JacobiOperator

1. Tridiagonal Matrix: The Jacobi operator ( J ) is a tridiagonal matrix, which has non-zero elements only on the main diagonal, the diagonal above it, and the diagonal below it. Its form is derived from the three-term recurrence relation of Jacobi polynomials.

[ J = \begin{pmatrix} -b_0 & \sqrt{-a_0 c_1} & 0 & 0 & \cdots \ \sqrt{-a_0 c_1} & -b_1 & \sqrt{-a_1 c_2} & 0 & \cdots \ 0 & \sqrt{-a_1 c_2} & -b_2 & \sqrt{-a_2 c_3} & \cdots \ 0 & 0 & \sqrt{-a_2 c_3} & -b_3 & \cdots \ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} ]

2. Eigenvalues and Eigenvectors:

   • The eigenvalues and eigenvectors of the Jacobi operator can be obtained through recursive relations stemming from the three-term recurrence relation of the Jacobi polynomials.

3. Diagonal Matrix:

   • Construct the diagonal matrix ( D ) by placing the eigenvalues on its main diagonal.

[ D = \begin{pmatrix} \lambda_0 & 0 & 0 & \cdots \ 0 & \lambda_1 & 0 & \cdots \ 0 & 0 & \lambda_2 & \cdots \ \vdots & \vdots & \vdots & \ddots \end{pmatrix} ]

4. Matrix of Eigenvectors:
   • Form the matrix ( P ) by arranging the eigenvectors as columns.

[ P = \begin{pmatrix} | & | & | & \ x_0 & x_1 & x_2 & \cdots \ | & | & | & \end{pmatrix} ]

5. Verification:
   • Verify the diagonalization by checking that ( P^{-1}JP = D ).

This process leverages the recursive structure of the Jacobi operator and the associated Jacobi polynomials to obtain an exact analytical diagonalization of ( J ).