Almost-sure quenched KAM tori for cubic NLS with a spatial white-noise potential

Source abstract

Let $$A_ω=-\partial_x^2+ρ\dot B_x(ω), \qquad ρ\ne0,$$ be the Dirichlet Schrödinger operator on $(0,π)$ with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here $\dot B_x$ denotes the distributional derivative of Brownian motion with respect to the spatial variable $x$. For $κ\ne0$, we consider the cubic nonlinear Schrödinger equation $$i u_t=A_ωu+κ|u|^2u.$$ We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set $J$ of cardinality $b$, we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant $b$-tori. The family is parametrized by a Cantor subset of $[ν,2ν]^b$ whose relative measure tends to one as $ν\to0$.