Star algebra spectroscopy

The spectrum of the infinite dimensional Neumann matrices M¹¹, M¹² and M²¹ in the oscillator construction of the three-string vertex determines key properties of the star product and of wedge and sliver states. We study the spectrum of eigenvalues and eigenvectors of these matrices using the derivation K₁ = L₁ + L _-1 of the star algebra, which defines a simple infinite matrix commuting with the Neumann matrices. By an exact calculation of the spectrum of K₁, and by consideration of an operator generating wedge states, we are able to find analytic expressions for the eigenvalues and eigenvectors of the Neumann matrices and for the spectral density. The spectrum of M ¹¹ is continuous in the range [-1/3, 0) with degenerate twist even and twist odd eigenvectors for every eigenvalue except for -1/3.