Hendrik Vogt

ON L p -SPECTRA AND ESSENTIAL SPECTRA OF SECOND-ORDER ELLIPTIC OPERATORS

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

A Weak Gordon Type Condition for Absence of Eigenvalues of One-dimensional Schrödinger Operators

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials.

A GENERALISATION OF THE FORM METHOD FOR ACCRETIVE FORMS AND OPERATORS

Abstract The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. M c Intosh  generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms. Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

Equivalence of Pointwise and Global Ellipticity Estimates

Defining an elliptic operator −∇ · (a∇) via the form method one normally imposes pointwise conditions on the matrix valued function a in order to get positivity, ellipticity and sectoriality of the form. In this note we show that the pointwise conditions on a are equivalent to the corresponding global ones on the form. MSC 1991: 35J20 (primary) Let Ω be an arbitrary open subset of R, a: Ω→ Cd×d a locally integrable, hermitian matrix valued function. Define the symmetric form τ in L2(Ω) by τ(u) := ∫ a∇u ·∇u on D(τ) := C∞ c (Ω). If τ is positive and closable then, by the form representation theorem, τ̄ is associated with a positive selfadjoint operator in L2(Ω) (which corresponds to Dirichlet boundary conditions). The main aim of this note is to show that the positivity of the form τ is equivalent to the positivity of the function a, i.e., a > 0 in the matrix sense a.e. A case of particular interest is the following: Let a1: Ω → Rd×d be locally integrable, symmetric matrix valued and locally strictly elliptic, i.e., for every compact set K ⊆ Ω there exists σ > 0 such that a1(x) > σ in the matrix sense for almost all x ∈ K. Then it is known that D(τN) := { u ∈ L2(Ω) ∩W 1 2,loc(Ω); τN(u) := ∫ a1∇u · ∇u <∞ } defines a symmetric Dirichlet form in L2(Ω) (cf. [1, Thm. 1.3.9]; one can show the closedness of τN like the completeness of the Sobolev space W 1 2 (Ω) because of the local strict ellipticity of a1). The associated selfadjoint operator in L2(Ω) corresponds to Neumann boundary conditions. Notation. Let |M | denote the Lebesgue measure of a measurable set M ⊆ R, χM the characteristic function of M . Sd−1 is the unit sphere of R, for the spectral radius of a hermitian matrix A ∈ Cd×d we write |A|(= supξ∈Sd−1 |Aξ · ξ|). For a function f : Ω → R we use the shorthand [f > 0] for the set {x ∈ Ω; f(x) > 0} (and similarly [f < g] etc). Q(f) denotes the form domain of the multiplication operator f in L2(Ω). Theorem. Let a: Ω→ Cd×d be a locally integrable hermitian matrix valued function, τ(u) := ∫ a∇u · ∇u for u ∈ C∞ c (Ω).

L 1 -estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

We investigate selfadjoint positivity preserving C0-semigroups that are dominated by the free heat semigroup on

A note on absorption semigroups and regularity

{\mathbb{R}^d}

L ∞ -Estimates for the Torsion Function and L ∞ -Growth of Semigroups Satisfying Gaussian Bounds

Rd. Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum, we prove estimates of their L1-norm in terms of the L2-norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup.

On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficients

For a general measure space (Ω,μ), it is shown that for every band M in Lp(μ) there exists a decomposition μ=μ′+μ′′ such that M=Lp(μ′)={f∈Lp(μ);f=0μ′′-a.e.}. The theory is illustrated by an example, with an application to absorption semigroups.