Shape of the ground state energy density of Hill ’ s equation with nice Gaussian potential

Consider Hill’s operator Q = − d2 dx2 + q(x) in which the potential q(x) is an almost surely continuous and rotation invariant Gaussian process on the circle x ∈ [0, 1). Viewing the classical Riccati map as a change of measure, we establish functional integral formulas for the probability density function of the ground state energy and also determine the density’s shape.