= Nonpositive modes in a Dirichlet-Robin interval expansion
{title2=$y(0)=0,\quad y'(1)=y(1)/\alpha,\quad\alpha>0$}
For $-y''=\lambda y$, positive <eigenvalues> have <eigenfunctions> $\sin(q_nx)$, where $\tan q_n=\alpha q_n$ and $q_n>0$. Their squared <norms> are $N_n=1/2-\sin(2q_n)/(4q_n)$. A zero <eigenvalue> occurs precisely at $\alpha=1$, with <eigenfunction> $x$. A negative <eigenvalue> occurs precisely at $0<\alpha<1$: it is $-\kappa^2$ with <eigenfunction> $\sinh(\kappa x)$, where $\tanh\kappa=\alpha\kappa$. There is exactly one such $\kappa$, since $\tanh\kappa/\kappa$ decreases strictly from $1$ to $0$. The <integration by parts> identity
$$
\langle y,-y''\rangle=\int_0^1|y'|^2dx-\frac{|y(1)|^2}{\alpha}
$$
and the <Cauchy-Schwarz inequality> show positivity for $\alpha>1$ and nonnegativity at equality. The complete expansion of a <square-integrable function> therefore includes the extra term $3x\int_0^1tf(t)dt$ when $\alpha=1$, or $\sinh(\kappa x)\int_0^1f(t)\sinh(\kappa t)dt/[\sinh(2\kappa)/(4\kappa)-1/2]$ when $\alpha<1$, in addition to $\sum_n[\int_0^1f(t)\sin(q_nt)dt/N_n]\sin(q_nx)$. Omitting that extra mode makes the system incomplete. In particular, $f=x$ at $\alpha=1$ is orthogonal to every positive sine mode and is an immediate counterexample to a sine-only expansion.
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