Elliptic Operators on Closed Manifolds

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

“…For Q ∈ ΨDO r cl (M ) with r > 0 such that Q is invertible and (2.1) holds in a sector {λ ∈ C|θ ≤ arg λ ≤ 2π − θ} for θ ∈ (π/2, π), we can form the heat operator e −tQ for t > 0 which belongs to ΨDO −∞ cl (M ), cf. Section 5.6 in [1]. We recall the definition of asymptotic expansions of real-valued functions.…”

“…This relation is typically exploited to investigate the on-diagonal behaviour of the heat kernel, see for example Section 5.5 of [1]. Lemma 4.1 allows us to link the singular structure of the kernel of the complex powers and the heat kernel asymptotics.…”

“…Here f is the Laplace exponent of the subordinator and A is the infinitesimal generator of Brownian motion given by 1 2 Δ. We illustrate this in Sect.…”

Integrated Density of States for the Periodic Schrödinger Operator in Dimension Two

“…On the basis of this result it is natural to conjecture that for general From the technical point of view the paper is a continuation of [25] where the density of states was studied in the case d = 1 for elliptic operators of a more general form than the Schrödinger operator. As in [25], our approach is a variant of the "near-similarity" method, which is usually applied in dimension one (see [18], [1] and [12], [13]). The central idea is to reduce the operator H with the help of a suitable similarity transformation, to an operator with constant coefficients.…”

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

“…For the case s ∈ R and Sobolev spaces on a compact closed manifold D see e.g. Proposition 5.4.1 in [Agr90]. For the general case we have not been able to find a proper reference and so we refer to our paper [ST12].…”