Appendix to Lecture 13

Analyzing Example 13.6 and Example 13.7. We claim that eigenvalues are monotone functions of $\alpha$, $\beta$.

To prove it we need without proof to accept variational description of eigenvalues of self-adjoint operators bounded from below (very general theory) which in this case reads as:

Theorem. Consider quadratic forms \begin{equation} Q (u)= \int _0^l |u'|^2\,dx + \alpha |u(0)|^2 + \beta |u(l)|^2 \end{equation} and \begin{equation} P (u)= \int _0^l |u|^2\,dx . \end{equation} Then there are at least $N$ eigenvalues which are less than $\lambda$ if and only iff there is a subspace $\mathsf{K}$ of dimension $N$ on which quadratic form \begin{equation} Q_\lambda (u)= Q(u)- \lambda P(u) \end{equation} is negative definite (i.e. $Q_\lambda (u)<0$ for all $u\in \mathsf{K}$, $u\ne 0$).

Note that $Q(u)$ is montone non-decreasing function of $\alpha,\beta$. Therefore $N(\lambda)$ (the exact number of e.v. which are less than $\lambda$) is montone non-increasing function of $\alpha,\beta$ and therefore $\lambda_N$ is montone non-decreasing function of $\alpha,\beta$.