On the Solution of Peierl’s Integral Equation by the Monte Carlo Method

The random walks used in [1] to evaluate the least eigenvalue of an integral equation can be used at the same time for computing the first eigenfunction.
In the case of an infinite cylindrical region equation (1) is transformed into (2). The computation of the kernel (3) is performed simultaneously with iterations. A numerical example shows that in general the variance does not decrease when the initial function is improved.