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

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

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

exactly solvable potentials and the concept of shape invariance the author analyses the relationship between shape invariance and exactly solvable potentials using earlier results he shows that all actually known shape invariant potentials with a single parameter can be recovered quite simply extension to the case of shape invariant potentials with two parameters as well as to non shape invariant but exactly solvable potentials are also possible

extending the class of solvable potentials i the infinite potential well with a sinusoidal bottom this is the first in a series of papers where we succeed in enlarging the class of exactly solvable potentials in one and three dimensions by obtaining solutions for new relativistic and nonrelativistic problems this is accomplished by constructing a matrix representation of the wave operator in a complete square integrable basis that makes it tridiagonal expanding the wave function in this basis makes the wave equation equivalent to a three term recursion relation for the expansion coefficients consequently finding solutions of the recursion relation is equivalent to solving the original problem doing so results in a larger class of solvable potentials the usual diagonal representation constraint results in a reduction from the larger class to the conventional class of solvable potentials giving the well known energy spectra and the corresponding wave functions moreover some of the new solvable problems show evidence of a klauder like phenomenon in the present work we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape

the entropy theorem via coding bounds the entropy theorem also known as the shannon mcmillan breiman theory or the asymptotic equipartition theorem asserts that for a stationary ergodic finite alphabet process the sequence 1 n log p x sub 1 sup n converges almost surely to the entropy rate h of the process the entropy theorem has been used to establish asymptotic bounds on the performance of noiseless codes here the coding theorems are established without using the entropy theorem and the coding theorems are then used to prove the entropy theorem the principle feature is the direct use of coding ideas to obtain the entropy theorem

conditionally exactly solvable potential and dual transformation in quantum mechanics we show that the conditionally exactly solvable potential of dutt et al 1995 j phys a math gen 28 l107 and the exactly solvable potential from which it is derived form a dual system

information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

the transformation of polynomial eigenfunctions of linear second order difference operators a special case of meixner polynomials as polynomial eigenfunctions of a linear second order difference operator a special case of the meixner polynomials m 2 c n x 1 is transformed explicitly into new complete sequences of non classical orthogonal polynomials

construction of solvable quantum mechanical potentials a method of generating some quantum mechanical exactly solvable potentials esps using the properties of classical orthogonal polynomials cops is presented it is illustrated using hypergeometric polynomial utilization of other cops to get more esps is indicated

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

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

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

superintegrable systems with position dependent mass first order integrals of motion for schrodinger equations with position dependent masses are classified eighteen classes of such equations with non equivalent symmetries are specified they include integrable superintegrable and maximally superintegrable systems among them is a system invariant with respect to the lie algebra of lorentz group and a system whose integrals of motion form algebra so 4 three of the obtained systems are solved exactly

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

quantization of the lienard ii equation and jacobi s last multiplier in this paper the role of jacobi s last multiplier in mechanical systems with a position dependent mass is unveiled in particular we map the lienard ii equation to a position dependent mass system the quantization of the lienard ii equation is then carried out using the point canonical transformation method together with the von roos ordering technique finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated laguerre and exceptional laguerre functions

generalized laguerre polynomials with position dependent effective mass visualized via wigner s distribution functions we construct analytically and numerically the wigner distribution functions for the exact solutions of the position dependent effective mass schrodinger equation for two cases belonging to the generalized laguerre polynomials using a suitable quantum canonical transformation expectation values of position and momentum operators are obtained analytically in order to verify the universality of heisenberg s uncertainty principle

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

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

exact solutions for anharmonic oscillators completes the author s investigation see phys lett a vol 81 p 17 8 1981 and presents exact solutions and eigenvalues for anharmonic oscillators the solutions being given in the form of definite integrals the conditions for the validity of these results are investigated

a decomposition theorem for banach space valued fuzzy henstock integral we establish the following decomposition theorem for fuzzy mappings with values in a banach space a fuzzy mapping is fuzzy henstock integrable if and only if it can be represented as a sum of a fuzzy mcshane integrable fuzzy mapping and of a fuzzy henstock integrable fuzzy mapping generated by a henstock integrable function

novel correlations in two dimensions two body problem we discuss a many body hamiltonian with two and three body interactions in two dimensions introduced recently by murthy bhaduri and sen apart from an analysis of some exact solutions in the many body system we analyse in detail the two body problem which is completely solvable we show that the solution of the two body problem reduces to solving a known differential equation due to heun we show that the two body spectrum becomes remarkably simple for large interaction strengths and the level structure resembles that of the landau levels we also clarify the ultraviolet regularization which is needed to define an inverse square potential properly and discuss its implications for our model

path integral solutions for deformed poschl teller like and conditionally solvable potentials i discuss in this paper the behaviour of the solutions of the so called q hyperbolic potentials i e poschl teller like and conditionally solvable potentials in terms of the path integral formalism the differences in comparison to the usual poschl teller like potentials are investigated including the discrete energy spectra and the bound state wavefunctions we also point out the relation of the q deformation with curvature on hyperboloids

method for generating additive shape invariant potentials from an euler equation in the supersymmetric quantum mechanics formalism the shape invariance condition provides a sufficient constraint to make a quantum mechanical problem solvable i e we can determine its eigenvalues and eigenfunctions algebraically since shape invariance relates superpotentials and their derivatives at two different values of the parameter a it is a non local condition in the coordinate parameter x a space we transform the shape invariance condition for additive shape invariant superpotentials into two local partial differential equations one of these equations is equivalent to the one dimensional euler equation expressing momentum conservation for inviscid fluid flow the second equation provides the constraint that helps us determine unique solutions we solve these equations to generate the set of all known independent shape invariant superpotentials and show that there are no others we then develop an algorithm for generating additive shape invariant superpotentials including those that depend on explicitly and derive a new dependent superpotential by expanding a scarf superpotential

approximation of fuzzy numbers using the convolution method in this study we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number we show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non differentiable points in previous studies this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers the possible non differentiable points of which were the two endpoints of the 1 cut the construction of smoothers is a key step in the process for producing approximations we also show that if appropriate smoothers are selected then we can use the convolution method to provide approximations that are differentiable lipschitz and that also preserve the core

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

conditionally solvable path integral problems some specific conditionally exactly solvable potentials are discussed within the path integral formalism they generalize the usually known potentials by the incorporation of fractional power behaviour and strongly anharmonic terms we find four different kinds of such potential the first is related to the coulomb potential the second is an anharmonic confinement potential and the third and fourth are related to the manning rosen potential

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

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

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

an exactly soluble schrodinger equation with smooth position dependent mass the one dimensional generalized schrodinger equation for a system with smooth potential and mass step is resolved exactly the wave function depends on the heun s function which is a solution of a second order fuchsian equation with four singularities the behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step two limiting cases are also studied when the width of the mass step is vanishing and when the smooth potential and mass step tend to an abrupt potential and mass step

a second class of solvable potentials related to the jacobi polynomials levai 1991 has used a simple method for generating solvable wave equations to construct potentials whose solutions include as factors jacobi polynomials of functions g x which solve specific differential equations application of this method to a previously unconsidered differential equation involving jacobi polynomials results in a new class of solvable potentials missing from earlier compilations an interesting property of these new potentials is that when two of their defining parameters are fixed they all possess the same energy eigenvalue spectrum independent of any change in a third parameter leading to an indefinite number of different potentials with the same energies

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

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

mapping of the five parameter exponential type potential model into trigonometric type potentials we choose the five parameter exponential type potential model as input and construct five trigonometric type potentials via point canonical transformations their energy spectra and wavefunctions are obtained in a unified manner by using the expressions for the energy spectra and wavefunctions of the five parameter exponential type potential model

the trigonometric rosen morse potential in the supersymmetric quantum mechanics and its exact solutions the analytic solutions of the one dimensional schrodinger equation for the trigonometric rosen morse potential reported in the literature rely upon the jacobi polynomials with complex indices and complex arguments we first draw attention to the fact that the complex jacobi polynomials have non trivial orthogonality properties which make them uncomfortable for physics applications instead we here solve the above equation in terms of real orthogonal polynomials the new solutions are used in the construction of the quantum mechanical superpotential

the ergodic and entropy theorems revisited a new proof of birkhoff s ergodic theorem is given using a sample path covering idea an idea created by ornstein and weiss in their extension of the information convergence theorem to random fields a sketch of the ornstein weiss proof for processes is included in the appendix

l2 series solutions of the dirac equation for power law potentials at rest mass energy we obtain solutions of the three dimensional dirac equation for radial power law potentials at rest mass energy as an infinite series of square integrable functions these are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the dirac operator is tridiagonal the wave equation results in a three term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials these are modified versions of the meixner pollaczek polynomials and of the continuous dual hahn polynomials the choice depends on the values of the angular momentum and the power of the potential

conditionally exactly solvable potentials and exceptional orthogonal polynomials it is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials some properties of such polynomials e g recurrence relation ladder operators differential equations etc have been obtained

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

isospectral hamiltonian for position dependent mass for an arbitrary quantum system and coherent states by means of the unitary transformation a new way for discussing the ordering prescription of the schrodinger equation with a position dependent mass pdm for isospectral hamiltonian operators is presented we show that the ambiguity parameter choices in the kinetic part of the hamiltonian can be explained through an exact susy qm symmetry as well as a consequence of an accidental symmetry under the z2 action by making use of the unitary transformation we construct coherent states for a family of pdm isospectral hamiltonians from a suitable choice of ladder operators we show that these states preserve the usual structure of klauder perelomov s states and thus saturate and minimize the position momentum uncertainty relation pmur under some special restrictions we show that pmur properties can be used to determine the sign of the superpotential

the w derivatives of fuzzy mappings in banach spaces this paper introduces w derivatives of set valued mappings and fuzzy mappings gives some w derivability properties of such mappings also shows the relations between the w derivatives and the weak integrals

shifted 1 n expansion and exact solutions for the potential v r z r gr λr2 the authors have found an infinite number of exact solutions for the hydrogenic atom in the external potential v r gr r2 not only for an s wave state but for higher waves as well from supersymmetric considerations the general schrodinger equation has been treated by the shifted 1 n expansion method the eigenvalues obtained from the shifted 1 n expansion are compared with those obtained by bessis et al 1987 and also with the supersymmetric exact values

factorization of sturm liouville operators solvable potentials and underlying algebraic structure in this paper a general method of factorization of sturm liouville sl operators is provided a method to solve sl eigenvalue problems is presented new classes of exactly solvable potentials are obtained the supersymmetry and shape invariance approaches are generalized to the sl operators it is shown that the sl shape invariance potentials have an underlying algebraic structure this algebra is in general infinite dimensional the condition of finite algebra is obtained

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

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

sample converses in source coding theory the rate and distortion performance of a sequence of codes along a sample sequence of symbols generated by a stationary ergodic information source are studied two results are obtained 1 the source sample sequence is encoded by an arbitrary sequence of block codes which operate at a fixed rate level r and a sample converse is obtained which states that with probability one the lower limit of the code sample distortions is lower bounded by d r the value of the distortion rate function at r 2 the source sample sequence is encoded by an arbitrary sequence of variable rate codes which operate at a fixed distortion level d and a sample converse is obtained which states that with probability one the lower limit of the code sample rates is lower bounded by r d the value of the rate distortion function at d a novel ergodic theorem is used to obtain both sample converses

random fuzzy number integrals in banach spaces abstract in this paper a new kind of random fuzzy number integral is introduced by the pettis aumann integral of the set valued functions in banach spaces the existence of the random fuzzy integral is also shown levi s convergence theorem of the integral is given a special mention

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

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

the interactions between ergodic theory and information theory information theorists frequently use the ergodic theorem likewise entropy concepts are often used in information theory recently the two subjects have become partially intertwined as deeper results from each discipline find use in the other a brief history of this interaction is presented in this paper together with a more detailed look at three areas of connection namely recurrence theory blowing up bounds and direct sample path methods

new class of conditionally exactly solvable potentials in quantum mechanics motivated by an idea of dutra 1993 we obtain a new class of one dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value it is shown that using shape invariant potentials as input one may generate different classes of such potentials even in more than one dimension we also illustrate that wkb and supersymmetry inspired wkb methods provide very good approximations for these potentials with the latter doing comparatively better

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

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

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them