Level surfaces of parametric eigenvalue problems (Q762669)

This paper concerns an eigenvalue problem containing the vector parameters \(\alpha\) and \(\beta\) : \(-(p(x,\alpha)u')'+q(x,\beta)u=\lambda u\), \(ru(0)-p(0,\alpha)u'(0)=0\), \(su(1)+p(1,\alpha)u'(1)=0\) where r and s are given numbers. We consider the eigenvalue as a function of the parameters \(\alpha\) and \(\beta\) and denote it by \(\lambda\) (\(\alpha\),\(\beta)\). Then its gradient can be evaluated by numerical procedures. If the gradient does not vanish, then the level surface for the function \(\lambda (\alpha,\beta)=\lambda_ c=const\). can be obtained numerically. In some cases the level surface splits the whole (\(\alpha\),\(\beta)\) space into two subspaces. When (\(\alpha\),\(\beta)\) are in different subspaces, the values of \(\lambda\) (\(\alpha\),\(\beta)\) are greater or less than \(\lambda_ c\), respectively. When \(\alpha\) and \(\beta\) are linear, then the smallest eigenvalue \(\lambda\) (\(\alpha\),\(\beta)\) is concave and its values in different subspaces split by the level surface can be controlled easily. These results are illustrated by 3 examples.