2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, exponential stability and norm continuity of the solutions. Applications to a reaction-diffusion equation with delay are given.

Let O c U n be an open set and let Tp — (Tp(t))>0 be consistent semigroups on L P (Q) for 1 ^p < oo, with generators Ap. Assume that Tp is an analytic semigroup of angle <p for some />oe(l, oo). It is then natural to ask under which conditions the semigroups Tp are analytic too. For the time being, suppose that Tp and Tp are

Let U = (U(t); t 0) be a substochastic strongly continuous semigroup on L 1 (X; m) where X is locally compact and m a Borel measure on X . We give conditions on absorption rates V implying that the (strong) Feller property carries over from U to U V . These conditions are essentially in terms of the Kato class associated with U . Preparing these results we discuss the perturbation theory of strongly continuous semigroups and properties of one-parameter semigroups on L1 (m). In the symmetric case of Dirichlet forms we generalize the results to measure perturbations. For the case of the heat equation on R d we show that the results are close to optimal. Introduction Let X be a locally compact space, m a Radon measure on X, U = (U(t); t 0) a strongly continuous symmetric sub-Markov semigroup on L 2 (m) (i.e., a semigroup associated with a Dirichlet form in L 2 (m).) Assume further that the semigroup U1 induced by U on L1 (m) satisfies the Feller property, i.e., the restrictio...

Let\Omega be an open subset of IR d and let T p for p 2 [1; 1) be consistent C 0 -semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting by A p their generators, we show that the spectrum oe(A p ) is independent of p 2 [1; 1). We also treat the case of weighted L p -spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight. We apply our result to Schrodinger operators, Petrovskij correct systems with Holder continuous coefficients, and elliptic operators in divergence form with real, but not necessarily symmetric coefficients and with complex coefficients. 1 Introduction and Main Results Let\Omega be an open subset of IR d and A be a closed linear operator in L 2(\Omega\Gamma := L 2(\Omega ; dx) where dx denotes Lebesgue measure. Assume that A generates a C 0 -semigroup T in L 2(\Omega\Gamma which induces consistent C 0 - semigroups T p with generators ...

Let fT p : q 1 p q 2 g be a family of consistent C 0 semigroups on L p(\Omega\Gamma8 with q 1 ; q 2 2 [1; 1) and\Omega ` R n open. We show that certain commutator conditions on T p and on the resolvent of its generator A p ensure the p independence of the spectrum of A p for p 2 [q 1 ; q 2 ]. Applications include the case of Petrovskij correct systems with Holder continuous coefficients, Schrodinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients. AMS Subject Classification: 35 J 45, 35 J 55, 47 D 06. Key Words: L p spectrum, spectral independence, elliptic systems. 0 Introduction Under what circumstances is the L p -spectrum of an elliptic operator acting in L p(\Omega\Gamma4 where \Omega ae R n is an open set, independent of p 2 [1; 1)? This question attracted new attention when B. Simon conjectured the p-independence of the spectrum of Schrodinger operators with Kato class potentials [...

Dedicated with great pleasure to Vladimir A. Marchenko on the occasion of his 80th birthday. Abstract. We characterize the spectrum of one-dimensional Schrödinger operators H = −d 2 /dx 2 + V in L 2 (R; dx) with quasi-periodic complex-valued algebro-geometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg–de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The corresponding problem appears to have been open since the mid-seventies. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green’s function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These

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