudo norm of the bound states and its sign is found to follow the 1 n rule similarly to the known scattering examples the reflection coefficients exhibit a handedness effect while the transmission coefficient picks up a complex phase factor when the direction of the incoming wave is reversed it is argued that the unusual findings might be caused by the asymptotically non vanishing though finite imaginary potential component comparison with the real rosen morse ii potential is also made

random growth of interfaces as a subordinated process we study the random growth of surfaces from within the perspective of a single column namely the fluctuation of the column height around the mean value y t identical with h t which is depicted as being subordinated to a standard fluctuation dissipation process with friction gamma we argue that the main properties of kardar parisi zhang theory in one dimension are derived by identifying the distribution of return times to y 0 0 which is a truncated inverse power law with the distribution of subordination times the agreement of the theoretical prediction with the numerical treatment of the 1 1 dimensional model of ballistic deposition is remarkably good in spite of the finite size effects affecting this model

dirac equation with position dependent effective mass and solvable potentials in the schrodinger equation we study the one dimensional non hermitian imaginary potential with a real energy spectrum in the framework of the position dependent effective mass dirac equation the dirac equation is mapped into the exactly solvable schrodinger like equation endowed with position dependent effective mass that we present a new procedure to solve it the point canonical transformation in non relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function the imaginary potential can be obtained the spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials we also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials

analytic and asymptotic properties of generalized linnik probability densities is a characteristic function of a symmetric probability density p since a 4 then the family of symmetric linnik densities p a g 0 2 had several a w x probabilistic applications see e g 2 7 analytic and asymptotic properw x ties of the densities p were studied in 8 a w x in 1984 klebanov maniya and melamed 9 introduced the concept of geometric strict stability and proved that the family of geometrically strictly stable densities coincides with the family of densities with characteristic functions

on solvable potentials related to so 2 2 scattering problems in one dimension associated with the so 2 2 group are studied the s matrices for the systems under consideration are calculated by using the theory of intertwining operators for semisimple lie groups the wave functions are expressed in terms of matrix elements of principal series representation of the so 2 1 group

a class of exactly solvable potentials related to the jacobi polynomials the author investigates a family of solvable potentials related to the jacobi polynomials this one dimensional potential family depends on three parameters and is restricted to the domain xo so it can be interpreted as the radial part of a central potential in three dimensions with l 0 closed expressions are obtained for the bound state energy spectrum and the wavefunctions the supersymmetric partner of this potential is also determined and it is found not to belong to the same potential family it is shown that this potential family is a special subclass of the general six parameter natanzon potential class and similarities with another subclass the ginocchio potentials are pointed out some aspects of supersymmetric quantum mechanics and shape invariance are also discussed in connection with the potential family under study

pt symmetry and its spontaneous breakdown in three dimensions the mechanism of introducing non hermiticity to non central symmetric potentials through both the azimuth and polar angles is discussed generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials such as the scarf i or rosen morse i potentials are used in the eigenvalue equation of the polar component it is shown that the spontaneous breakdown of symmetry can also occur in these non central potentials several potentials are proposed in the azimuthal eigenvalue equation too where the use of periodic boundary conditions is essential possible generalizations of the results are outlined

on supersymmetries in nonrelativistic quantum mechanics one dimensional nonrelativistic systems are studied when time independent potential interactions are involved their supersymmetries are determined and their closed subsets generating kinematical invariance lie superalgebras are pointed out the study of even supersymmetries is particularly enlightened through the already known symmetries of the corresponding schrodinger equation three tables collect the even odd and total supersymmetries as well as the invariance super algebras

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

two singular potentials the space splitting effect the quantum mechanical problem of a particle moving in one dimension 1d under the influence of a singular potential is reexamined an attractive delta function potential of infinite strength and the 1d coulomb potential is considered in particular in both cases the singularity acts as an impenetrable barrier thus dividing space into two independent regions analysis of the scattering states gives results in agreement with this conclusion the implications that the bound states of the 1d coulomb potential cannot be classified according to parity even though the potential is symmetric is also discussed herein

susyqm and other symmetries in quantum mechanics the relation of supersymmetric quantum mechanics susyqm is discussed with two other symmetry based approaches symmetry and a differential realization of the su 1 1 and su 2 algebra it is demonstrated that symmetry imposes conditions on the even and odd parts of the real and imaginary components of the superpotential w x and these are expressed in terms of a system of first order linear differential equations which is homogeneous when the factorization energy is real and inhomogeneous when it is complex the formal solution of this system is presented for various special cases as well as for the general case it is shown that a trivial solution of this system corresponds to the unbroken symmetry for the scarf ii potential the formalism of susyqm is also linked with that of the potential algebra approach and it is demonstrated that the j and j ladder operators of some su 1 1 or su 2 algebras act on series of degenerate levels of different potentials essentially in the same way as the a and a shift operators of susyqm examples are presented for su 1 1 and su 2 potential algebras as well as for spectrum generating algebras of the same type possible generalizations of this construction are also pointed out

matrix elements for powers of x dependent operators for the hyperbolic poschl teller potentials analytical matrix elements of the position operator xn n 0 and of the first and second order differential operators are derived using the eigenfunctions of two hyperbolic poschl teller potentials these general relations are written in closed form and calculated from the properties of the integral powers of hypergeometric series additionally an explicit expression is derived for the successive derivatives of the beta function convergence and reliability of these results will be discussed and an application to the calculation of the rovibrational energy spectrum of co will be considered

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

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

algebraic solution for the natanzon confluent potentials an algebraic method is used to solve the bound state problem for the most general natanzon potential for which the schrodinger equation can be reduced to the confluent hypergeometric form hence confluent potentials the solution is obtained straightforwardly a simple argument is given to sustain that the natanzon potential is the most general confluent potential

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

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

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

exact solutions of deformed schrodinger equation with a class of non central physical potentials in this paper we present exact solutions of schrodinger equation for a class of non central physical potentials within the formalism of position dependent effective mass the energy eigenvalues and eigenfunctions of the bound states for the schrodinger equation are obtained analytically by means of asymptotic iteration method and easily calculated through a new generalized decomposition of the effective potential allowing easy separation of the coordinates our results are in excellent agreement with other works in the literature

factorization of spin dependent hamiltonians we generalize the formalism of the factorization method by introducing spin degrees of freedom in addition to local potential terms we compare this construction with conventional factorization techniques of non relativistic hamiltonians and the formalism of supersymmetric quantum mechanics the characteristic isospectrality of the hamiltonians h1 qr and h2 rq is maintained in our more general framework as illustrative examples we consider two exactly solvable models related to the harmonic oscillator and the coulomb problems and some quasi exactly solvable problems in all three cases the zero energy ground state of h1 is found to be infinitely degenerate further possible generalizations of our approach are also outlined

common scaling patterns in intertrade times of u s stocks we analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the u s economy over a period of 4 yrs for all companies we find that i the probability density function of intertrade times may be fit by a weibull distribution ii when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form iii the intertrade times exhibit power law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales in agreement with the correlation behavior of the absolute price returns for the corresponding company and iv the magnitude series of intertrade time increments is characterized by long range power law correlations suggesting the presence of nonlinear features in the trading dynamics while the sign series is anticorrelated at small scales our results suggest that independent of industry sector market capitalization and average level of trading activity the series of intertrade times exhibit possibly universal scaling patterns which may relate to a common mechanism underlying the trading dynamics of diverse companies further our observation of long range power law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock support the hypothesis that the dynamics of transaction times may play a role in the process of price formation

fredholm theory for pairs of closed subspaces of a banach space we study the fredholm theory for pairs of closed subspaces of a banach space developed by kato we define the strictly singular and the strictly cosingular pairs of subspaces and we show that some of the results of operator theory can be extended to this context however there are some notable differences on the one hand the perturbation classes problem has a positive answer in this context the upper and lower semi fredholm pairs are stable under strictly singular and strictly cosingular perturbations respectively and this stability characterizes the strictly singular and the strictly cosingular pairs note that it has been proved recently that the perturbation classes problem for continuous semi fredholm operators has a negative answer on the other hand unlike in the case of operators the fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs we also show the stability under composition of the compact the strictly singular and the strictly cosingular pairs of subspaces

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

the perturbation classes problem in fredholm theory abstract we give a negative answer to the perturbation classes problem the perturbation class of the upper semi fredholm operators contains properly the strictly singular operators and the perturbation class of the lower semi fredholm operators contains properly the strictly cosingular operators

non relativistic treatment of diatomic molecules interacting with a generalized kratzer potential in hyperspherical coordinates we investigate solutions of a non relativistic wave equation in hyperspherical coordinates for a diatomic molecule system interacting with a generalized kratzer potential rovibrational eigenvalues and corresponding wavefunctions of non relativistic diatomic molecules have been determined within the framework of the asymptotic iteration method certain fundamental conditions for the applications of the asymptotic iteration method such as a suitable asymptotic form for the wave function and the termination condition for the iteration process are discussed n dimensional bound state eigenfunction solutions used in studying the dynamical variables of diatomic molecules are obtained in terms of a confluent hypergeometric function and a generalized laguerre polynomial this systematic approach is tested by calculating the rovibrational energy spectra of hydrogen and sodium chloride molecules

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

analytical solutions of a generalized non central potential in n dimensions we present that n dimensional non relativistic wave equation for the generalized non central potential with arbitrary angular momentum is analytically solvable in the hyperspherical coordinates asymptotic iteration method as a different approach is applied to obtain n dimensional energy eigenvalues and the corresponding eigenfunctions in hyperspherical coordinates the wave function solutions are obtained in terms of hypergeometric functions and jacobi polynomials the bound states of quantum systems under consideration for some special cases such as hartmann and makarov potentials have been discussed in n dimensions

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

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

an sl 4 r lie algebraic treatment of the first family of poschl teller potentials an sl 4 r dynamical potential algebra containing the so 4 potential algebra is constructed for the two parameter poschl teller potentials of the first kind for this purpose the relation between the wigner rotation matrices and the solutions of the first poschl teller equation is used explicit expressions are given for the sl 4 r generators as well as for their action on the normalised solutions all the hamiltonian eigenstates corresponding to the family of potentials with quantised potential strengths m m differing by integers are proved to belong to a single sl 4 r unitary irreducible representation of the ladder series for integral values of m and m the latter is dladd 0 0 eta while for half integral values it is dladd 1 2 1 2 eta where eta is some real parameter both irreducible representations are also shown to be characterised by generalised young pattern labels pqr0 where p 2 1 2i eta and q r 0

comment on penetrability of a one dimensional coulomb potential by m moshinsky see ibid vol 26 p 2445 1993 the analysis of the one dimensional schrodinger equation with the potential lambda x given is found to be incorrect contrary to the author s claim the origin is indeed impenetrable

spectral analysis of the complex cubic oscillator using the exact semiclassical analysis we study the spectrum of a one parameter family of complex cubic oscillators the invariance property of the complex hamiltonians and the reality property of the spectrum are discussed analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated the global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by bender and wu for the quartic oscillator

comprehensive analysis of conditionally exactly solvable models we study a quantum mechanical potential introduced previously as a conditionally exactly solvable ces model besides an analysis following its original introduction in terms of the point canonical transformation we also present an alternative supersymmetric construction of it we demonstrate that from the three roots of the implicit cubic equation defining the bound state energy eigenvalues there is always only one that leads to a meaningful physical state finally we demonstrate that the present ces interaction is in fact an exactly solvable natanzon class potential

green s matrix from jacobi matrix hamiltonian we propose two ways for determining the green s matrix for problems admitting hamiltonians that have infinite symmetric tridiagonal i e jacobi matrix form on some basis representation in addition to the recurrence relation coming from the jacobi matrix the first approach also requires the matrix elements of the green s operator between the first elements of the basis in the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane both approaches are illustrated with the non trivial but calculable example of the d dimensional coulomb green s matrix we give the corresponding formulas for the d dimensional harmonic oscillator as well

on fractional tempered stable processes and their governing differential equations we derive the governing equation of the tempered stable subordinator hereafter tss which generalizes the space fractional differential equation satisfied by the law of the stable subordinator itself this equation is expressed in terms of the shifted fractional derivative of order 0 1 coinciding with the stability parameter we then generalize this equation by introducing a time fractional derivative of order 0 1 resp 1 1 and we prove that it is satisfied by the law of a tss time changed by the inverse of a stable subordinator resp by the stable subordinator itself the corresponding processes can therefore be called fractional ts processes finally we provide fractional extensions of the relativistic stable processes which we define as a brownian motion with a random time argument represented by independent fractional ts processes of order resp 1

numerical study of transport properties in continuum percolation the authors present numerical simulations of ac conductance for a random resistor capacitor network the conductance obeys a probability density function p g varies as g alpha 0 alpha 1 they use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice at low frequencies they find that for the concentration p of conducting bonds less than the percolation threshold pc the imaginary part of conductance is proportional to frequency im geff approximately omega and the real part of conductance shows an anomalous frequency dependence re geff approximately omega 2 alpha the results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the maxwell garnett approximation they also calculate the non universal dc conductivity exponents in continuum percolation the results are consistent with earlier theoretical predictions and numerical calculations

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

potential theory of geometric stable processes in this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate brownian motions with geometric stable subordinators more precisely we establish the asymptotic behaviors of the green function and the levy density of symmetric geometric stable processes the asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes the green function behaves near zero as 1 x d log 2 x while the levy density behaves like 1 x d we also study the asymptotic behaviors of the green function and levy density of subordinate brownian motions with iterated geometric stable subordinators as an application we establish estimates on the capacity of small balls for these processes as well as mean exit time estimates from small balls and a harnack inequality for these processes

single variable realisation of the su 1 1 spectrum generating algebra and discrete eigenvalue spectra of a class of potentials a wide range of operators can be expressed in terms of the generators of the su 1 1 algebra the spectral properties of which are well known we attempt to find a realisation of the algebra in its most general form and thus evolve a unified approach to the problem of finding the spectra of hamiltonians amenable to the technique rather than taking each case separately we extend the analysis to obtain the spectra of potentials whose coordinate dependence is implicit

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

application of nonlinear deformation algebra to a physical system with poschl teller potential nonlinear deformation algebra are realized in a physical system with p schl teller potential the raising and lowering operators satisfying this algebra are constructed from which the eigenproblem of the system can be exactly solved by the operator method the physical meaning of two deforming functions involved in this algebra is also found in addition algebra is obtained naturally and discussions on the coherent state are also made

pt symmetric potentials and the so 2 2 algebra starting from a differential realization of the generators of the so 2 2 algebra we connect the eigenvalue equation of the casimir invariant either with the hypergeometric equation or the schr odinger equation in the latter case we consider problems for whichso 2 2 appears as a potentia la lgebra connecting states with the same energy in different potentials we analyse the role of the two so 2 1 subalgebras and point out their importance for pt symmetric problems where the doubling of bound states is known to occur we present two mechanisms for this and illustrate them with the example of the scarf and the p oschl teller ii potentials we also analyse scattering states transmission

linnik levy process and some extensions abstract in the literature the linnik mittag leffler laplace and asymmetric laplace distributions are the most known examples of geometric stable distributions the geometric stable distributions are especially useful in the modeling of leptokurtic data with heavy tailed behavior they have found many interesting applications in the modeling of several physical phenomena and financial time series in this paper we define the linnik levy process llp through the subordination of symmetric stable levy motion with gamma process we discuss main properties of llp like probability density function levy measure and asymptotic forms of marginal densities we also consider the governing fractional type fokker planck equation to show practical applications we simulate the sample paths of the introduced process moreover we give a step by step procedure of the parameters estimation and calibrate the parameters of the llp with the arconic inc equity data taken from yahoo finance further some extensions of the introduced process are also discussed

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 lie algebraic approach to effective mass schrodinger equations we use lie algebraic techniques to obtain exact solutions of the effective mass schrodinger equation in particular we use the su 1 1 algebra both as a spectrum generating algebra and as a potential algebra to obtain exact solutions of effective mass schrodinger equations corresponding to a number of potentials we also discuss the construction of isospectral hamiltonians for which both the mass and the potential are different

algebra representations on eigenfunctions of the rosen morse potential the bound state eigenfunctions of the rosen morse n rosen and p m morse phys rev 42 210 1932 potential are investigated using ladder operators that give a representation of the algebra a1 the representations are generally infinite dimensional and indecomposable the representation space containing unbounded functions as well as the normalizable eigenfunctions operators giving eigenfunctions of a potential with different strength are also found giving a representation of d2 the a1 or su2 representations are identified in terms of a classification proposed by sannikov s s sannikov yad fiz 6 1294 1967 sov j nucl phys 6 939 1968 and the connection with representations obtained by gruber and klimyk b gruber and a u klimyk j math phys 19 2009 1978 25 755 1984 is given the evaluation of matrix elements is considered

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

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case

on the nonlinear fock description of quantum systems with quadratic spectra the nonlinear version of the fock approach developed previously for systems with extended equidistant spectra eleonsky v m and korolev v g 1995 j phys a math gen 28 4973 is also shown to describe systems with quadratic spectra some general properties of shift operators for such spectra are studied it is found in particular that families of shift operators related to the generators of the algebra su 1 1 as well as supersymmetry ladder operators arise naturally in that approach in a unified way