Abstract. Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ⊂ R d is deformed. For a family of domains {Ωt} t∈ [a,b] we prove that the Morse index of L on Ωa differs from the Morse index of L on Ω b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ωa is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on Ω b ) and the "simplified" problem (on Ωa). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.

We study the Schrödinger operator L = −∆ + V on a star-shaped domain Ω in R d with Lipschitz boundary ∂Ω. The operator is equipped with quite general Dirichlet-or Robin-type boundary conditions induced by operators between H 1/2 (∂Ω) and H −1/2 (∂Ω), and the potential takes values in the set of symmetric N × N matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian Grassmannian of the subspace of H 1/2 (∂Ω) × H −1/2 (∂Ω) corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of L (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative eigenvalues of a bilinear form related to the boundary operator.

The dual fluorescence of 4-(N,N-dimethylamino)benzonitrile and 4-(N,N-diethylamino)benzonitrile (DMABN and DEABN, respectively), has been studied in aqueous solutions of cyclodextrins. Fluorescence parameters (peak maximum, lifetime, and relative intensity) have been measured and are found to be consistent with the formation of complexes of probe and cyclodextrin.Enhanced emission of the twisted internal charge transfer state (TICT) fluorescence is observed in cyclodextrin. with the greatest effect for the DMABN/a-cyclodextrin system. Our results promote further understanding of both the photophysical behavior of DMABN and the complexation of probe with cyclodextrin. The use of dialkylaminobenzonitriles as polarity probes is discussed.

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schrödinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.

Protoporphyrin IX and its various ester derivatives have been previously shown to undergo self‐sensitized photooxygenation to yield hydroxyaldehydes (photoprotoporphyrin) and mono‐ and diformyl deuteroporphyrin derivatives. In the present study the photoreactions of these products in the presence of oxygen have been investigated. All of the photooxidation products are themselves good sensitizers of singlet oxygen. In addition spin trapping experiments indicate these products can produce superoxide in low‐to‐moderate efficiency by an excited state electron transfer process. The photo‐products themselves are somewhat more stable to photooxidation than protoporphyrin IX itself. The two monoformyl‐monovinyl deuteroporphyrins have been found to undergo further photooxidation at the vinyl groups to yield primarily monoformyl hydroxyaldehydes in a reaction mainly involving singlet oxygen analogous to the initial reaction of protoporphyrin IX.

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Abstract. The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold M divided into components Ω 1 and Ω 2 by a separating hypersurface Σ. A homotopy argument relates the spectrum of a second-order elliptic operator on M to its Dirichlet and Neumann spectra on Ω 1 and Ω 2 , with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on Σ. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant's nodal domain theorem is given, with an explicit formula for the nodal deficiency.

Supplementary Material Available: Thermal and positional parameters for phase I after two-site refinement, atomic coordinates and thermal -parameters after single-site refinement, bond lengths and angles from single-site refinement, and thermal parameters and structure factors for phase II UDD (62 pp). Ordering information is given on any current masthead page.