Boundary least squares method for the solution of Schrödinger's equation in quantum wires

We present a method for the approximate semianalytiral calculation of wave functions and eigenenergies in systems consisting of domains with known bulk solutions of the corresponding Schrödinger equation (e.g. with piecewise constant potential). The trial wave function is written as a normalized linear combination of several bulk solutions for the given energy. The coefficients of the linear combination are found by minimization of the integral of square mismatch along the boundaries. Mathematically the problem is equivalent to the minimization of the Rayleigh quotient or solution of the generalized eigenproblem for the vector of coefficients. The value of the residual provides an estimation of the accuracy of the results and gives the possibility to choose an optimal set of trial functions. We illustrate the use of the method by calculation of eigenfunction of infinitely long triangular and quadrilateral wires.