discrete collocation methods for second kind fredholm integral equations the numerical solution of second kind fredholm integral equations by the discrete collocation method and a discrete iterated collocation method is studied orders of convergence for the two discrete methods are given for the case in which an interpolatory quadrature rule of appropriate precision is used if the kernel is sufficiently smooth these orders of convergence will be the same as those of the exact methods some numerical results are given

numerical analysis of an inverse problem for the eikonal equation we are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface this is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so called soner boundary condition the state equation is discretised by a suitable finite difference scheme for which we obtain existence uniqueness and an error bound we set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero the derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques finally we describe some numerical results

infinite horizon problems on stratifiable state constraints sets this paper deals with a state constrained control problem it is well known that unless some compatibility condition between constraints and dynamics holds the value function has not enough regularity or can fail to be the unique constrained viscosity solution of a hamilton jacobi bellman hjb equation here we consider the case of a set of constraints having a stratified structure under this circumstance the interior of this set may be empty or disconnected and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints in such situations the classical pointing qualification hypothesis are not relevant the discontinuous value function is then characterized by means of a system of hjb equations on each stratum that composes the state constraints this result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur

high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

on global superconvergence of iterated collocation solutions to linear second kind volterra integral equations it is well known that in contrast to fredholm integral equations iterated collocation solutions based on collocation at the gauss points to volterra integral equations of the second kind exhibit optimal discrete superconvergence only at the mesh points here we show that some degree of global superconvergence is possible on the entire interval

nonlinear optimal control via occupation measures and lmi relaxations we consider the class of nonlinear optimal control problems ocps with polynomial data i e the differential equation state and control constraints and cost are all described by polynomials and more generally for ocps with smooth data in addition state constraints as well as state and or action constraints are allowed we provide a simple hierarchy of lmi linear matrix inequality relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value under some convexity assumptions the sequence converges to the optimal value of the ocp preliminary results show that good approximations are obtained with few moments

an lp approach to dynamic programming principles for stochastic control problems with state constraints abstract we study a class of nonlinear stochastic control problems with semicontinuous cost and state constraints using a linear programming lp approach first we provide a primal linearized problem stated on an appropriate space of probability measures with support contained in the set of constraints this space is completely characterized by the coefficients of the control system second we prove a semigroup property for this set of probability measures appearing in the definition of the primal value function this leads to dynamic programming principles for control problems under state constraints with general bounded costs a further linearized dpp is obtained for lower semicontinuous costs

regime switching constrained viscosity solutions approach for controlling dam reservoir systems abstract a new stochastic control problem of a dam reservoir system installed in a river is analyzed both mathematically and numerically water balance dynamics of the reservoir are piece wise deterministic and are driven by a stochastic regime switching inflow process the system is controlled to balance among the operation purpose and the internal and downstream environmental conditions finding the optimal operation policy of the system reduces to solving an optimality equation with a discontinuous hamiltonian which is a system of nonlinear degenerate parabolic or hyperbolic equations we show that the optimality equation has at most one constrained viscosity solution and find the solution explicitly under certain conditions the model is applied to numerical computation of the operation policy of an existing dam reservoir system using a high order finite difference scheme the computational results can suggest how the operation policy should be adapted according to the environmental concerns of the river

on reachable sets for a class of nonlinear systems with constraints the determination of the reachable set for a class of nonlinear systems with control and state trajectory constraints is investigated the main result links this problem with the determination of the set of admissible controls for which procedures already exist the paper also gives a procedure to generate an admissible control which steers the system to a reachable state c 1999 academic press

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

necessary optimality conditions for infinite dimensional state constrained control problems abstract this paper is concerned with first order necessary optimality conditions for state constrained control problems in separable banach spaces assuming inward pointing conditions on the constraint we give a simple proof of pontryagin maximum principle relying on infinite dimensional neighboring feasible trajectories theorems proved in 20 further we provide sufficient conditions guaranteeing normality of the maximum principle we work in the abstract semigroup setting but nevertheless we apply our results to several concrete models involving controlled pdes pointwise state constraints as positivity of the solutions are allowed

optimal stock liquidation in a regime switching model with finite time horizon this paper is concerned with a finite horizon optimal selling rule a set of geometric brownian motions coupled by a finite state markov chain is used to characterize stock price movements given a fixed transaction fee the optimal selling rule can be obtained by solving an optimal stopping problem the corresponding value function is shown to be the unique viscosity solution to the associated hjb equations numerical solutions to these equations and their convergence are obtained a numerical example is presented to illustrate the results

state constrained stochastic optimal control problems via reachability approach this paper deals with a class of stochastic optimal control problems socps in the presence of state constraints it is well known that for such problems the value function is in general discontinuous and its characterization by a hamilton jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system here we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions to this end the socp is first translated into a state constrained stochastic target problem then a level set approach is used to describe the backward reachable sets of the new target problem it turns out that these backward reachable sets describe the value function the main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalization however the target problem involves a new state variable and a new control variable that is u

the pontry agin maximum principle for minimax problems of optimal control the subject of this paper is the pontryagin maximum principle for the minimax problem of optimal control that is we will derive the necessary conditions which a control must satisfy in order to be optimal for the problem in which the cost is measured pointwise in time by the essential supremum over t of some function of time the state and the control it is desired to use controls to minimize this cost

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

viscosity solutions and optimal control problems with integral constraints abstract we discuss optimal control problems with integral state control constraints we rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system and prove that the value function although possibly discontinuous is the unique viscosity solution of the constrained boundary value problem for the corresponding hamilton jacobi equation the state constraint is the epigraph of the minimal solution of a second hamilton jacobi equation our framework applies for instance to systems with design uncertainties

uniqueness of constrained viscosity solutions in hybrid control systems we study constrained viscosity solutions with an unbounded growth for a class of first order hamilton jacobi bellman equations arising in hybrid control systems to deal with the boundary constraint and rapid growth of the solutions we construct a particular set of test functions and under very mild conditions establish a comparison theorem which gives the estimate of distance between the subsolution and the supersolution the comparison theorem implies uniqueness of the constrained viscosity solution if its existence is ensured and under some additional assumptions we give an existence result by showing that the value function is a constrained viscosity solution we then apply the obtained uniqueness results to an optimal scheduling problem and finally to stochastic manufacturing systems

a moment equation reformulation of rayleigh ritz theory we present a moment equation rayleigh ritz rr variational theory for the ground state energy an extended fourier transform space can be defined through the moment equation corresponding to a given schrodinger hamiltonian within this extended space we can implement a variational ansatz with respects to configurations of the form in which the energy dependent functions i satisfy the momentum space schrodinger equation ii are uniquely prescribed iii are most likely non integrable and iv yield the physical solution for physical e and values on the basis of this representation one can minimize the energy expectation value with respects to the e and missing moment variables the proposed approach is in sharp contrast to traditional configuration space rr implementations in which the selection of a variational basis is not manifest a priori particularly for problems of spatial dimension greater than one the analysis of one and two dimension problems is presented

superconvergence of the iterated collocation methods for hammerstein equations in this paper we analyse the iterated collocation method for hammerstein equations with smooth and weakly singular kernels the paper expands the study which began in 16 concerning the superconvergence of the iterated galerkin method for hammerstein equations we obtain in this paper a similar superconvergence result for the iterated collocation method for hammerstein equations we also discuss the discrete collocation method for weakly singular hammerstein equations some discrete collocation methods for hammerstein equations with smooth kernels were given previously in 3 18

rapid wavefunction reconstruction through hankel hadamard moments analysis studies by handy and bessis 1985 on the effectiveness of a hankel hadamard determinant quantisation analysis are generalised to include wavefunction reconstruction for a given bosonic ground state wavefunction psi chi es x excellent pointwise convergence to ds d chi can be readily achieved with very few moments this approach also yields very good lower and upper bounds to ds d chi

linear estimates for trajectories of state constrained differential inclusions and normality conditions in optimal control abstract we consider solutions to a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second approximating trajectory x 0 t which satisfies the linear estimate x x ac 0 t k e if one of the following two cases occurs i the initial datum x 0 x 0 is given but lies in a compact set containing only points where the boundary is smooth ii the initial point x 0 of the approximating trajectory x can be chosen arbitrarily subsequently we employ these linear ac estimates to establish conditions for normality of the generalized euler lagrange condition for optimal control problems with state constraints in which we have an integral term in the cost we finally provide an illustrative example which underlines the fact that if conditions i and ii above are not satisfied then we can find a degenerate minimizer

continuous viscosity solutions for nonlocal dirichlet problems with coercive gradient terms in this paper we study existence of solutions of nonlocal dirichlet problems that include a coercive gradient term whose scaling strictly dominates the one of the integro differential operator for such problems the stronger effect of the gradient term may give rise to solutions not attaining the boundary data or discontinuous solutions on the boundary our main result states that under suitable conditions over the right hand side and boundary data there is a unique holder continuous viscosity solution attaining the boundary data in the classical sense this result is accomplished by the construction of suitable barriers which as a byproduct lead to regularity results up to the boundary for the solution

an asymptotic analysis of hierarchical control of manufacturing systems under uncertainty this paper presents an asymptotic analysis of a hierarchical manufacturing system with machines subject to breakdown and repair the rate of change in machine states is much larger than the rate of fluctuation in demand and the rate of discounting of costs and this gives rise to a limiting problem in which the stochastic machine availability is replaced by the equilibrium mean availability the value function for the original problem converges to the value function of the limiting problem moreover the control for the original problem can be constructed from the optimal controls of the limiting problem in a way which guarantees asymptotic optimality of the value function the limiting problem is computationally more tractable and sometimes has a closed form solution

superconvergence of a collocation type method for simple turning points of hammerstein equations approximation d un point de retournement simple de l equation d hammerstein dependant d un parametre

on nonlinear optimal control problems with state constraints this paper is concerned with an optimal control problem where the state is constrained to stay either in a smooth open set omega or in its closure overline omega under a higher order sufficient condition for the viability of omega and overline omega we prove that the optimal cost function v omega is the unique continuous constrained solution of the hamilton jacobi bellman equation furthermore we show that v omega coincides with the optimal cost function v overline omega on omega

large time behavior of solutions of hamilton jacobi equations with periodic boundary data abstract we study the large time behavior of viscosity solutions of hamilton jacobi equations with periodic boundary data on bounded domains we establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity

singular control with state constraints on unbounded domain we study a class of stochastic control problems where a cost of the form e 0 e s l x s ds h y o s d y s is to be minimized over control processes y whose increments take values in a cone y of r p keeping the state process x x b gy in a cone x of r k k p here x e x b is a brownian motion with drift b and covariance e g is a fixed matrix and y is the radon nikodym derivative dy d y let 1 2 trace d 2 b d where d denotes the gradient solutions to the corresponding dynamic programming pde f l v sup g df h 0 y g 1 on x are considered with a polynomial growth condition and are required to be supersolution up to the boundary corresponding to a state constraint boundary condition on 3x under suitable conditions on the problem data including continuity and nonnegativity of l and h and polynomial growth of l our main result is the unique viscosity sense solvability of the pde by the control problem s value function in appropriate classes of functions in some cases where uniqueness generally fails to hold in the class of functions that grow at most polynomially e g when h 0 our methods provide uniqueness within the class of functions that in addition have compact level sets the results are new even in the following special cases 1 the one dimensional case k p 1 x y r 2 the first order case 0 3 the case where l and h are linear the proofs combine probabilistic arguments and viscosity solution methods our framework covers a wide range of diffusion control problems that arise from queueing networks in heavy traffic

shifted 1 n expansion approach to the interaction v r r2 λr2 1 gr2 the authors use the shifted 1 n expansion method to calculate the energy eigenvalues of the non polynomial oscillator v r r2 r2 1 gr2 the results thus obtained are in excellent agreement with exact numerical results it is also shown that the non polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1 n method matches extremely well with the exact analytic results given by supersymmetry

singular perturbations in manufacturing an asymptotic analysis for a large class of stochastic optimization problems arising in manufacturing is presented a typical example of the problems considered in this paper is a production planning problem with random capacity and demand in this example it is assumed that the capacity of the system fluctuates faster than the other quantities the general model considered here also has a fast controlled markov process in its state description by using the difference in the time scales of different quantities the problem is simplified by averaging out the fast process then asymptotically optimal strategies are constructed from the optimal solutions of the limiting problems the proofs of these results use the theory of viscosity solutions to dynamic programming equations however the formal construction of the asymptotically optimal strategies does not require knowledge of this theory

optimal switching in finite horizon under state constraints we study an optimal switching problem with a state constraint the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain we prove that the value function associated with this problem is the limit of value functions associated with unconstrained switching problems with penalized coefficients as the penalization parameter goes to infinity this convergence allows one to set a dynamic programming principle for the constrained switching problem we then prove that the value function is a solution to a system of variational inequalities svi in the constrained viscosity sense we finally prove that uniqueness for our svi cannot hold and we give a weaker characterization of the value function as the maximal solution to this svi all our results are obtained without any regularity assumption on the constraint domain

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

turnpike sets and their analysis in stochastic production planning problems this paper considers optimal infinite horizon stochastic production planning problems with capacity and demand to be finite state markov chains the existence of the optimal feedback control is shown with the aid of viscosity solutions to the dynamic programming equations turnpike set concepts are introduced to characterize the optimal inventory levels it is proved that the turnpike set is an attractor set for the optimal trajectories provided that the capacity is assumed to be fixed at a level exceeding the maximum possible demand conditions under which the optimal trajectories enter the convex closure of the set in finite time are given the structure of turnpike sets is analyzed last but not least it is shown that the turnpike sets exhibit a monotone property with respect to capacity and demand it turns out that the monotonicity property helps in solving the optimal production problem numerically and in some cases analytically

lipschitz regularity for censored subdiffusive integro differential equations with superfractional gradient terms in this paper we are interested in integro differential elliptic and parabolic equations involving nonlocal operators with order less than one and a gradient term whose coercivity growth makes it the leading term in the equation we obtain lipschitz regularity results for the associated stationary dirichlet problem in the case when the nonlocality of the operator is confined to the domain feature which is known in the literature as censored nonlocality as an application of this result we obtain strong comparison principles which allow us to prove the well posedness of both the stationary and evolution problems and steady ergodic large time behavior for the associated evolution problem

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

asymptotic control for a class of piecewise deterministic markov processes associated to temperate viruses we aim at characterizing the asymptotic behavior of value functions in the control of piecewise deterministic markov processes pdmp of switch type under nonexpansive assumptions for a particular class of processes inspired by temperate viruses we show that uniform limits of discounted problems as the discount decreases to zero and time averaged problems as the time horizon increases to infinity exist and coincide the arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics the approach strongly relies on viscosity techniques linear programming arguments and coupling via random measures associated to pdmp as an intermediate step in our approach we present the approximation of discounted value functions when using piecewise constant in time open loop policies

collocation methods using piecewise polynomials for second kind integral equations abstract we consider the numerical solution of second kind fredholm integral equations in one dimension by using the collocation method and its iterated variant the collocation solution will be sought in a space of piecewise polynomials of order r superconvergence results for the iterated collocation solution are known when discontinuous piecewise polynomials are used and the collocation points are taken to be the r gaussian points shifted to each subinterval here we give a corresponding superconvergence result for the iterated collocation solution when continuous piecewise polynomials with no continuity requirements on the derivatives are used the collocation points are taken to be the knots plus the r 2 lobatto points shifted to each subinterval some numerical results are also given

moment problem formulation of the simplified ideal magnetohydrodynamics ballooning equation a fundamentally new method for determining the eigenvalues of linear differential operators is presented the method involves the application of moment analysis and affords a fast and precise numerical algorithm for eigenvalue computation particularly in the intermediate and strong coupling regimes the most remarkable feature of this approach is that it provides exponentially converging lower and upper bounds to the eigenvalues the effectiveness of this method is demonstrated by applying it to an important magnetohydrodynamics problem recently studied by paris auby and dagazian j math phys 27 2188 1986 through the very precise lower and upper bounds obtained this approach gives full support to their analysis

hankel hadamard analysis of quantum potential x2 λx2 1 gx2 the hankel hadamard moment determinant analysis of handy and bessis 1985 is applied to the potential x2 lambda x2 1 gx2 rapidly convergent lower and upper bounds to the ground state energy and first excited state are obtained application of a novel type of pade analysis allows the determination of all other excited states through an orthogonality quantisation prescription

bound state method with elementary product wavefunctions we propose a new version of the multipoint pade technique to solve schrodinger equations in their first order riccati form our compact product representation of wavefunctions interpolates or matches available approximants at several points the mechanism of matching determines the energies and eliminates spurious solutions on v x mu 2x2 lambda x2 1 gx2 its efficiency is illustrated numerically

an escape time criterion for queueing networks asymptotic risk sensitive control via differential games we consider the problem of risk sensitive control of a stochastic network in controlling such a network an escape time criterion can be useful if one wishes to regulate the occurrence of large buffers and buffer overflow in this paper a risk sensitive escape time criterion is formulated which in comparison to the ordinary escape time criteria penalizes exits that occur on short time intervals more heavily the properties of the risk sensitive problem are studied in the large buffer limit and related to the value of a deterministic differential game with constrained dynamics we prove that the game has value and that the value is the viscosity solution of a pde for a simple network the value is computed demonstrating the applicability of the approach

value functions for differential games and control systems with discontinuous terminal cost this paper deals with mayer s problem for control systems and differential games with discontinuous terminal cost there are two main results in the paper the first one says that the value function for control systems can be characterized as the unique solution in suitable sense to the hamilton jacobi bellman equation without any regularity assumptions on the terminal cost for differential games satisfying isaacs s minmax condition the second main result says that the value function is the unique solution to the hamilton jacobi isaacs equation when the terminal cost is semicontinuous this allows to prove the existence of the value under isaacs s condition this paper extends some results already well known in the continuous case

trajectories of differential inclusions with state constraints abstract the paper deals with solutions of a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second trajectory x 0 t which satisfies x x w 1 1 c e 1 ln e as shown by an earlier counterexample this bound is sharp

semicontinuous solutions of hamilton jacobi bellman equations with degenerate state constraints abstract in this paper the value function of a bolza optimal control problem with state constraints is characterized as the unique lower semicontinuous solution of a hamilton jacobi equation the state constraints are given by an arbitrary closed set with possibly empty interior in particular soner s inward pointing condition is extended here to the case of degenerate state constraints

infinite reload options pricing and analysis infinite reload options allow the user to exercise his reload right as often as he chooses during the lifetime of the contract each time a reload occurs the owner receives new options where the strike price is set to the current stock price we consider a modified version of the infinite reload option contract where the strike price of the new options received by the owner is increased by a certain percentage we refer to this new contract as an increased reload option the pricing problem for this modified contract is characterized as an impulse control problem resulting in a hamilton jacobi bellman equation we use fully implicit timestepping and prove that the discretized equations are monotone stable and consistent implying convergence to the viscosity solution we also derive a globally convergent iterative method for solving the non linear discrete equations numerical examples show that both the exercise policy and the option value are very sensitive to the percentage increase in the reload strike

on the schrodinger equation for the interaction x2 λx2 1 gx2 by forming the 6 6 pade approximants to the energy perturbation series for the interaction x2 lambda x2 1 gx2 as obtained from the hypervirial relations the ground and the first three excited state energy levels are calculated for g or 2 the results of the present calculation are in good agreement with those of the variational calculations the authors also present another set of exact solutions to the schrodinger equation for the interaction x2 lambda x2 1 gx2