Spectrum of the Kerzman?Stein Operator for Model Domains

Abstract

For a domain Ω ⊂ C, the Kerzman-Stein operator is the skewhermitian part of the Cauchy operator acting on L 2 (bΩ), which is defined with respect to Euclidean measure. In this paper we compute the spectrum of the Kerzman-Stein operator for three domains whose boundaries consist of two circular arcs: a strip, a wedge, and an annulus. We also treat the case of a domain bounded by two logarithmic spirals. (2000). 45E05, 45E10, 30C40. Mathematics Subject Classification

The Kerzman–Stein operator for piecewise continuously differentiable regions

“…Interestingly, the spectrum is purely continuous as it was for the example of the unbounded wedge as shown in [3]. Other examples showing this behavior include an infinite strip and logarithmic sector [3]. The point spectrum, however, seems to be affected more by the global geometry of the boundary.…”

“…Subsequent work on the problem was concerned with giving a complete description of the spectrum for model domains [3], asymptotics of eigenvalues for ellipses with small eccentricity [5], and norm estimates that are invariant with respect to Möbius transformation [1,4]. For a disc or halfplane, there is complete cancellation of singularities and the Kerzman-Stein operator is trivial [13].…”

Spectrum of the Kerzman-Stein Operator for the Ellipse

“…If Ω is unbounded or if ∂Ω has a corner, however, this is not necessarily true. See [4] for specific examples when A is non-compact.…”

“…So in a previous article [4] the author computed the spectrum for domains bounded by two circular arcs or two logarithmic spirals-logarithmic spirals are known to have constant inversive curvature. The ellipse, then, is the first example for which there is no apparent Möbius symmetry.…”

A Lower Estimate for the Norm of the Kerzman-Stein Operator

“…In Section 3, we prove Theorem 1; we do this in the context of Singh's problem so that the similarity with the Bergman-Schiffer estimate is clearly evident. In Section 4, we evaluate the estimates for examples the author studied previously [3,4]. In certain cases the estimates are sharp.…”