Fundamental solutions to Pell equation with prescribed size

We prove that the number of parameters $D$ up to a fixed $x\geq2$ such that the fundamental solution $\varepsilon_D$ to the Pell equation $T^2-DU^2=1$ lies between $D^{\frac12+\alpha_1}$ and $D^{\frac12+\alpha_2}$ is greater than $\sqrt x\log^2x$ up to a constant as long as $\alpha_1<\alpha_2$ and $\alpha_1<3/2$. The starting point of the proof is a reduction step already used by the authors in earlier works. This approach is amenable to analytic methods. Along the same lines, and inspired by the work of Dirichlet, we show that the set of parameters $D\leq x$ for which $\log\varepsilon_D$ is larger than $D^\frac14$ has a cardinality essentially larger than $x^\frac14\log^2x$.