Spectral theory of higher order differential operators by examples

Lp SPECTRAL THEORY OF HIGHER-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

Topics from spectral theory of differential operators

Introduction 2 1. Self-adjointness of Schrödinger operators 3 1.0. Solving the Schrödinger equation 3 1.1. Linear operators in Hilbert space 7 1.2. Criteria for (essential) self-adjointness 13 1.3. Application to Schrödinger operators 18 2. Hardy-Rellich inequalities 21 2.0. Relative boundedness 21 2.1. Weighted estimates 24 2.2. Explicit bounds 26 3. Spectral properties of radially periodic Sc...

Operators of higher order

Motivated by results on interactive proof systems we investigate the computational power of quantifiers applied to well-known complexity classes. In special, we are interested in existential, universal and probabilistic bounded error quantifiers ranging over words and sets of words, i.e. oracles if we think in a Turing machine model. In addition to the standard oracle access mechanism, we also ...

Spectral Order and Isotonic Differential Operators of Laguerre-pólya Type

Abstract. The spectral order on Rn induces a natural partial ordering on the manifold Hn of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre-Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space l of real bounded sequences. As a consequen...

Spectral Theory Examples

Example 1 (Spectrum of Multiplication Operators) Let • (X, M, µ) be a σ–finite (actually semifinite will do) measure space, • 1 ≤ p ≤ ∞ and • a : X → C be a bounded measurable function on X. (Aϕ)(x) = a(x)ϕ(x) Point spectrum: Let λ ∈ C. Then λ1l − A is injective ⇐⇒ ϕ ∈ L p (X, M, µ), (λ − a(x))ϕ(x) = 0 a.e. =⇒ ϕ(x) = 0 a.e. ⇐⇒ λ − a(x) = 0 a.e.