  We prove that all the zeros of certain meromorphic functions are on the
critical line $\text{Re}(s)=1/2$, and are simple (except possibly when
$s=1/2$). We prove this by relating the zeros to the discrete spectrum of an
unbounded self-adjoint operator. Specifically, we show for $h(s)$ a meromorphic
function with no zeros in $\text{Re}(s)>1/2$ and no poles in
$\text{Re}(s)<1/2$, real-valued on $\R$, $\frac{h(1-s)}{h(s)}\ll
|s|^{1-\epsilon}$ in $\text{Re}(s)>1/2$ and $\frac{h(1-s)}{h(s)}\notin
L^2(1/2+i\R)$, the only zeros of $h(s)\pm h(1-s)$ are on the critical line. One
instance of such a function $h$ is $h(s)=\xi(2s)$, the completed zeta-function.
We use spectral theory suggested by results of Lax-Phillips and Colin de
Verdi\`{e}re. This simplifies ideas of W. M\"{u}ller, J. Lagarias, M. Suzuki,
H. Ki, O. Vel\'{a}squez Casta\~{n}\'{o}n, D. Hejhal, L. de Branges and P.R.
Taylor.

Number Theory;Functional Analysis