High-energy asymptotics for finite-interval Schrödinger operators with Gaussian white-noise potential

We study the one-dimensional Schrödinger operator on a fixed interval with Gaussian white-noise potential, $$H_{\omega }=-\frac{{\text{dd}}^2}{\text{dd}{x}^2}+\rho {\dot{B}}_x(\omega ),$$ under Dirichlet boundary conditions. The operator is defined pathwise through the quasi-derivative realization of Sturm–Liouville operators with distributional potentials. Let ${\lambda }_n$ be the Dirichlet eigenvalues, ${\lambda }_n^{+}=\max \{{\lambda }_n,0\}$, and $k_n=\sqrt{{\lambda }_n^{+}}$. For every finite $p$, we prove the high-energy expansion $$k_n=\frac{n\pi }{L}+\frac{\rho }{n\pi }{\int }_0^L{\sin }^2\left(\frac{n\pi s}{L}\right)\text{dd}{B}_s+O_{L^p(\mathrm{\Omega })}(n^{-2})\mathrm{.}$$ Consequently, almost surely, ${\lambda }_n>0$ for all sufficiently large $n$ and, for every $\epsilon >0$, $$k_n=\frac{n\pi }{L}+O(n^{-1+\epsilon })\mathrm{.}$$ We also obtain first-order eigenfunction asymptotics with explicit Brownian stochastic-integral corrections. In particular, for the $L^2(0,L)$-normalized Dirichlet eigenfunction ${\phi }_n$, with a fixed sign convention, $$\underset{0\le x\le L}{\sup }\mid {\phi }_n(x)-\sqrt{\frac{2}{L}}\sin (k_{n}x)\mid =O(n^{-1+\epsilon })$$ almost surely. The proofs use stochastic Prüfer coordinates, stochastic Volterra expansions, the Burkholder–Davis–Gundy inequality, and a Borel–Cantelli argument. The estimates provide a first step toward KAM-type small-divisor analysis for Hamiltonian PDEs with white-noise spatial potentials.