= Finite-interval Airy spectral determinant
{title2=$\Delta(k)=e^{-ikL}+\alpha e^{-i\alpha kL}+\alpha^2e^{-i\alpha^2 kL}$}
Here $\alpha=e^{2\pi i/3}$. The missing-trace matrix has determinant $-\sqrt3k\Delta(k)$. Its nonzero zeros are the <eigenvalues> in the parametrization $Av=ik^3v$ of $A=-d^3/dx^3$ with $v(0)=v(L)=v'(L)=0$. <Integration by parts> gives $\operatorname{Re}\langle v,Av\rangle=-|v'(0)|^2/2$. No nonzero zero lies in $\operatorname{Re}(-ik^3)\leq0$: on its boundary the determinant equation reduces, after rotation, to an equality between a unit-modulus exponential and $\cosh u-i\sqrt3\sinh u$, whose modulus exceeds one unless $u=0$. The zero at the origin is removable in the original boundary problem. There are nevertheless infinitely many genuine negative <eigenvalues> of $A$, obtained from $e^{-3s/2}=2\cos(\sqrt3s/2+\pi/3)$, $s>0$, with $k=-is/L$. Absence of contour poles in the <Fokas method> does not imply absence of <discrete spectrum> of the operator.
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