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Register a new userWeighted analyticity of Hartree-Fock eigenfunctions
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We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical solution of such eigenvalue problems.
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We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrodinger equation with power-like nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein-Gordon equations with power nonlinearity. Finally, we discuss the link of this approach with inverse scattering, and with complete integrability.
We consider an electronic bound state of the usual, non-relativistic, molecular Hamiltonian with Coulomb interactions and fixed nuclei. Away from appropriate collisions, we prove the real analyticity of all the reduced densities and density matrices, that are associated to this bound state. We provide a similar result for the associated reduced current density.
In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in [5,19] as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.
We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.
Consider the Dirichlet-Laplacian in $Omega:= (0,L)times (0,H)$ and choose another open set $omegasubset Omega$. The estimate $0<C_{omega}leq R_{omega}(u):=frac{Vert uVert^{2}_{L^{2}(omega)}}{Vert uVert^{2}_{L^{2}(Omega)}}leq frac{vol(omega)}{vol(omega)}$, for all the eigenfunctions, is well known. This is no longer true for an inhomogeneous elliptic selfadjoint operator $A$. In this work we create a partition among the set of eigenfunctions: $forall omega$, the eigenfunctions satisfy $R_{omega}>C_{omega}>0,exists omega, omega ot=emptyset$, such that $inf R_{omega}(u)=0$,and we wish to characterize these two sets. For two patterns we give a sufficient condition, sometimes necessary. As our operator corresponds to a layered media we can give another representation of its spectrum: i.e. a subset of points of $Rtimes R$ that leads to the suggested partition and others connected results: micro local interpretation, default measures,... Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added.
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