fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

derivation of the time dependent propagator for the three dimensional schrodinger equation with one point interaction an explicit formula for the time dependent propagator of the schrodinger equation with one point interaction in three dimensions is given the derivation is based on the inverse laplace transformation applied to the corresponding resolvent

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

accelerated gradient methods for nonconvex nonlinear and stochastic programming in this paper we generalize the well known nesterov s accelerated gradient ag method originally designed for convex smooth optimization to solve nonconvex and possibly stochastic optimization problems we demonstrate that by properly specifying the stepsize policy the ag method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first order information similarly to the gradient descent method we then consider an important class of composite optimization problems and show that the ag method can solve them uniformly i e by using the same aggressive stepsize policy as in the convex case even if the problem turns out to be nonconvex we demonstrate that the ag method exhibits an optimal rate of convergence if the composite problem is convex and improves the best known rate of convergence if the problem is nonconvex based on the ag method we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization to the best of our knowledge this is the first time that the convergence of the ag method has been established for solving nonconvex nonlinear programming in the literature

the functional analytic versus the functional integral approach to quantum hamiltonians the one dimensional hydrogen atoma the capabilities of the functional analytic and of the functional integral approach for the construction of the hamiltonian as a self adjoint operator on hilbert space are compared in the context of non relativistic quantum mechanics differences are worked out by taking the one dimensional hydrogen atom as an example that is a point mass on the euclidean line subjected to the inverse distance potential this particular choice is made with the intent to clarify a long lasting discussion about its spectral properties in fact for the four parameter family of possible hamiltonians the corresponding energy dependent green functions are derived in closed form the multiplicity of hamiltonians should be kept in mind when modeling certain experimental situations as for instance in quantum wires

spectral splitting method for nonlinear schrodinger equations with singular potential we consider the time dependent one dimensional nonlinear schrodinger equation with pointwise singular potential by means of spectral splitting methods we prove that the evolution operator is approximated by the lie evolution operator where the kernel of the lie evolution operator is explicitly written this result yields a numerical procedure which is much less computationally expensive than multi grid methods previously used furthermore we apply the lie approximation in order to make some numerical experiments concerning the splitting of a soliton interaction among solitons and blow up phenomenon

homotopy algorithm for symmetric eigenvalue problems the homotopy method can be used to solve eigenvalue eigenvector problems the purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results in practice it is rerely of any interest to compute all the eigenvalues the homotopy method having the order preserving property can provide any specific eigenvalue without calculating any other eigenvalues besides this advantage we note that the homotopy algorithm is to a large degree a parallel algorithm numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices

explicit derivation of the propagator for a dirac delta potential an explicit derivation of the propagator is given for a particle in a dirac delta potential lambda delta x by making a direct application of a variational technique followed by a systematic use of laplace transforms although the solution of the corresponding schrodinger equation has been known for years the solution of the propagator has been obtained only recently

a new class of point interactions in one dimension we present a class of self adjoint extensions of the symmetric operator c 0 r1 0 which correspond formally to perturbations of the laplacian by pseudopotentials involving 2 these operators which provide new examples of generalized point interactions in the sense of seba are defined by the boundary conditions 0 e z 0 r 0 0 ez r 0 0 for z c r r we calculate their spectra resolvents and scattering matrices and show that they can be realized as limits of schrodinger operators with local short range potentials

boundary conditions in path integrals from point interactions for the path integral of the one dimensional dirac particle a proposal is given of how to implement point interactions and boundary conditions in the path integral the starting point is a path integral formulation of a dirac particle in one dimension the implementation of a point interaction yields by means of a perturbation expansion the corresponding green function for a relativistic particle in the non relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen by a proper combination of the various possibilities of implementing the point interaction the whole range of the four parameter family of boundary conditions for point interactions can be exploited in the relativistic case as well as in the non relativistic limit in addition making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line in particular dirichlet and neumann boundary conditions emerge the method is illustrated with some examples

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

distribution theory for schrodinger s integral equation much of the literature on point interactions in quantum mechanics has focused on the differential form of schrodinger s equation this paper in contrast investigates the integral form of schrodinger s equation while both forms are known to be equivalent for smooth potentials this is not true for distributional potentials here we assume that the potential is given by a distribution defined on the space of discontinuous test functions first by using schrodinger s integral equation we confirm a seminal result by kurasov which was originally obtained in the context of schrodinger s differential equation this hints at a possible deeper connection between both forms of the equation we also sketch a generalisation of kurasov s j math anal appl 201 1 297 323 1996 result to hypersurfaces second we derive a new closed form solution to schrodinger s integral equation with a delta prime potential this potential has attracted considerable attention including some controversy interestingly the derived propagator satisfies boundary conditions that were previously derived using schrodinger s differential equation third we derive boundary conditions for super singular potentials given by higher order derivatives of the delta potential these boundary conditions cannot be incorporated into the normal framework of self adjoint extensions we show that the boundary conditions depend on the energy of the solution and that probability is conserved this paper thereby confirms several seminal results and derives some new ones in sum it shows that schrodinger s integral equation is a viable tool for studying singular interactions in quantum mechanics

statistical inference for the population landscape via moment adjusted stochastic gradients modern statistical inference tasks often require iterative optimization methods to compute the solution convergence analysis from an optimization viewpoint informs us only how well the solution is approximated numerically but overlooks the sampling nature of the data in contrast recognizing the randomness in the data statisticians are keen to provide uncertainty quantification or confidence for the solution obtained by using iterative optimization methods the paper makes progress along this direction by introducing moment adjusted stochastic gradient descent a new stochastic optimization method for statistical inference we establish non asymptotic theory that characterizes the statistical distribution for certain iterative methods with optimization guarantees on the statistical front the theory allows for model misspecification with very mild conditions on the data for optimization the theory is flexible for both convex and non convex cases remarkably the moment adjusting idea motivated from error standardization in statistics achieves a similar effect to acceleration in first order optimization methods that are used to fit generalized linear models we also demonstrate this acceleration effect in the non convex setting through numerical experiments

proper incorporation of the self adjoint extension method to the green function formalism one dimensional function potential case the one dimensional function potential is discussed in the framework of the green function formalism without invoking perturbation expansion it is shown that the energy dependent green function for this case is crucially dependent on the boundary conditions which are provided by the self adjoint extension method the most general green function which contains four real self adjoint extension parameters is constructed the relation between the bare coupling constant and the self adjoint extension parameter is also derived

dispersion and strichartz estimates for some finite rank perturbations of the laplace operator we consider the dispersion properties in lp spaces of schrodinger hamiltonians with a large number of obstacles modelled by rank one perturbations we obtain both for the dispersion an strichartz estimates nonperturbative results with respect to the coupling constants

time dependent propagator with point interaction we compute the time dependent schrodinger propagator with a point interaction in dimension n or 3 including the new cases of n 2 and the most general interaction supported by a point for n 1 we also give the small time asymptotics for n or 3 the case n 2 has the peculiarity of involving logarithmic terms in the expansion

the transition from diffusion to blow up for a nonlinear schrodinger equation in dimension 1 we consider the time dependent one dimensional nonlinear schrodinger equation with a pointwise singular potential we prove that if the strength of the nonlinear term is small enough then the solution is well defined for any time regardless of the choice of initial data in contrast if the nonlinearity power is larger than a critical value for some initial data a blow up phenomenon occurs in finite time in particular if the system is initially prepared in the ground state of the linear part of the hamiltonian then we obtain an explicit condition on the parameters for the occurrence of the blow up

heat kernel expansion for semitransparent boundaries we study the heat kernel for an operator of laplace type with a function potential concentrated on a closed surface we derive the general form of the small t asymptotics and explicitly calculate several first heat kernel coefficients

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

global convergence of online limited memory bfgs global convergence of an online stochastic limited memory version of the broyden fletcher goldfarb shanno bfgs quasi newton method for solving optimization problems with stochastic objectives that arise in large scale machine learning is established lower and upper bounds on the hessian eigenvalues of the sample functions are shown to suffice to guarantee that the curvature approximation matrices have bounded determinants and traces which in turn permits establishing convergence to optimal arguments with probability 1 experimental evaluation on a search engine advertising problem showcase reductions in convergence time relative to stochastic gradient descent algorithms