We study the scattering problem for the Schrodinger equation on the half-line with the
Robin boundary condition at the origin. We derive an expression for trace of the difference
of perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation
for the perturbation determinant and trace formulas of Buslaev-Faddeev type. We further
generalize the method used for obtaining trace formulas to matrix-valued Schrodinger
operator. We derive trace formulas for a star graph which satisfies Kirchhoff vertex condition
at origin. Finally, we apply the commutation method to matrix-valued Schrodinger
operator defined on the half-line with the Robin boundary condition at zero. We also obtain
sharp Lieb-Thirring inequalities and show how they can be used for related problems.