High-energy asymptotics for finite-interval Schr\"odinger operators with Gaussian white-noise potential
We study the one-dimensional Schr\"odinger operator on a fixed interval with Gaussian white-noise potential, \[ H_\omega=-\frac{\dd^2}{\dd x^2}+\rho\dot B_x(\omega), \] 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 \[ k_n=\frac{n\pi}{L} +\frac{\rho}{n\pi}\int_0^L \sin^2\left(\frac{n\pi s}{L}\right)\,\dd B_s +O_{L^p(\Omega)}(n^{-2}). \] Consequently, almost surely, $\lambda_n>0$ for all sufficiently large $n$ and, for every $\varepsilon>0$, \[ k_n=\frac{n\pi}{L}+O(n^{-1+\varepsilon}). \] We also obtain first-order eigenfunction asymptotics with explicit Brownian stochastic-integral corrections. In particular, for the $L^2(0,L)$-normalized Dirichlet eigenfunction $\varphi_n$, with a fixed sign convention, \[ \sup_{0\le x\le L} \left|\varphi_n(x)-\sqrt{\frac{2}{L}}\sin(k_n x)\right| =O(n^{-1+\varepsilon}) \] almost surely. The proofs use stochastic Pr\"ufer 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.