Inverse spectral problems for tridiagonal N by N real Hamiltonians with spectral parameter in the initial conditions

In this paper, the concept of generalized spectral function is introduced for finite order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data and normalizing numbers of the matrix. Inverse problems are investigated for tridiagonal N by N complex hamiltonians. Also the most important feature of the problem is that there is initial conditions, the spectral parameter linearly.