= Dirichlet-Robin eigenvalues on a finite interval
{c}
{title2=$-y''=\lambda y,\quad y(0)=0,\quad y'(\ell)+ky(\ell)=0$}
For $\ell>0$ and $k>0$, the <Sturm-Liouville problem> with one <Dirichlet boundary condition> and one <Robin boundary condition> has positive <eigenvalues>. Indeed, <integration by parts> gives
$$
\lambda\int_0^\ell y^2\,dx=\int_0^\ell(y')^2\,dx+k y(\ell)^2>0
$$
for every nonzero <eigenfunction>. Let $z=\ell\sqrt\lambda$ and $\kappa=k\ell$. The <eigenfunctions> are proportional to $\sin(zx/\ell)$, with
$$
\tan z=-\frac z\kappa.
$$
For each $n\geq1$, there is exactly one root $z_n$ in $((n-1/2)\pi,n\pi)$: on that interval $\tan z+z/\kappa$ increases strictly from $-\infty$ to a positive value. There are no other positive roots. The apparent root $z=0$ gives only the zero sine function and is not an <eigenvalue>.
For fixed $k,\ell$, write $z_n=(n-1/2)\pi+\varepsilon_n$. The root equation becomes $\kappa\cot\varepsilon_n=(n-1/2)\pi+\varepsilon_n$, hence
$$
\varepsilon_n=\frac{\kappa}{(n-1/2)\pi}+O(n^{-3}),\qquad
\lambda_n=\frac{(n-1/2)^2\pi^2}{\ell^2}+\frac{2k}{\ell}+O(n^{-2}).
$$
An optional unit normalization uses $\int_0^\ell\sin^2(\sqrt{\lambda_n}x)\,dx=\ell/2+k/[2(k^2+\lambda_n)]$. The positive sign of the Robin boundary term matters: reversing it can introduce nonpositive modes, as in <nonpositive modes in a Dirichlet-Robin interval expansion>.
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