Semi-classical analysis of Schrödinger operators and compactness in the ∂-Neumann problem

Abstract

We study the asymptotic behavior, in a "semi-classical limit", of the first eigenvalues (i.e. the groundstate energies) of a class of Schrödinger operators with magnetic fields and the relationship of this behavior with compactness in thē ∂-Neumann problem on Hartogs domains in C 2 .

The ∂-cohomology Groups, Holomorphic Morse Inequalities, and Finite Type Conditions

“…It is noteworthy that whereas the underpinning of Witten's approach is a semi-classical analysis of Schrödinger operators without magnetic fields, Demailly's holomorphic Morse inequality is connected to Schrödinger operators with strong magnetic fields. (Interestingly, a related phenomenon also occurs in compactness in the ∂-Neumann problem for Hartogs domains in C 2 (see [FS02,CF05]): Whereas Catlin's property (P ) can be phrased in terms of semi-classical limits of non-magnetic Schrödinger operators, compactness of the ∂-Neumann operator reduces to Schrödinger operators with degenerated magnetic fields.) More recently, Berman established a local version of holomorphic Morse inequalities on compact complex manifolds [Ber04] and generalized Demailly's holomorphic Morse inequalities to complex manifolds with non-degenerated boundaries [Ber05].…”

Complex tangential flows and compactness of the ∂-Neumann operator

“…These include global regularity [Kohn and Nirenberg 1965], the Fredholm theory of Toeplitz operators [Henkin and Iordan 1997], and the (non)existence of solution operators to ∂ with well-behaved solution kernels [Hefer and Lieb 2000]. There are also interesting connections to the theory of Schrödinger operators [Fu and Straube 2002;Christ and Fu 2005]. Catlin [1984] gave a sufficient condition, which he called property (P): near the boundary, there should exist plurisubharmonic functions bounded between 0 and 1 with arbitrarily large Hessians.…”

“…Property (P) was studied in detail (under the name B-regularity) by Sibony [1987]; see also [Sibony 1991]. On sufficiently regular domains, property (P) is equivalent to a quantitative version of Oka's lemma [Harrington 2007]. A version of property(P), called condition ( P), was introduced, and shown to still imply compactness, by McNeal [2002].…”

“…Thus the potential theory, the analysis, and the geometry mesh perfectly on locally convexifiable domains. It is also known that on smooth bounded Hartogs domains in ‫ރ‬ 2 , compactness of N 1 is equivalent to (P 1 ) and to ( P 1 ); see [Fu and Straube 2002;Christ and Fu 2005]. However, it is well understood that the boundaries of convex (hence of locally convexifiable) domains do not exhibit some of the more intriguing aspects of the interaction with the ambient space that occur on general pseudoconvex boundaries [D 'Angelo 1993].…”

“…q-dimensional varieties in the boundary obstruct property (P q ); see [Sibony 1987;Fu and Straube 2001;Straube ≥ 2007]. Analytic discs in the boundary have been shown to obstruct condition ( P 1 ) in [Straube 2006].…”