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.