We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the n-th eigenfunction is bounded below by n− l, where l is the number of links that distinguish the graph from a tree. Our results apply to operators on bo...

An ion moving in a constant magnetic field is considered as the generic example of two-dimensional topological (Chern-Simons) theory. The eigenfunctions for the generic particle are determined by solving Schrödinger’s equation in the secular and non-secular forms of the theory, corresponding to the inclusion and non-inclusion respectively in the Lagrangian of the particle’s kinetic energy. A no...

We consider the zeros on the boundary ∂Ω of a Neumann eigenfunction φλj of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(λj) where −∆φλj = λ2jφλj . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(λj). It follows that the number of nodal lines of φλj (components of the nodal set) which touch the b...

Stress intensities of a singular near-tip field around the vertex of a triple junction wedge in multilevel thin film packages are calculated using the two-state M-integral. For this calculation the existence of a simple auxiliary field in the sense of the M-integral associated with every eigenfunction for the triple junction vertex is first verified numerically. The auxiliary field is then empl...

The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u=u∈(X/T ). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/∈ and slow...

Nonlinear stochastic optimal control problems are fundamental in control theory. A general class of such problems can be reduced to computing the principal eigenfunction of a linear operator. Here, we describe a new method for finding this eigenfunction using a moving least-squares function approximation. We use efficient iterative solvers that do not require matrix factorization, thereby allow...

Marseille Cedex 9 France One-dimensional quantum scattering for potentials de ned as measures Charles-Antoine GUERIN1 Abstract We generalize the basic one-dimensional scattering formalism to potentials de ned as measures and retrieve the classical results that hold for smooth potentials. We introduce a set of generalized eigenfunctions for the corresponding Schr odinger operator and study thei...

A wave function of the N -component KP Hierarchy with continuous flows determined by an invertible matrix H is constructed from the choice of an MN -dimensional space of finitely-supported vector distributions. This wave function is shown to be an eigenfunction for a ring of matrix differential operators in x having eigenvalues that are matrix functions of the spectral parameter z. If the space...

Generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator H = √ −∆ + V (x) with |V (x)| ≤ C〈x〉−σ, σ > 3/2, are considered. We compute the integral kernels of the boundary values R± 0 (λ) = ( √ −∆− (λ± i0))−1, and prove that the generalized eigenfunctions φ±(x, k) are bounded on R x × {k | a ≤ |k| ≤ b}, where [a, b] ⊂ (0,∞)\σp(H), and σp(H) is the set of eigenvalues of ...