Orthogonality of generalized eigenfunctions in Weyl's expansion theorem

Generalized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.∗

There is a well developed theory (see [5, 9]) of analysis on certain types of fractal sets, of which the Sierpinski Gasket (SG) is the simplest non-trivial example. In this theory the fractals are viewed as limits of graphs, and notions analogous to the Dirichlet energy and the Laplacian are constructed as renormalized limits of the corresponding objects on the approximating graphs. The nature ...

Quasi-orthogonality on the boundary for Euclidean Laplace eigenfunctions

Consider the Laplacian in a bounded domain in Rd with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are ‘quasi-orthogonal’ on the boundary with respect to a certain norm. Boundary orthogonality is proved asymptotically within a narrow eigenvalue window of width o(E1/2) centered about E, as E → ∞. For the special case of Dirichlet boundary conditions, the norm...

A Theorem on Spin-Eigenfunctions

A theorem is proved to the effect that a wave function for a set of N spins,, which is a product of single-spin wave functions and which is an eigenstate of the square of the total spin S*, must be a state with the maximum possible value of S* and of S 3 z being an arbitrary direction. This theorem has z been applied in a separate work by the authors to show strikingly that the imposition of sy...

Generalized eigenfunctions of relativistic Schrödinger operators I

Generalized eigenfunctions of the 3-dimensional relativistic Schrödinger operator √ ∆+ V (x) with |V (x)| ≤ C〈x〉−σ, σ > 1, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernel of ( √ −∆ − z), z ∈ C \ [0, +∞), which has nothing in common with the integral kernel of (−∆− z), but the leadin...

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle