The central limit theorem for the spectrum of the random one-dimensional Schrödinger operator

Abstract

Let HL = −d2/dt2+q(t,ω) be an one-dimensional random Schrodinger operator in ξ2(−L, L) with the classical boundary conditions. The random potential q(t,ω) has a form q(t, ω)=F(xt), where xt is a Brownian motion on the Euclidean v-dimensional torus, F∶Sv→ R1 is a smooth function with the nondegenerated critical points, minsv F = 0. Let\(N_L (\lambda ) = \sum _{\lambda _i (L) \leqslant \lambda } 1(\lambda _i (L,\omega )\) are the eigenvalues of HL) be a spectral distribution function in the “volume” [− L,L] and N(λ) = limL→∞(1/2L)NL(λ) be a corresponding limit distribution function.