For the first time, we establish the Lippmann-Schwinger equations for a system of multiband energy-eigenvalue spectrum so that the involved time-independent Schrödinger wavefunctions become matrix elements. As a matter of fact, a supremum-seminorm of the total wavefunction matrix is estimated in terms of the same seminorm for the partial wavefunction matrix and for two key matrices of operators. In addition, an associated matrix-tensor formalism is presented.
We consider the initial value problem for systems of one dimensional nonlinear Schrödinger equations with long-long or long-short range interactions. Global existence and time decay of small solutions are presented under mass resonance conditions.
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