In recent times, considerable attention was focused on the computation of the eigenvalues and eigenfunctions of the Finite (truncated) Fourier Transform (FFT). An incentive to that a series of papers by Slepian and his collaborators at Bell labs was. The latest results published by Walter and Shen on sampling with the prolate spheroidal functions — the 1D FFT eigenfunctions — will necessary pro...

We study the behavior of eigenfunctions in the semiclassical limit for Schrödinger operators with a simple well potential and a (non-zero) constant magnetic field. We prove an exponential decay estimates on the low-lying eigenfunctions, where the exponent depends explicitly on the magnetic field strength.

We give an explicit formula for the number of nodal domains of certain eigenfunctions on a flat torus. We apply this to an isospectral but not isometric family of pairs of flat four-dimensional tori constructed by Conway and Sloane, and we show that corresponding eigenfunctions have the same number of nodal domains. This disproves a conjecture by Brüning, Gnutzmann, and Smilansky.

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The eigenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU (2) group with half-integer angular momenta. They are special linear combinations of conventional spherical spinors.

A framework of a general type of Petrovsky equation is formulated. The characteristic equation for eigenvalues of the system is derived and the associated eigenfunctions are found. For N connected beams with linear feedbacks at joint points, the asymptotic expansions of eigenvalues and eigenfunctions are further developed, and the Riesz basis property and the exponential stability are then conc...