| dc.description |
We calculate the spectrum of the matrix M′ of Neumann coefficients of the Witten vertex, expressed in the oscillator basis including the zero-mode a<sub>0</sub>. We find that in addition to the known continuous spectrum inside [−1/3,0) of the matrix M 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 b<sub>0</sub> = 8 ln 2 on which the forms of the eigenvectors depend. |
