A generalized eigenfunction expansion may be regarded as a representation for the spectral theorem by a transform technique. These representations have been presented in many forms, an early version of which was the von Neumann "direct integral" decomposition for a class of operator algebras [l9]. In 1953 [17], Mautner applied the von Neumann technique to the class of operators acting in an L2-...

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

In this paper we study eigenfunction expansions associated with the Schrodinger operator with a singular potential. In the paper it is obtained sufficient conditions for localization and uniformly convergence of the regularizations of the corresponding expansions

For |Ψ an eigenfunction of the Hamiltonian operatorˆH = ˆ T + ˆ V , wherê T is the kinetic energy operator andˆV the potential energy operator , the following relations between expectation values are satisfied:

The asymptotic behaviour of a finite energy pseudoholomorphic strip with Lagrangian boundary conditions in a symplectic manifold is determined by an eigenfunction of the linearized operator at the (transverse) intersection.

A general Casoratian formulation is proposed for the 2D Toda lattice equation, which involves coupled eigenfunction systems. Various Casoratian type solutions are generated, through solving the resulting linear conditions and using a Bäcklund transformation.