Two Parameter Eigenvalue Problems

Start: 03/05/2018 - 4:15pm
End  : 03/05/2018 - 5:15pm

Applied Math Seminar

Stephen B. Robinson (Wake Forest University)


Consider a small variation of a standard first semester linear algebra problem. Find the {\em eigenpairs} $(\lambda,\mu)$ such that the problem

\begin{equation} \label{e.matrixproblem}

Ax \; = \; \lambda\, Bx \, + \, \mu\, x


has a nontrivial vector solution $x\in\mathbb{R}^N$, where $A$ is a symmetric positive definite matrix and $B$ a symmetric matrix. The eigenpairs are on {\em eigencurves} in the plane.

When $A, B$ are, for example, the $3\times 3$ matrices given by


A\; = \; \begin{bmatrix}

2 & -1 & 0 \\

-1 & 2 & -1 \\

0 & -1 & 2


\qquad \text{and}\qquad

B\; = \; \begin{bmatrix}

2 & -1 & 0 \\

-1 & 2 & -1 \\

0 & -1 & 0



the eigencurves are easily determined and are plotted in Figure \ref{matrix}.


\begin{figure}[h] %%% GENERALIZED MATRIX FIGURE


\includegraphics[width=110mm, height=75mm]{matrix}

\caption{Eigencurves for the matrix eigenproblem \eqref{e.matrixproblem}-\eqref{e.matrixvalues}.}





The ideas that arise in this finite dimensional example provide good motivation for a more complicated version of the eigenpair problem.



-\Delta u(x) \; = & \; \mu\, m_0(x)\, u(x) \quad\text{for }x\in \Omega \\

\frac{\partial u}{\partial\nu}(x) \; + \; c(x)\,u(x)\; = & \; \lambda\, b_0(x)\, u(x)\quad \text{for }x\in \partial\Omega,



where $c, b_0, m_0$ are given functions in appropriate $L^p$-spaces on a smooth bounded region $\Omega$ in $\mathbb{R}^N$, and $\lambda, \mu$ are real eigenparameters.

Here, $m_0$ is assumed to be strictly positive, $b_0$ may be sign-changing, and $\nu$ denotes the outward normal vector.

The weak formulation of this problem leads to an analysis of abstract eigencurve problems associated with triples $(a, b, m)$ of continuous symmetric bilinear forms on a real Hilbert space $V$.


In this talk I will decribe how the eigenpairs form {\em variational eigencurves} with nice properties. In particular, the curves satisfy some convexity properties that are easy to describe. For example, the first eigencurve is convex, i.e. any straight line intersects the curve at most twice, and the second eigencurve is a little less convex, i.e. any straight line interesects the curve at most four times. In general, any line intersects the $n$th eigencurve at most $2n$ times.


Emmy Noether Rm Millikan 1021 Pomona College

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