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35
Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
, 2007
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ON SOME SCHRÖDINGER AND WAVE EQUATIONS WITH TIME DEPENDENT POTENTIALS
"... Abstract. The existence and uniqueness of the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a b ..."
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Abstract. The existence and uniqueness of the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate. 1.
Finite element approximations for Schrödinger equations with applications to electronic structure computations
 J. Comput. Math
"... Dedicated to the 70th birthday of Professor Junzhi Cui In this paper, both the standard finite element discretization and a twoscale finite element discretization for Schrödinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Sc ..."
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Dedicated to the 70th birthday of Professor Junzhi Cui In this paper, both the standard finite element discretization and a twoscale finite element discretization for Schrödinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schrödinger equations. Very satisfying applications to electronic structure computations are provided, too.
Safronov Absolutely continuous spectrum of a class of random non ergodic Schrödinger operators Int
 Math. Res. Not. vol
, 2005
"... ABSTRACT. We consider a class of random Schrödinger operators with nondecaying potentials and prove that their absolutely continuous spectrum almost surely fills the positive halfline. We establish the existence of the wave operators using the trace class scattering theory. 1. ..."
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ABSTRACT. We consider a class of random Schrödinger operators with nondecaying potentials and prove that their absolutely continuous spectrum almost surely fills the positive halfline. We establish the existence of the wave operators using the trace class scattering theory. 1.
Numerical Solutions of the Spectral Problem for arbitrary SelfAdjoint Extensions of the OneDimensional Schrödinger Equation
 SIAM J. Numer. Anal
"... Abstract. A numerical algorithm to solve the spectral problem for arbitrary self–adjoint extensions of 1D regular Schrödinger operators is presented. The construction of all self–adjoint extensions of the symmetric Schrödinger operator on a compact manifold of arbitrary dimension with boundary is ..."
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Abstract. A numerical algorithm to solve the spectral problem for arbitrary self–adjoint extensions of 1D regular Schrödinger operators is presented. The construction of all self–adjoint extensions of the symmetric Schrödinger operator on a compact manifold of arbitrary dimension with boundary is discussed. The self–adjoint extensions of such symmetric operators are shown to be in one–to–one correspondence with the group of unitary operators on a Hilbert space of boundary data, refining in this way wellknown theorems on the existence of self–adjoint extensions for Laplace–Beltrami operators. The corresponding self–adjoint extensions are characterized by a generalized class of boundary conditions that include the well–known Dirichlet, Neumann, Robin boundary conditions, etc. Only the numerical solution of 1D regular cases are consider in this paper. They constitute however a non–trivial problem. The corresponding numerical algorithms are constructed and their convergence is proved. An appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions as described by the general theory. Some significant numerical experiments are also discussed. Contents
May 2006A Wavelet Based Sparse Grid Method for the Electronic Schrödinger Equation
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A mathematical introduction to HartreeFock SCF methods in Quantum Chemistry
, 2008
"... It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the manybody problem. — Max Born, 1960 We present here an introduction to HartreeFock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discuss ..."
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It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the manybody problem. — Max Born, 1960 We present here an introduction to HartreeFock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the BornOppenheimer approximation, we arrive to the Hartree and HartreeFock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. Additionally, the discretization of the oneelectron orbitals space is reviewed and the RoothaanHall formalism introduced. Finally, the basic underlying concepts related to the construction and selection of Gaussian basis sets are discussed focusing in algorithmic efficiency issues. The whole work is intentionally introductory and rather selfcontained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.