Pointwise Properties of Eigenfunctions and Heat Kernels of Dirichlet–Schrödinger Operators

Let D be an open domain, a second order elliptic operator with continuous coefficients, and let be a Schrödinger operator associated with H₀, acting on L²(D), with Dirichlet boundary conditions. We provide in this paper both L² and pointwise bounds for the eigenfunctions of H, in term of the Agmon's metric of q and of the quasi-hyperbolic geometry of D. At least when H₀=-, we show that the pointwise bounds obtained for the ground state eigenfunction of H are qualitatively sharp either when q diverges sufficiently fast at the boundary, or in planar regular domains. We also give applications to the intrinsic ultracontractivity of H. Finally, we prove a result concerning the pointwise decay at the boundary of the heat kernel of H in -regular domains.     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.     

Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. In the case of real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.     

On Pointwise Convergence for Schrödinger Operator in a Convex Domain

Journal of Fourier Analysis and Applications - In this paper, we prove that the maximal inequality $$begin{aligned} big Vert sup _{|t|<1}|e^{itDelta _D}f(x,y)|big Vert...     

Some Properties of Eigenfunctions of the Schrödinger Operator in a Magnetic Field

We study the behavior of eigenfunctions corresponding to a positive point spectrum of the Schrödinger operator with magnetic and electric potentials.     

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Abstract Theory of Pointwise Decay with Applications to Wave and Schrödinger Equations

We prove pointwise in time decay estimates via an abstract conjugate operator method. This is then applied to a large class of dispersive equations.     

Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schr?dinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr?dinger-type equations. In particular, the sharp G?rding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Received: 14 December 1998     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.     

Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

Loss of regularity for supercritical nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional à la Brenier.     

Kernel estimates for a class of Schrödinger semigroups

We prove short time estimates for the heat kernels of certain Schr?dinger operators with unbounded potentials in . The asymptotic distribution of the eigenvalues is also considered.