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

nonlinear error bounds for lower semicontinuous functions on metric spaces as a development of the theory of linear error bounds for lower semicontinuous functions defined on complete metric spaces introduced in aze et al nonlinear anal 49 643 670 2002 and refined in aze and corvellec esaim control optim calc var 10 409 425 2004 we propose a similar approach to nonlinear error bounds based on the notion of strong slope the variational principle and the change of metric principle the latter allowing to obtain sharp estimates for such error bounds through a reduction to the linear case

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

local completeness drop theorem and ekeland s variational principle by using a very general drop theorem in locally convex spaces we obtain some extended versions of ekeland s variational principle which only need assume local completeness of some related sets and improve hamel s recent results from this we derive some new versions of caristi s fixed points theorems in the framework of locally convex spaces we prove that danes drop theorem ekeland s variational principle caristi s fixed points theorem and phelps lemma are equivalent to each other

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

some equivalent formulations of the generalized ekeland s variational principle and their applications abstract in this paper we first give a generalized takahashi s existence theorem from the existence theorem we establish some equivalence relations between generalized caristi s fixed point theorems and generalized ekeland s variational principles some applications to the existence theorems of weak sharp minima and global error bounds for lower semicontinuous functions are also given

analytic representations based on su 1 1 coherent states and their applications we consider two analytic representations of the su 1 1 lie group the representation in the unit disc based on the su 1 1 perelomov coherent states and the barut girardello representation based on the eigenstates of the su 1 1 lowering generator we show that these representations are related through a laplace transform a weak resolution of the identity in terms of the perelomov su 1 1 coherent states is presented which is valid even when the bargmann index k is smaller than various applications of these results in the context of the two photon realization of su 1 1 in quantum optics are also discussed

mapping theorems for gateaux differentiable and accretive operators let p be a nonlinear operator mapping the banach space x into the banach space y there are many approaches to studying solvability of the equation px y for y e y a considerable number of which involve local or infinitesimal assumptions on the operator p this is due in part to the fact that p frequently arises from a differential or integral operator which has monotonicity or smoothness properties for which local apriori estimates are readily obtainable in this paper we derive mapping theorems for operators p satisfying two different kinds of local assumptions differentiability assumptions and monotonicity assumptions our approach in the former case relies on an abstract newton kantorovich scheme which was pioneered by m altman see for example l and further developed in 19 in the latter case we extend some results of f e browder using a combination of both new and standard techniques in each case our basic technique involves an application of a new version of ekeland s theorem which we derive from a fundamental maximal principle of h brezis and f e browder in section 2 of this paper we derive our new version of ekeland s theorem and in section 3 apply this result to nonlinear operators using different and by now standard techniques we prove a domain invariance result in section 4 needed for our result an accretive operators

ekeland s variational principle for vectorial multivalued mappings in a uniform space abstract in this paper we establish ekeland s variational principle and an equilibrium version of ekeland s variational principle for vectorial multivalued mappings in the setting of separated sequentially complete uniform spaces our approaches and results are different from those in chen et al 2008 hamel 2005 and lin and chuang 2010 13 14 15 as applications of our results we study vectorial caristi s fixed point theorems and takahashi s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated sequentially complete uniform space we also apply our results to study maximal element theorems which are unified methods of several variational inclusion problems our results contain many known results in the literature fang 1996 21 and will have many applications in nonlinear analysis

ekeland s variational principle minimax theorems and existence of nonconvex equilibria in complete metric spaces abstract in this paper we introduce the concept of function which generalizes the concept of w distance studied in the literature we establish a generalized ekeland s variational principle in the setting of lower semicontinuous from above and functions as applications of our ekeland s variational principle we derive generalized caristi s common fixed point theorems a generalized takahashi s nonconvex minimization theorem a nonconvex minimax theorem a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces we also prove that these theorems also imply our ekeland s variational principle

on some nonlinear problems induced by an abstract maximal element principle abstract in this paper by applying an abstract maximal element principle on quasi ordered sets established by lin and du we obtain a generalized brezis browder principle system vectorial versions of ekeland s variational principle and maximal element principle and a vectorial version of takahashi s nonconvex minimization theorem moreover we investigate the equivalence between scalar versions and vectorial versions of these results

generalized intelligent states and squeezing the robertson schrodinger uncertainty relation for two observables a and b is shown to be minimized in the eigenstates of the operator a ib being a complex number such states called generalized intelligent states gis can exhibit arbitrarily strong squeezing of a or b the time evolution of gis is stable for hamiltonians which admit linear in a and b invariants systems of gis for the su 1 1 and su 2 groups are constructed and discussed it is shown that su 1 1 gis contain all the perelomov coherent states cs and the barut and girardello cs while the spin cs are a subset of su 2 gis cs for an arbitrary semisimple lie group can be considered as a gis for the quadratures of the weyl generators

nonlinear deformations of su 2 and su 1 1 generalizing witten s algebra nonlinear deformations of su 2 and su 1 1 involving two deforming functions f j0 and g j0 are considered for g j0 1 they reduce to some algebras first studied by polychronakos 1990 and rocek 1991 spatial emphasis is laid on the case where g j0 is a linear function of j0 it is shown that for any lambda 2 3 there exist lambda 1 parameter algebras that are deformations of su 2 or su 1 1 respectively and for which f j0 is a polynomial of degree lambda for lambda 2 such algebras are equivalent to witten s 1990 first deformation of su 2 or su 1 1 for any lambda the spectrum of j0 is exponential instead of linear as in the case where g j0 1

equivalents to ekeland s variational principle in uniform spaces abstract a minimal element theorem on sequentially complete uniform spaces is presented that generalizes earlier results of pacific j math 55 2 1974 335 341 proc amer math soc 108 3 1990 707 714 two more equivalent formulations of the minimal element theorem are given it is applied to derive new versions of ekeland type theorems which improve results of j math anal appl 202 1996 398 412 finally the minimal element theorem is used to obtain an ekeland type theorem for functions with values in a linear space without topological structure

equivalent formulations of ekeland s variational principle abstract we prove that for some 0 inf x f inf x f e there exists y x such that 0

a drop theorem without vector topology abstract danes drop theorem is extended to bornological vector spaces an immediate application is to establish ekeland type variational principle and its equivalence caristi fixed point theorem in bornological vector spaces meanwhile since every locally convex space becomes a convex bornological vector space when equipped with the canonical von neumann bornology qiu s generalization of danes work to locally convex spaces is recovered

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

the first order necessary conditions for nonsmooth variational and control problems the first order necessary conditions for nonsmooth problems of lagrange bolza multiple integral and optimal control are given the necessary conditions are expressed in terms of differential inclusions we deal with some new differentials that generalize the differential of frechet in particular the pontryagin maximum principle is proved without the assumption of continuous differentiability

a near subtraction result in metric spaces the locally closed graph case abstract this paper concerns a near subtraction result for multifunctions with locally closed graph in metric spaces

two point eigenvalue correlation for a class of non selfadjoint operators under random perturbations we consider a non selfadjoint h differential model operator ph in the semiclassical limit h rightarrow 0 subject to random perturbations with a small coupling constant assume that rm e frac 1 ch text 0 suitably large let be the closure of the range of the principal symbol

poschl teller paradoxes in the well known poschl teller trigonometric potential well a symmetric regularization x x ie of the impenetrable end point barriers is performed this leads to the four different solvable generalizations of the model as a byproduct the scheme clarifies certain apparent paradoxes encountered in the classically forbidden coupling regime

stability results for ekeland s e variational principle and cone extremal solutions given x a banach space and f x a a proper lower semicontinuous function which is bounded from below the ekeland s e variational principle asserts the existence of a point xi in x which we call e extremal with respect to f which satisfies fu fxi e u xi for all u x u xi by using set convergence notions kuratowski painleve mosco bounded hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e ext f the set of such e extremal points the key for the geometrical understanding of such properties is to consider the equivalent phelps extremization principle which given a closed set d in x and a partial ordering with respect to a pointed cone associates the set of elements of d maximal with respect to this order direct or potential applications are given in various fields multicriteria optimization numerical algorithmic calculus of variations

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

existence of minimizers on drops for a boundedly generated drop a e a property which holds for instance whenever e is bounded where a belongs to a real banach space x and e subset x is a nonempty convex set we show that for every lower semicontinuous function h x longrightarrow mathbb r cup infty that satisfies sup delta 0 inf x in e delta b x h x h a b x being the unitary open ball in x there exists bar x in a e such that h a ge h bar x and bar x is a strict minimizer of h on the drop bar x e

the drop theorem the petal theorem and ekeland s variational principle on montre que le principe variationnel d ekeland ou plus precisement une forme legerement modifiee est une consequence d un resultat geometrique connu sous le terme de theoreme de la goutte

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

some generalizations of ekeland type variational principle with applications to equilibrium problems and fixed point theory abstract in this paper we introduce the concept of a q function defined on a quasi metric space which generalizes the notion of a function and a w distance we establish ekeland type variational principles in the setting of quasi metric spaces with a q function we also present an equilibrium version of the ekeland type variational principle in the setting of quasi metric spaces with a q function we prove some equivalences of our variational principles with caristi kirk type fixed point theorems for multivalued maps the takahashi minimization theorem and some other related results as applications of our results we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps we also extend the nadler s fixed point theorem for multivalued maps to a q function and in the setting of complete quasi metric spaces as a consequence we prove the banach contraction theorem for a q function and in the setting of complete quasi metric spaces the results of this paper extend and generalize many results appearing recently in the literature

polynomial families and schrodinger equation one example for nonhypergeometric type of correspondence we discuss the explicit construction of the schrodinger equations admitting representation through some family of general nonorthogonal polynomials the specific choice of the third order polynomial coefficient functions that lead to quasi solvable families of schrodinger potentials is considered in detail

phase operators temporally stable phase states mutually unbiased bases and exactly solvable quantum systems we introduce a one parameter generalized oscillator algebra that covers the case of the harmonic oscillator algebra and discuss its finite and infinite dimensional representations according to the sign of the parameter we define an hamiltonian operator associated with and examine the degeneracies of its spectrum for the finite when 0 and the infinite when 0 representations of we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators to overcome the difficulties related to the phase operator in the infinite dimensional case and to avoid the degeneracy problem for the finite dimensional case we introduce a truncation procedure which generalizes the one used by pegg and barnett for the harmonic oscillator this yields a truncated generalized oscillator algebra where s denotes the truncation order we construct two types of temporally stable states for as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of two applications are considered in this paper the first concerns physical realizations of and in the context of one dimensional quantum systems with finite morse system or infinite poschl teller system discrete spectra the second deals with mutually unbiased bases used in quantum information

application of polynomial su 1 1 algebra to poschl teller potentials two novel polynomial su 1 1 algebras for the physical systems with the first and second poschl teller pt potentials are constructed and their specific representations are presented meanwhile these polynomial su 1 1 algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the hamiltonians associated with the first and second pt potentials the algebraic approach explores an appropriate new pair of raising and lowing operators k of polynomial su 1 1 algebra as a pair of shift operators of our hamiltonians in addition two usual su 1 1 algebras associated with the first and second pt potentials are derived naturally from the polynomial su 1 1 algebras built by us

ekeland s variational principle in locally convex spaces and the density of extremal points in this paper we prove a general version of ekeland s variational principle in locally convex spaces where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function from this we obtain a number of special versions of ekeland s principle which include all the known extensions of the principle in locally convex spaces moreover we give a general criterion for judging the density of extremal points in the general ekeland s principle which extends and improves the related known results

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