On the essential spectrum of Schrödinger operators on Riemannian manifolds

The main result of this paper is a lower bound for the essential spectrum of Schrödinger operators −Δ+V on Riemannian manifolds. In particular, we obtain conditions on V which imply the discreteness of the spectrum, or equivalently, the compactness of the resolvent.     

Fractional integration associated with second order divergence operators on R~n

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (-Δ) -α/2 are extended to the generalised fractional integrals L-α/2 for 0 < α < n, where L =-div A is a uniformly complex elliptic operator with bounded measurable coefficients in Rn.     

On the Fučik spectrum of non-local elliptic operators

In this article, we study the Fu?ik spectrum of the fractional Laplace operator which is defined as the set of all \({(\alpha, \beta)\in \mathbb{R}^2}\) such that $$\quad \left.\begin{array}{ll}\quad (-\Delta)^s u = \alpha u^{+} - \beta u^{-} \quad {\rm in}\;\Omega \\ \quad \quad \quad u = 0 \quad \quad \quad \qquad {\rm in}\; \mathbb{R}^n{\setminus}\Omega.\end{array}\right\}$$ has a non-trivial solution u, where \({\Omega}\) is a bounded domain in \({\mathbb{R}^n}\) with Lipschitz boundary, n > 2s, \({s \in (0, 1)}\) . The existence of a first nontrivial curve \({\mathcal{C}}\) of this spectrum, some properties of this curve \({\mathcal{C}}\) , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fu?ik spectrum.     

Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

By variational methods and Morse theory, we prove the existence of uncountably many \((\alpha ,\beta )\in \mathbb R ^2\) for which the equation \(-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}\) in \(\Omega \) , has a sign changing solution under the Neumann boundary condition, where a map \(A\) from \(\overline{\Omega }\times \mathbb R ^N\) to \(\mathbb R ^N\) satisfying certain regularity conditions. As a special case, the above equation contains the \(p\) -Laplace equation. However, the operator \(A\) is not supposed to be \((p-1)\) -homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator \(-\mathrm{div}\, A(x, \nabla u)\) on \(W^{1,p}(\Omega )\) contains some open unbounded subset of \(\mathbb R ^2\) .     

Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods

Given an open domain (possibly unbounded) Ω?R ⁿ , we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L ¹(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.     

A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators

Summary An integral inequality for the eigenfunctions of linear second order elliptic operators in divergence form is proved. The result is a generalization of the Payner-Rayner inequality.

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Résumé On démontre une inégalité intégrale pour les fonctions propres d'une classe d'opérateurs linéaires élliptiques du deuxième ordre. Le résultat est une généralization de l'inégalité de Payner-Rayner.

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This study was performed within the G.N.A.F.A. of the Italian C.N.R.     

Toeplitz operators on hardy spaceH p (S) with 0<p≤1

LetB ⁿ be the unit ball inC ⁿ ,S is the boundary ofB ⁿ . We letL ^p (S) denote the usual Lebesgue spaces overS with respect to the normalized surface measure,H ^p (B ⁿ ) is its usua holomorphic subspace.H ^p (S) denotes the atomic Hardy spaces defined in [GL]. LetPL ² (S)H ²(B ⁿ ) denote the orthogonal projection. For eachfL (S), we useM _f L ^p (S)L ^p (S) to denote the multiplication operator, and we define the Toeplitz operatorT _f =PM _f . The paper gives a characterization theorem onf such that the Toeplitz operatorsT _f and are bounded fromH ^p (S)H ^p (B ⁿ ) with 0<p1. Also several equivalent conditions are given.     

Essential spectrum of Schrödinger operators with δ-interactions on the union of compact Lipschitz hypersurfaces

In this note we prove that the essential spectrum of a Schrödinger operator with δ-potential supported on a finite number of compact Lipschitz hypersurfaces is given by [0, +∞). We emphasize that the union of a family of Lipschitz hypersurfaces is in general not Lipschitz. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On essential self-adjointness for Schrödinger operators with wildly oscillating potentials

A sufficient condition is given for the operator T₀: C^∞₀(R^m) → L²(R^m) given by

$\text{T}{}_0\text{∑}\text{K?1}\text{M}\text{(}\text{i?}\text{?x}{}_1\text{+b}{}_1\text{)a}{}_{\mathrm{1K}}\text{(}\text{i?}\text{?N}{}_{\mathrm{k}}\text{+b}{}_{\mathrm{k}}\text{)+q}$

to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞.     

Compactness and essential norms of composition operators on Orlicz–Lorentz spaces

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.     

The spectrum is continuous on the set of p–hyponormal operators

In this paper it is shown that the spectrum , a set valued function, is continuous when the function is restricted to the set of all p-hyponormal operators on a Hilbert space. Received November 9, 1998; in final form August 6, 1999 / Published online July 3, 2000     

Second order Hamilton–Jacobi–Bellman equations with an unbounded operator

This work is devoted to the study of a class of Hamilton–Jacobi–Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique.     

Weighted eigenvalue problems for quasilinear elliptic operators with mixed Robin–Dirichlet boundary conditions

We investigate the existence of principal eigenvalues type problems with weights for the quasilinear operator −_Δp+V_ψp$-{\mathrm{\Delta }}_p+V{\psi }_p$ with mixed weighted Robin–Dirichlet boundary conditions in a bounded regular domain. We also give some results on the existence of nonprincipal eigenvalues.     

Strongly irreducible operators and Cowen-Douglas operators on c 0, l p (1 ? p < ��)

This paper gives some sufficient conditions for the strongly irreducibility of operators which have the forms of upper triangular operator matrices on Banach spaces. Based on these results, strongly irreducible Cowen-Douglas operators of index n are constructed on c₀, l_p (1≤p<∞) for all 1≤n≤∞.     

The spectral asymptotics of elliptic operators of Schrödinger type on a hyperbolic space

For self-adjoint second-order elliptic differential operators that satisfy the non-trapping condition on the n-dimensional hyperbolic space H ⁿ and coincide with the operator in a neighborhood of infinity, where is the Laplace-Beltrami operator on H ⁿ ,we obtain the complete asymptotic expansion of the spectral function as +.For self-adjoint operators of the form (–) +Q _m–r,where Q _m–r is a pseudodifferential operator of order m–r that is automorphic with respect to a discrete group of isometries of the spaceH ⁿ whose fundamental domain has finite volume, we introduce the spectral distribution function N(),which is the analog of the integrated state density, and we find its asymptotics up to order O(^(n–r)/m)as +.Bibliography: 49 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 4–32, 1991.     

Cosine families generated by second-order differential operators on W1,1(0, 1) with generalized Wentzell boundary conditions

Using the abstract framework [Bátkai, A. and Engel, K.-J., 2004, Abstract wave equations with generalized Wentzell boundary conditions. Journal of Differential Equations, 207, 1–20.] we show that certain second-order differential operators with generalized Wentzell boundary conditions generate cosine families and hence also analytic semigroups on W^1,1(0,1). This complements the main result [Favini, A., Ruiz Goldstein, G., Goldstein, J.A., Obrecht, E. and Romanelli, S., 2003, General Wentzell boundary conditions and analytic semigroups on W^{1, p} (0,1). Applicable Analysis, 82, 927–935.] on the generation of an analytic semigroup by the second derivative with generalized Wentzell boundary conditions on W^{1, p} (0,?1) for 1<p<∞.