A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

Strong coupling between surface plasmon polaritons and Sulforhodamine 101 dye

We demonstrate a strong coupling between surface plasmon polaritons and Sulforhodamine 101 dye molecules. Dispersion curves for surface plasmon polaritons on samples with a thin layer of silver covered with Sulforhodamine 101 molecules embedded in SU-8 polymer are obtained experimentally by reflectometry measurements and compared to the dispersion of samples without molecules. Clear Rabi splittings, with energies up to 360 and 190 meV, are observed at the positions of the dye absorption maxima. The split energies are dependent on the number of Sulforhodamine 101 molecules involved in the coupling process. Transfer matrix and coupled oscillator methods are used to model the studied multilayer structures with a great agreement with the experiments. Detection of the scattered radiation after the propagation provides another way to obtain the dispersion relation of the surface plasmon polaritons and, thus, provides insight into dynamics of the surface plasmon polariton/dye interaction, beyond the refrectometry measurements