Estimates of initial scales for layers with small random negative-definite perturbations

In this work, we consider a Schrödinger operator in a multi-dimensional layer with small random perturbations. The perturbations are distributed in periodicity cells of an arbitrarily chosen periodic lattice. To each cell, we put in correspondence a random variable; these random variables are independent and have the same distributions. The perturbations are described by the same abstract symmetric operator depending on the random variable multiplied by a global small parameter. We consider the case where the perturbations shift the bottom of the spectrum of the unperturbed operator to the left by a quantity of order of the square of the small parameter. Under these conditions, we establish the main result, which is the estimate of initial scales. We also present particular examples that demonstrate the main result.