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Register a new userConcentration and confinement of eigenfunctions in a bounded open set (version 2)
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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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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.
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 study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for any given probability density $e(x)$, the diffusion coefficient can be built to have $e(x)$ as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density $e(x) $, a polynomial rate of convergence to equilibrium.Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.
It is known from clever mathematical examples cite{Ca10} that the Monge ansatz may fail in continuous two-marginal optimal transport (alias optimal coupling alias optimal assignment) problems. Here we show that this effect already occurs for finite assignment problems with $N=3$ marginals, $ell=3$ sites, and symmetric pairwise costs, with the values for $N$ and $ell$ both being optimal. Our counterexample is a transparent consequence of the convex geometry of the set of symmetric Kantorovich plans for $N=ell=3$, which -- as we show -- possess 22 extreme points, only 7 of which are Monge. These extreme points have a simple physical meaning as irreducible molecular packings, and the example corresponds to finding the minimum energy packing for Frenkel-Kontorova interactions. Our finite example naturally gives rise, by superposition, to a continuous one, where failure of the Monge ansatz manifests itself as nonattainment and formation of microstructure.
In this article we discuss generalized harmonic confinement of massless Dirac fermions in (2+1) dimensions using smooth finite magnetic fields. It is shown that these types of magnetic fields lead to conditional confinement, that is confinement is possible only when the angular momentum (and parameters which depend on it) assumes some specific values. The solutions for non zero energy states as well as zero energy states have been found exactly.
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