Courant's Theorem Research Articles

On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I

According to Courant's theorem, an eigenfunction associated with the n -th eigenvalue \lambda_n has at most n nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to \lambda_n . We call this assertion the Extended Courant Property. In this paper, we propose new, simple and explicit examples for which the extended Courant property is false: convex domains in \mathbb{R}^n (hypercube and equilateral triangle), domains with cracks in \mathbb{R}^2 , on the round sphere \mathbb{S}^2 , and on a flat torus \mathbb{T}^2 . We also give numerical evidence that the extended Courant property is false for the equilateral triangle with rounded corners, and for the regular hexagon.

Nodal decompositions of graphs

A nodal domain of a function is a maximally connected subset of the domain for which the function does not change sign. Courant's nodal domain theorem gives a bound on the number of nodal domains of eigenfunctions of elliptic operators. In particular, the k-th eigenfunction contains no more than k nodal domains. We prove a generalization of Courant's theorem to discrete graphs. Namely, we show that for the k-th eigenvalue of a generalized Laplacian of a discrete graph, there exists a set of corresponding eigenvectors such that each eigenvector can be decomposed into at most k nodal domains. In addition, we show this set to be of co-dimension zero with respect to the entire eigenspace.

Pleijel’s theorem for Schrödinger operators with radial potentials

In 1956, ˚ A. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunction as the eigenvalue tends to +∞. A similar question occurs naturally for the case of the Schrodinger operator. The first significant result has been obtained recently by the first author for the case of the harmonic oscilllator. The purpose of this paper is to consider more general potentials which are radial. We will analyze either the case when the potential tends to +∞ or the 1 case when the potential tends to zero, the considered eigenfunctions being associated with the eigenvalues below the essential spectrum.

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $\nu_n$ of the $n$-th eigenfunction satisfies $n\ge \nu_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-\nu_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

On the realization of holonomic constraints

The idea of realizing holonomic constraints by means of elastic forces was proposed by Lecornu, Klein and Prandtl /1/ when dealing with the paradox of dry friction discovered by Painleve. The general theorem on the realization of holonomic constraints with the help of elastic forces directed towards the configurational manifold of a constrained system was proposed by Courant and was proved in /2/. The generalization of Courant's theorem was considered in /3–5/ by studying the passage to the limit in the case when the velocity of the system at the initial instant is transverse to the manifold defined by the constraint equations. In /2–5/ the assumption that the system in question in conservative is used to a considerable degree.

On positive differential operators (deficiency indices, factorization, perturbations)

SynopsisA short survey is given of some recent results. The perturbations and stability of discrete spectrum and the problems of resolvent convergence of densely or non-densely defined elliptic differential operators are considered. The Courant theorem on variations of the domain is generalized. In connection with Berezanskiy's theorem on essential self-adjointness, the test for the finite velocity of propagation is extended. The Frobenius and the Krein-Heinz-Rellich factorization theorems and the Etgen Pawlowski oscillation criterion are generalized for equations of any order with operator-valued coefficients. Brusentsev's recent example of a two term fourth order differential operator with deficiency index 4 is discussed.