On spectrum Lévy constants for quadratic irrationalities and class numbers of real quadratic fields

Let $h(d)$ be the class number of the field $\mathbb Q(\sqrt d)$ and let $\beta(\sqrt d)$ be the Lévy constant. A connection between these constants is studied. It is proved that if d is large, then the value $h(d)$ increases, roughly speaking, at the rate $\exp\beta(\sqrt d)/\beta^2(\sqrt d)$ as $\beta(\sqrt d)$ grows. A similar result is obtained in the case where the value $\beta(\sqrt d)$ is close to $\log(1+\sqrt5)/2)$, i.e., to the least possible value. In addition, it is shown that the interval $[\log(1+\sqrt5)/2),\log(1+\sqrt3)/\sqrt2))$ contains no values of $\beta(\sqrt p)$ for prime $p$ such that $p\equiv3\mod4$. As a corollary, a new criterion for the equality $h(d)=1$ is obtained.