The Spectrum of the Neumann Matrix with Zero Modes

We calculate the spectrum of the matrix M ′ of Neumann coefficients of the Witten vertex, expressed in the oscillator basis including the zero-mode a0. We find that in addition to the known continuous spectrum inside [− 3 , 0) of the matrixM without the zero-modes, there is also an additional eigenvalue inside (0, 1). For every eigenvalue, there is a pair of eigenvectors, a twist-even and a twist-odd. We give analytically these eigenvectors as well as the generating function for their components. Also, we have found an interesting critical parameter b0 = 8 ln 2 on which the forms of the eigenvectors depend.