The Lippmann–Schwinger equation for quantum wires

We consider the discrete Schrodinger operator with a potential of a special form defined on a graph whose nodes lie on the union of two intersected straight lines. We prove that there exist unique quasi-levels (eigenvalues or resonances) in the neighborhoods of the point $\pm2$ (these points consist a boundary of the essential spectrum). The asymptotic formulae for quasi-levels are obtained. We find the conditions for the coefficient of reflection is equal to zero.