On L-spectra and essential spectra of second order elliptic operators
نویسندگان
چکیده
The aim of this paper is to investigate spectral properties of second order elliptic operators with measurable coefficients. Namely, we study the problems of L-independence of the spectrum and stability of the essential spectrum. The problem of L-independence of the spectrum for elliptic operators has a long history going back to B. Simon [30] where the question was posed for Schrödinger operators. The main breakthrough was made by R. Hempel and J. Voigt [14] who answered the question in the affirmative for the case that the negative part of the potential is from the Kato class. This result was a starting point for many extensions in different directions [2, 9, 10, 15, 17, 25, 26, 27] (the list is by no means complete). Most of the results deal with cases when the operators are selfadjoint in L and can be defined in all L, 1 6 p <∞. Under these conditions an abstract approach based on a functional calculus was developed by E. B. Davies [9]. In [26] L-independence was established for the Schrödinger operator with form bounded negative part of the potential. In this case the operator exists only in L for p from a certain interval around p = 2. The ideas from [26] were put in a more general context in [25]. Further progress was made by Yu. Semenov [27] who treated selfadjoint elliptic operators with unbounded coefficients, adapting the method from [26]. In the non-symmetric case the
منابع مشابه
Stability of essential spectra of bounded linear operators
In this paper, we show the stability of Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectrum of bounded linear operators on Banach spaces which remain invariant under additive perturbations belonging to a broad classes of operators $U$ such $gamma(U^m)
متن کاملCompact weighted Frobenius-Perron operators and their spectra
In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.
متن کاملPerturbation of Essential Spectra of Exterior Elliptic Problems
For a second-order symmetric strongly elliptic differential operator on an exterior domain in Rn it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptoti...
متن کاملOn the Lp-theory of C0-semigroups associated with second order elliptic operators. II
We study positive C0-semigroups on Lp associated with second order uniformly elliptic divergence type operators with singular lower order terms, subject to a wide class of boundary conditions. We obtain an interval (pmin, pmax) in the Lp-scale where these semigroups can be defined, including the case 2 6∈ (pmin, pmax). We present an example showing that the result is optimal. We also show that ...
متن کامل