Related papers: A Borg-Levinson theorem for higher order elliptic …

This paper proves new results on spectral and scattering theory for matrix-valued Schr\"odinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson…

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in $\mathbb{R}^n$…

We consider dissipative Schroedinger operators of the form $H=-\Delta+V(x)$ on $L^2(\mathbb R^3)$, with $V(x)$ a complex, bounded and decaying potential with a non-positive imaginary part. We prove a topological version of Levinson's…

In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic…

We give a simple way to extend index-theoretical statements from partial differential operators with smooth coefficients to operators with coefficients of finite Sobolev order.

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order…

Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable,…

We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…

In this paper we present examples of nondivergence form second order elliptic operators with continuous coefficients such that $L$ has an irregular boundary point that is regular for the Laplacian. Also for any eigenvalue spread <1 of the…

The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the…

In this paper we classify M\"{o}bius invariant differential operators of second order in two dimensional Euclidean space, and establish a Liouville type theorem for general M\"{o}bius invariant elliptic equations.

We prove a variant of the so-called bilinear embedding theorem for operators in divergence form with complex coefficients and with nonnegative locally integrable potentials, subject to mixed boundary conditions, and acting on arbitrary open…

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…

In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We consider the two closely related topics of inhomogeneous problems and problems with boundary data in fractional…

We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the…

A Borg-type uniqueness theorem for matrix-valued Schr\"odinger operators is proved. More precisely, assuming a reflectionless potential matrix and spectrum a half-line $[0,\infty)$, we derive triviality of the potential matrix. Our approach…

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases…

In this paper, we study an inverse spectral problem for the fourth-order differential equation $y^{(4)} - (p y')' + q y = \lambda y$ with real-valued coefficients $p$ and $q$ of $L^2(0,1)$. We prove that, for near-constant coefficients, the…

We propose to interpret Levinson's theorem as an index theorem. This exhibits its topological nature. It furthermore leads to a more coherent explanation of the corrections due to resonances at thresholds.