# Quartic SYK spectral edge — H1

For \(x\ge0\) and \(0\le u\le1\), put
\[
k_x(u)=\frac{\cosh(x(1/2-u))}{2\cosh(x/2)},\qquad
G_\mu(u)=\int_0^\infty k_x(u)\,\mu(dx),\qquad
\Sigma_\rho(u)=\int_0^\infty k_x(u)\,\rho(dx),
\]
where \(\mu\) is a probability measure and \(\rho\) is a finite positive measure on \([0,\infty)\).

A pair \((\mu,\rho)\) satisfies the Dyson equation if
\[
\omega\int_0^\infty\frac{\mu(dx)}{\omega^2+x^2}
=\frac1{\omega+\omega\int_0^\infty\frac{\rho(dx)}{\omega^2+x^2}}\qquad(\omega>0).
\]

For every \(\beta>0\), there is exactly one such pair satisfying
\[
\Sigma_\rho(u)=\beta^2G_\mu(u)^3\qquad(0\le u\le1).
\]