Trace Moments for Schr\"odinger Operators with Matrix White Noise and the Rigidity of the Multivariate Stochastic Airy Operator

We study the semigroups of random Schr\"odinger operators of the form Hf=-12f''+(V+ξ)f, where f:I→ Fʳ (F= R, C, H) are vector-valued functions on a possibly infinite interval I⊂ R that satisfy a mix of Robin and Dirichlet boundary conditions, V is a deterministic diagonal potential with power-law growth at infinity, and ξ is a matrix white noise. Our main result consists of Feynman-Kac formulas for trace moments of the form E[∏ₖ₌₁ⁿTr[ e^-tₖH]] (n∈ N, tₖ>0). One notable example covered by our main result consists of the multivariate stochastic Airy operator (SAO) of Bloemendal and Vir\'ag (Ann. Probab., 44(4):2726-2769, 2016), which characterizes the soft-edge eigenvalue fluctuations of critical rank-r spiked Wishart and GO/U/SE random matrices. As a corollary of our main result, we prove that if V's growth is at least linear (this includes the multivariate SAO), then H's spectrum is number rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789-1858, 2017). Together with the rigidity of the scalar SAO, this completes the characterization of number rigidity in the soft-edge limits of Gaussian β-ensembles and their finite-rank spiked versions.