Solutions of the three-magnon bound state equation. III. The physical eigenstate

The physical eigenvalue and eigenfunction are found analytically by directly solving the three-magnon bound state equation in one dimension. The structure of the equation suggests a simple ansatz for the eigenfunction, and the calculation is reduced to the solution of a finite number of linear algebraic equations. The method is elementary, but the explicit solution is found after a long calculation. The connection of the wavefunction with previous results is also established.