We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrodinger equationi∂u/∂t=(−1/2)Δu+V(t,x)u,u(0)=u
0. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u
0eC
1(ℝ,L
2) ∩C
0(ℝH
2(ℝ
 n
 )) foru
0eH
2(ℝ
 n
 ). The conditions are general enough to accommodate moving singularities of type ∣x∣−2+ɛ(n≧4) or ∣x∣−n/2+ɛ(n≧3).

/pdf/existence-of-solutions-for-schrodinger-evolution-equations-jtjbfvmvwd.pdf

Existence of solutions for Schrödinger evolution equations

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.

/pdf/a-stable-multi-scale-kernel-for-topological-machine-learning-17swwheaff.pdf

A stable multi-scale kernel for topological machine learning

Publisher Summary This chapter discusses basic modeling. The chapter illustrates that quantum chemistry aims at understanding the properties of matter through the modeling of its behavior at a subatomic scale, where matter is described as an assembly of nuclei and electrons. At this scale, the equation that rules the interactions between these constitutive elements is the Schrodinger equation. It can be considered as a universal model for at least three reasons. First, it contains all the physical information of the system under consideration so that any of the properties of this system can be deduced in theory from the Schrodinger equation associated to it. Second, the Schrodinger equation does not involve any empirical parameter, except some fundamental constants of Physics; it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types.

Computational quantum chemistry: A primer

We present an overview of some works on the models of computational quantum chemistry. We examine issues such as the existence of ground states (both for the electronic structure and the configuration of nuclei), the foundations of the models of the crystalline phase, and the macroscopic limits. We emphasize the connections between the physical modelling, the numerical concerns and the mathematical analysis of the problems. 1. Outline This article presents an expository review of some collective work by a group of researchers (X. Blanc, E. Cancès, I. Catto, M. Esteban, E. Séré) to which we belong on the mathematical aspects of the models used for the simulation of matter at the microscopic scale. The relation of these models to models for the continuum description of matter at the macroscopic scale will also be examined. The mathematical material reported on here covers almost two decades, starting in the late 1980s with our contribution to a general endeavour (mainly initiated by E.H. Lieb, B. Simon, W. Thirring) to put the models of quantum chemistry on a sound mathematical basis (see Sections 3 to 5). It continues in the 1990s with our work on the thermodynamic limit of molecular models in order to establish and/or justify models for the crystal phase (see Section 6). The most recent step (in fact hardly addressed here, only outlined in Section 7, and postponed until future publications) is the definition of the energy of general infinite microscopic sets of particles and the passage to the macroscale with the tentative definition of the density of mechanical energy on the basis of models at the atomistic scale. The program described above is ambitious. However, we are in a position where on the one hand some building blocks of the whole program are indeed known from the mathematical standpoint for most of the models used in practice, and on the other hand where we are able to accomplish all the steps for some models, rather academic in nature but nevertheless illustrative and of some physical relevance. Therefore, we believe that this is the proper time for such a survey article. Enough convincing arguments are available to prove that it is indeed possible to analyze models of microscopic matter and to determine the macroscopic limit, on a mathematical ground, provided one restricts one’s ambition. A large body Received by the editors November 20, 2004. 2000 Mathematics Subject Classification. Primary 35Bxx, 35Jxx, 35Pxx, 49Kxx, 81Q05, 81Q10, 82Bxx. This article is an extended version by the two authors of notes based upon a series of lectures given by PLL at Collège de France during the fall semester of the academic year 2003/04. c ©2005 American Mathematical Society Reverts to public domain 28 years from publication

/pdf/from-atoms-to-crystals-a-mathematical-journey-3isxfztnzz.pdf

From atoms to crystals: a mathematical journey

This paper is concerned with the stability of periodic travelling-wave solutions of the Korteweg-de Vries equation $$ u_t+uu_x+u_{xxx}=0. $$ Here, $u$ is a real-valued function of the two variables $x,t\in{\mathbb R}$ and subscripts connote partial differentiation. These special solutions were termed cnoidal waves by Korteweg and de Vries. They also appear in earlier work of Boussinesq. It is shown that these solutions are stable to small, periodic perturbations in the context of the initial-value problem. The approach is that of the modern theory of stability of solitary waves, but adapted to the periodic context. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

/pdf/stability-of-cnoidal-waves-40uz1e7awr.pdf

Stability of cnoidal waves

Part I. Fourier Series and Periodic Distributions: 1. Preliminaries 2. Fourier series: basic theory 3. Periodic distributions and Sobolev spaces Part II. Applications to Partial Differential Equations: 4. Linear equations 5. Nonlinear evolution equations 6. The Korteweg-de Vries Part III. Some Nonperiodic Problems: 7. Distributions, Fourier transforms and linear equations 8. KdV, BO and friends Appendix A. Tools from the theory of ODEs Appendix B. Commutator estimates Bibliography Index.

Fourier analysis and partial differential equations

We discuss the local Cauchy problem for the generalised Kadomtsev–Petviashvili equation, namely , in the periodic and nonperiodic settings.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500021740

On equations of KP-type

The Benjamin-Ono equation in weighted Sobolev spaces

KDV, BO and friends in weighted Sobolev spaces

We prove existence and uniqueness of solutions of i (∂ψ/∂ t ) = (−Δ+ x 1 g ( t )+ q ( x ))ψ, ψ( x , s ) = ψ s ( x ) in ℝ 3 for potentials q ( x ) including the Coulomb case. Existence and completeness of the wave operators is established for g ( t ) periodic with zero mean and q ( x ) short-range, smooth in the x 1 direction. We characterize scattering and bound states in terms of the period operator.

Rafael José Iorio

Papers

Fourier analysis and partial differential equations

On equations of KP-type

The Benjamin-Ono equation in weighted Sobolev spaces

KDV, BO and friends in weighted Sobolev spaces

On the Schrödinger equation with time-dependent electric fields