conditionally solvable path integral problems ii natanzon potentials new classes of exactly solvable potentials are discussed within the path integral formalism they are constructed from the hypergeometric and confluent natanzon potentials respectively it is found that they allow incorporation of four free parameters which give rise to fractional power behaviour long range and strongly anharmonic terms we find six different classes of such potentials

algebraic shape invariant potentials in two steps the simplified algebraic structure of the shape invariance condition in two steps is developed by imposing an extra relation on the two superpotentials of the corresponding two step potentials this simplified version of potential algebra is found to be similar to that of the shape invariance condition in one step the solvable potentials of shape invariance in two steps with a translation change of parameters a1 a0 are shown to possess such a simplified version of potential algebra the condition under which the potential algebra of shape invariance in more than two steps becomes simplified is also discussed

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

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

fuzzy n cell numbers and the differential of fuzzy n cell number value mappings in this paper we give the definition of a special kind of n dimension fuzzy numbers fuzzy n cell numbers discuss their operations and representation theorems define a complete metric on the fuzzy n cell number space and prove that the metric is equivalent to the supremum metric derived by the hausdorff metric between the level sets of the n dimension fuzzy numbers and obtain an embedding theorem of the fuzzy n cell number space isometrically embeds it into a concrete banach space we also consider the differential of the fuzzy mappings from an interval into the fuzzy n cell number space by using the embedding theorem

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

partially regular solution to landau lifshitz maxwell equations in two dimensions abstract the present paper is concerned with the existence uniqueness and singularities of the landau lifshitz equation coupled with maxwell equations on riemannian manifold in two dimensions it is shown that there exists a weak solution which is regular with exception of at most finitely many singular points

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

algebraic solution for the natanzon hypergeometric potentials an algebraic method based on a strategy that makes use of a realization of the algebra so 2 1 in terms of differential operators is used to solve the bound state problem for the most general natanzon potentials for which the schrodinger equation can be reduced to hypergeometric form hence hypergeometric potentials

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

on scattering systems related to the so 2 1 group ii we study a class of solvable hamiltonians h which have strongly broken potential group structure however the scattering matrices of the systems under consideration are also related to the intertwining operator of so 2 1

mapping of shape invariant potentials under point canonical transformations the authors give explicit point canonical transformations which map twelve types of shape invariant potentials which are known to be exactly solvable into two potential classes the eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively

supersymmetry operator transformations and exactly solvable potentials a large class of potentials can be solved algebraically by using supersymmetry and shape invariance the authors apply operator transformations f transformations to these algebraically solvable problems to obtain a larger class of solvable potentials the natanzon class of potentials which are not shape invariant the important condition which is independent of supersymmetry for finding new potentials from old ones using operator transformations is that the resulting schrodinger equation has a potential which does not depend on the state as a special case of the f transformation they rederive the previously known connection between the 3d harmonic oscillator the hydrogen atom and the morse potential they also discuss the lack of commutivity of susy and the f transformations

path integral treatment for the one dimensional natanzon potentials the green s function relative to the one dimensional natanzon potentials is exactly calculated by the path integral approach the energy spectrum the wave functions and the scattering function s are deduced

notes on fuzzy complex analysis several fuzzy complex analysis problems are discussed pointing out some errors in literature and some weaknesses of introduced concepts a modified approach to fuzzy complex analysis is proposed and its impact is discussed

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

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

a 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

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

connection between type a and e factorizations and construction of satellite algebras recently we introduced a new class of symmetry algebras called satellite algebras which connect with one another wavefunctions belonging to different potentials of a given family and corresponding to different energy eigenvalues here the role of the factorization method in the construction of such algebras is investigated a general procedure for determining an so 2 2 or so 2 1 satellite algebra for all the hamiltonians that admit a type e factorization is proposed such a procedure is based on the known relationship between type a and e factorizations combined with an algebraization similar to that used in the construction of potential algebras it is illustrated with examples of the generalized morse potential the rosen morse potential and the kepler problem in a space of constant negative curvature and in each case the conserved quantity is identified it should be stressed that the method proposed is fairly general since the other factorization types may be considered as limiting cases of type a or e factorizations

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

fuzzy complex analysis ii integration abstract this paper continues the author s research in fuzzy complex analysis where we now define and study the basic properties of a fuzzy contour integral for fuzzy mappings which map a rectifiable curve in the complex plane into fuzzy complex numbers

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

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

generators for nonlinear canonical transformations canonical transformations whose generators are linear in the basic operators pj and arbitrary in the canonically conjugate operators qj are explicitly constructed it is shown that they correspond to gauge transformations and changes of variables some applications are mentioned in the one dimensional case

on the generalized morse potential the equivalence between the generalized morse gmp and eckart potentials is shown the study of the hypergeometric natanzon potentials using so 2 1 techniques is applied to compute the eigenfunctions and eigenvalues of the eckart gmp potential the action of the group generators is studied with the result that a family of eckart potentials is obtained which is different from the one obtained in susyqm

measure valued solution to the strongly degenerate compressible heisenberg chain equations in this paper we are concerned with the existence of solutions to the compressible heisenberg chain equations by the vanishing viscosity method we prove that this system admits at least one measure valued solution

potentials of the heun class we review different methods of generating potentials such that the one dimensional schrodinger equation odse can be transformed into the hypergeometric equation we compare our results with previous studies and complement the subject with new findings our main result is to derive new classes of potentials such that the odse can be transformed into the heun equation and its confluent cases the generalized heun equation is also considered

contraction of superintegrable hamiltonian systems we investigate the contraction of a class of superintegrable hamiltonians by implementing the contraction of the underlying lie groups we also discuss the behavior of the coordinate systems that separate their equations of motion the motion constants as well as the corresponding solutions along such a process

explicit wavefunctions for shape invariant potentials by operator techniques the authors obtain explicit expressions for the wavefunctions of all the known shape invariant potentials by using the operator method

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

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

ancfis a neurofuzzy architecture employing complex fuzzy sets complex fuzzy sets cfss are an extension of type 1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane although there has been considerable progress made in determining the properties of cfss and complex fuzzy logic there has yet to be any practical application of this concept we present the adaptive neurocomplex fuzzy inferential system ancfis which is the first neurofuzzy system architecture to implement complex fuzzy rules and in particular the signature property of rule interference we have applied this neurofuzzy system to the domain of time series forecasting which is an important machine learning problem we find that ancfis performs well in one synthetic and five real world forecasting problems and is also very parsimonious experimental comparisons show that ancfis is comparable with existing approaches on our five datasets this work demonstrates the utility of complex fuzzy logic on real world problems

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

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

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

some exact blow up solutions of landau lifshitz equations without gilbert term in this paper we construct some exact blow up solutions of multi dimensional space time landau lifshitz equations under the radially symmetric coordinates among these solutions two blow up finite energy solutions on s 2 are proposed the question of whether a solution of the inhomogeneous isotropic landau lifshitz equations can develop a finite time singularity on s 2 from smooth initial data with finite energy is not clear our results show that the blow up phenomenon can really happen on this initial data

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

some remarks for fuzzy complex analysis abstract this paper gives some results for fuzzy complex sets generalize and improve the corresponding work of fuzzy complex numbers in 1 we also introduce the derivative and integral of functions mapping complex numbers into the set of generalized fuzzy complex numbers and it is different to the definitions in 2 4 all this may be a foundation for researching fuzzy complex analysis

on starshaped fuzzy sets in this paper we firstly define some new types of fuzzy starshapedness discuss the relationships among the different types of fuzzy starshapedness and then present some basic properties of them finally we develop several important results on the shadows of starshaped fuzzy sets

group theoretical approach to reflectionless potentials we examine the general form of potentials with zero reflection coefficient in one dimensional hamiltonians connected with casimir invariants of non compact groups

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

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

on scattering systems related to the group scattering systems related to the noncompact groups g in the sense that the hamiltonian of the system can be written as a function of the casimir operator of g are considered the s matrix for such systems are defined in terms of an intertwining operator of underling symmetry group g the s matrices for one dimensional scattering systems with symmetry group are classified

fuzzy complex numbers representations operations and its analysis abstract as a key theoretical basis of fuzzy complex analysis a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form trigonometric form and algebraic form of fuzzy complex numbers and some arithmetic operations are investigated the concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions the metric between two fuzzy complex numbers the derivative differentiability and analyticity of fuzzy complex valued functions are defined and characterized the necessary and sufficient conditions of the analyticity of fuzzy complex valued functions are investigated furthermore the integral of fuzzy complex valued functions and cauchy integral theorem are given and discussed

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

generalized morse potential symmetry and satellite potentials we study in detail the bound state spectrum of the generalized morse potential gmp which was proposed by deng and fan as a potential function for diatomic molecules by connecting the corresponding schrodinger equation with the laplace equation on the hyperboloid and the schrodinger equation for the poschl teller potential we explain the exact solvability of the problem by an symmetry algebra and obtain an explicit realization of the latter as we prove that some of the generators connect among themselves wavefunctions belonging to different gmps called satellite potentials the conserved quantity is some combination of the potential parameters instead of the level energy as for potential algebras hence belongs to a new class of symmetry algebras we also stress the usefulness of our algebraic results for simplifying the calculation of frank condon factors for electromagnetic transitions between rovibrational levels based on different electronic states

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

satellite potentials for hypergeometric natanzon potentials as a result of the so 2 1 treatment of the hypergeometric natanzon potentials vn a set of potentials related to a given one is determined these are the satellite potentials and all belong to the natanzon class the set arises as the result of the action of the so 2 1 generators on the carrier space of an irreducible representation the results are compared to those obtained from supersymmetric quantum mechanics susyqm for some natanzon potentials the chains of natanzon potentials constructed using the idea of a satellite potential are in most cases different from the susyqm chains

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

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

algebraic approach to non central potentials an algebraic approach to class of separable non central hamiltonians is presented we show that the bound states of quantum systems under consideration are described by unitary representations of the algebra

algebraic shape invariant potentials as the generalized deformed oscillator within the framework of supersymmetric quantum mechanics we study the simplified version of the potential algebra of the shape invariance condition in k steps where k is an arbitrary positive integer the associated potential algebra is found to be equivalent to the generalized deformed oscillator algebra that has a built in zk grading structure the algebraic realization of the shape invariance condition in k steps is therefore formulated by the method of the zk graded deformed oscillator based on this formulation we explicitly construct the general algebraic properties for shape invariant potentials in k steps in which the parameters of partner potentials are related to each other by the translation a1 a0 the obtained results include the cyclic shape invariant potentials of period k as a special case

smooth solution to the one dimensional inhomogeneous non automorphic landau lifshitz equation using the differential difference method and viscosity vanishing approach we obtain the existence and uniqueness of the global smooth solution to the periodic initial value problem of the inhomogeneous non automorphic landau lifshitz equation without gilbert damping terms in one dimension to establish the uniform estimates we use some identities resulting from the fact jz x t jz 1 and the fact that the vectors fz z x z z xg form an orthogonal base of the space r 3

the general besselian and legendrian path integrals the general besselian and legendrian path integrals based on the confluent and hypergeometric natanzon potentials are calculated these two solutions cover all other path integral representations which are related to the radial harmonic oscillator and the modified poschl teller path integral

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

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

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

global weak solutions of the fractional landau lifshitz maxwell equation abstract the fractional landau lifshitz maxwell equation is considered in this paper we show the global existence of suitable weak solutions by galerkin method and the vanishing viscosity method the main difficulties in this study are due to the loss of compactness of this system and the fact that the nonlinear term is nonlocal and of the same order of the equation to overcome these difficulties we introduce the commutator estimates and some cancellation properties to this equation which prove to be proper tools in the study of landau lifshitz type equations

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

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

models with sl 2 c symmetry and their s matrices we study quantum systems whose scattering modes are governed by unitary representation of the algebra the s matrices of the systems under consideration are defined from intertwining relations for the weyl equivalent representations of the group sl 2 c or its lie algebra

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