n |sn−sn+1|, is presented. This Ising-like model is an interesting example of a graduate-level statistical mechanics problem where the eigenfunctions and eigenvalues of an integral operator, the transfer integral, can be determined exactly. This can be contrasted with the determination of eigenfunctions and eigenvalues of differential operators, such as the quantum mechanical finite-depth squar...

Nonlinear variational methods have become very powerful tools for many image processing tasks. Recently a new line of research has emerged, dealing with nonlinear eigenfunctions induced by convex functionals. This has provided new insights and better theoretical understanding of convex regularization and introduced new processing methods. However, the theory of nonlinear eigenvalue problems is ...

In the present paper, we study surfaces invariant under the 1-parameter subgroup in Sol space $rm Sol_3$. Also, we characterize the surfaces in $rm Sol_3$ whose coordinate functions of an immersion of the surface are eigenfunctions of the Laplacian $Delta$ of the surface.

The article describes the eigenvalue and eigenfunction problems. Basic properties, some applications and examples in system analysis are provided.

We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections of high powers of positive line bundles over Kahler manifolds, and of Laplace eigenfunctions of large eigenvalue on compact Riemannian manifolds. We describe statistical patterns in the zeros, critical points and L norms of random polynomials and holomorphic sections, and the i...

This equation is regular and already in normal form. We obtain the eigenvalues from the characteristic equation: α + ω = 0 α = −ω α = ±iω This gives {e} as eigenfunctions. However, since the original equation has real coefficients, we would like a basis of real-valued eigenfunctions. Since Re(e) = cosωx and Im(e) = sinωx, we take {cosωx, sinωx} as a basis. Thus the eigenfunctions of (?1) are al...

This paper addresses the problem of using the Karhunen-Lo eve transform with partial data. Given a set of empirical eigenfunctions we show how to recover the modal coeecients for each gappy snapshot by a least-squares procedure. This method gives an unbiased estimate of the data that lay in the gaps and permits gaps to be lled in a reasonable manner. In addition, a scheme is advanced for nding ...

Abstract An extended Harry Dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the Harry Dym hierarchy associated with the pseudo-differential operator L = u∂ +u0 +u1∂−1 + · · ·. The corresponding Lax presentation possesses a self-consistent source involving squared eigenfunctions. The resulting extended Harry Dym hierarchy is reduced to t...

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic surfacess arising as a quotient of the upper half-plane by a discrete “arithmetic” subgroup of SL2(R) (for example, SL2(Z), and in this case the corr...

We study oscillations in the eigenfunctions for a fractional Schrödinger operator on the real line. An argument in the spirit of Courant’s nodal domain theorem applies to an associated local problem in the upper half plane and provides a bound on the number of nodal domains for the extensions of the eigenfunctions. Using the combinatorial properties of noncrossing partitions, we turn the nodal ...