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

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

efficient computation of modal sensitivities for systems with repeated frequencies the physical origin and ambiguity of repeated frequency modes is discussed followed by a mathematical derivation of the governing equations for the partial derivatives of repeated eigenvalues with respect to changes in the system parameters this is followed by a derivation for the associated eigenvector derivative equations physical interpretations are included to help explain the analytical results an efficient computation procedure which preserves the bandwidth for very large systems is also proposed for systems with repeated or closely spaced eigenvalues two small examples to illustrate the solution procedure are presented 8 references

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

scattering and bound state green s functions on a plane via so 2 1 lie algebra we calculate the green s functions for the particle vortex system for two anyons on a plane with and without a harmonic regulator and in a uniform magnetic field these green s functions which describe scattering or bound states depending on the specific potential in each case are obtained exactly using an algebraic method related to the so 2 1 lie group from these green s functions we obtain the corresponding wave functions and for the bound states we also find the energy spectra

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

construction of the operator for a symmetric model we obtain a closed form expression of the operator for the symmetric scarf i potential it is also shown that the eigenfunctions form a complete set

quantum and fisher information from the husimi and related distributions the two principal immediate influences which we seek to interrelate here upon the undertaking of this study are papers of yczkowski and slomczynski j phys a 34 6689 2001 and of petz and sudar j math phys 37 2262 1996 in the former work a metric the monge one specifically over generalized husimi distributions was employed to define a distance between two arbitrary density matrices in the petz sudar work completing a program of chentsov the quantum analog of the classically unique fisher information monotone metric of a probability simplex was extended to define an uncountable infinitude of riemannian also monotone metrics on the set of positive definite density matrices we pose here the questions of what is the specific unique fisher information metric for the classically defined husimi distributions and how does it relate to the infinitude of quantum metrics over the density matrices of petz and sudar we find a highly proximate small relative entropy relationship bet

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

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

characteristics of sensitivity analysis of repeated frequencies 5huang x z and zan s wing and fin buffet on the standard dynamic model rto rto tr 26 ac 323 avt tp 19 neuilly sur seine france oct 2000 6goura g s l time marching analysis of flutter using computational fluid dynamics ph d dissertation dept of aerospace engineering univ of glasgow glasgow scotland u k sept 2001 7goura g s l badcock k j woodgate m a and richards b e a data exchange method for fluid stucture interaction problems aeronautical journal vol 105 no 1046 2001 pp 215 221 8goura g s l badcock k j woodgate m a and richards b e implicit method for the time marching analysis of flutter aeronautical journal vol 105 no 1046 2001 pp 199 214 9melville r nonlinear simulation of aeroelastic instability aiaa paper 2001 0570 jan 2001 e livne associate editor

localization of multi constrained optima and avoidance of local optima in structural optimization problems abstract this paper describes two new methods of structural optimization the first method is seen primarily in conjunction with optimality criteria algorithms it proposes a new procedure for the solution of multi constrained optimization problems based on the calculation of partial optima the partial optima refer to solutions of the optimization problem only for part of the constraints in the simplest case only for a single constraint and can be easily determined by already known procedures the second method is a new mathematical programming method designed to combine a high convergence speed together with a high reliability with respect to the avoidance of local minima

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

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

factorization of complex canonical transformations we investigate the lie series representation of the canonical transformations in a finite dimensional complex phase space it is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function a holomorphic function and an element of the cyclic group c4 the imaginary function can be considered as an observable in the sense of classical mechanics some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system moreover a special kind of mappings is studied which provides a link between entropy action and state functions the occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case

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

inhomogeneous boson realization of indecomposable representations of lie algebras by making use of the differential realization of lie algebras in the space of inhomogeneous polynomials of a certain number of variables the corresponding inhomogeneous boson realization of lie algebras is given a new kind of indecomposable representations of lie algebras are studied on the universal enveloping algebra of heisenberg weyl algebra its subspaces and its quotient spaces the finite dimensional representations are naturally obtained on the subspaces of fock space as an example the indecomposable and irreducible representations of the lie algebra su 2 are discussed in detail

non bijective canonical transformations in quantum mechanics proposes a new formulation for the representation of non bijective canonical transformations in quantum mechanics these are represented by a unitary operator acting on a hilbert space h x h similar to the one used for the description of a two dimensional system

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

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

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

fisher information kinetic energy and uncertainty relation inequalities by interpolating between fisher information and mechanical kinetic energy we introduce a general notion of kinetic energy with respect to a parameter of schrodinger wavefunctions from a statistical inference perspective kinetic energy is the sum of fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase a family of integral inequalities concerning kinetic energy and moments are established among which the cramer rao inequality and the weyl heisenberg inequality are special cases in particular the integral inequalities involving the negative order moments are relevant to the study of electron systems moreover by specifying the parameter to a scale we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity

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

on the pseudo norm and admissible solutions of the symmetric scarf i potential the physically admissible solutions of the symmetric scarf i potential are identified in the domain of real and complex energies it is found that generally there are no admissible complex energy solutions and there is one with real energy in a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi parity q however the two sets are not orthogonal in the domain of real energies the sign of the pseudo norm of states with real energy is found to oscillate as 1 n in accordance with the example of other symmetric potentials possessing an infinite number of discrete levels it is argued that the spontaneous breakdown of symmetry cannot be defined for the scarf i potential a comparison with some symmetric extensions of the infinite square well is presented

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

general potentials described by so 2 1 dynamical algebra in parabolic coordinate systems the authors propose general three dimensional potentials in rotational and cylindrical parabolic coordinates which are generated by direct products of the so 2 1 dynamical group then they construct their green functions algebraically and find their spectra particular cases of these potentials which appear in the literature are also briefly discussed

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

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

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

some integral identities and inequalities for entire functions and their application to the coherent state transform let be an entire function on cn and for any h 0 and r 0 define fr z r e 2 z 2h let d h denote h n times lebesgue measure on cn frs2 2 d h n sh frsd h from this and a logarithmic sobolev inequality we easily deduce qnq fr q pnp fr p for all 0 p q t8 where the lp norms are taken with respect to the measure d h above we apply these results to the study of the spaces ap consisting of all entire functions satisfying z pe 2 z 2h d h obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in ap for 1 p we then apply our results to the coherent state transform extending and simplifying some previously known results

fisher information matrix of husimi distribution we calculate the fisher information matrix of husimi distribution in the fock bargmann representation it turns out that the fisher information of the position and that of the momentum move in opposite directions and that a weighted trace of the fisher information matrix is a constant independent of the husimi distribution this may be interpreted as a kind of uncertainty relation in the spirit of heisenberg uncertainty principle from the statistical inference point of view

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

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

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

nonlinear canonical transformations i the linear canonical transformation q aq bp p cq dp ad bc 1 q p constant is extended to include terms of the second third and fourth degree products of second degree transformations are used to construct third and fourth degree transformations the question of factorisability of fourth degree transformations is discussed

solvable potentials generated by sl 2 r canonical transformations are used to build realisations of sl 2 r in terms of the basic quantum mechanical operators q and p the results are used to construct solvable potentials in the framework of one dimensional quantum mechanics

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

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

admissible large perturbations in structural redesign a new algorithm for structural redesign by perturbation is developed it allows for large changes between baseline and objective structures and can satisfy both modal and static displacement objectives it is faster and more accurate than previously developed perturbation resizing methods because it advances incrementally to the objective structure using large admissible perturbations the computer code developed is used as postprocessor to general or special purpose finite element codes to improve upon the design of structures with unacceptable modal and or static displacement response depending on the relation betweem the number of redesign goals allowable structural changes and admissibility constraints the desired redesign may be feasible either locally optimal or unique or may not exist in the latter case a minimum error inadmissible design is produced several numerical examples are used to study the effects of definition and relaxation of the redesign admissible domain and demonstrate the accuracy of the new redesign algorithm an offshore tower with repeated eigen values and 192 degrees of freedom is also redesigned subject to frequency static displacement goals and admissibility constraints 22 refs

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

calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues summary an algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues in the proposed method the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution by introducing an additional normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution the method is numerically stable and the homogeneous solutions are computed by the second order derivatives of the eigenequations two numerical examples are used to illustrate the validity of the proposed method copyright 2015 john wiley sons ltd

complex variable method for eigensolution sensitivity analysis the application of complex variable method cvm for eigenvalue and eigenvector sensitivity analysis is presented gradi ent based methods usually used in structural optimization require accurate sensitivity information cvm is accurate robust and easy to implement as compared to other approximate sensitivity analysis methods by cvm the first order modal sensitivity can b e calculated using an existing structural analysis package this is achieved by making a small complex perturbation to a design variable the resulting eigensolutions would be complex the sensitivity of eigensolutions with respect to that design variable can then be obtained by taking the complex part of the eigensolutions divided by the perturbation since there is no subtraction involved in the sensitivity calculation the approximation is not sensitive to step size thus unlike finite difference method s cvm does not suffer the effect of subtraction cancellation error when a very small step size is used the computation of 1 st order modal sensitivity is not hampered by repeated eigenvalue problem as is the case with other sensitivity analysis methods for repeated eigenvalues cvm automatically provides correct eigenvalue sensitivity and the associated eigenvector sensitivity for a differentiable mode a 5 dof system with an eigenvalue of multiplicity 3 is used to illustrate the salient features of cvm in eigen solution sensitivity analysis nomenclature k stiffness matrix for spring mass system m mass matrix for spring mass system eigenvalue for spring mass system eigenvector for spring mass system derivative of w r t design variable

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

an iterative method for finite dimensional structural optimization problems with repeated eigenvalues a numerical method for solving finite dimensional structural optimization problems with constraints on natural frequency and on buckling is formulated the method treats nondifferentiable repeated eigenvalues that have been shown to systematically arise in structural optimization recent results on differentiability of eigenvalues are used to develop a generalized gradient projection method for structural optimization the algorithm is shown to overcome technical difficulties associated with nondifferentiability of repeated eigenvalues the method is used to solve buckling and vibration optimization problems in which repeated eigenvalues occur

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

new inhomogeneous boson realizations and inhomogeneous differential realizations of lie algebras the inhomogeneous boson realizations ihbr and the corresponding inhomogeneous differential realizations ihdr of lie algebras which play an important role in the search of quasi exactly solvable problems qesp of quantum mechanics are studied all possible ihdr of semisimple lie algebras can be obtained in this way as examples the ihbr and the corresponding ihdr of lie algebras su 2 and su 3 are studied in detail

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

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

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

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

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

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

optimal inertia lumping from modal mass matrices for structural dynamics an optimal inertia lumping formulation using modal mass matrices obtained from structural analysis is presented enabling recovery of lumped inertia information even when substructuring has been employed the method is implemented with a commercial optimization program and uses reduced mass and stiffness formulations employed in commercial finite element analysis programs the result is lumped inertia models of complex flexible bodies for use in structural dynamics including flexible multibody dynamic simulation a method for dealing with nondifferentiability of repeated eigenvalues that arise in the optimization process is derived and implemented in the proposed formulation numerical examples are presented to validate the method relative to conventional mass lumping approaches