On a method for computing eigenvalues and eigenfunctions of linear differential operators

Hydrogen Atom in a Finite Linear Space

“…For other cases, the method requires one to take into account the behavior of the solution at the singularities of the differential equation or a further modification for nonseparable problems. 2 Actually, the sum of the squares of these expressions gives a matrix (L ) 2 different from L 2 , as is shown in the Appendix. Since the difference is essentially a projection matrix, the dimension of the linear space in which the use of (L ) 2 yields exact results becomes smaller than the one corresponding to the use of L 2 .…”

“…2 Actually, the sum of the squares of these expressions gives a matrix (L ) 2 different from L 2 , as is shown in the Appendix. Since the difference is essentially a projection matrix, the dimension of the linear space in which the use of (L ) 2 yields exact results becomes smaller than the one corresponding to the use of L 2 . This difference can be compensated by increasing M.…”

“…it is necessary to extract from ψ(r, θ, ϕ) the singular behavior at the endpoints of the intervals (see [2,7,8]). This can be done in part through the change of variable…”

“…. , N. L 2 is given by (2), and 1 is the identity matrix of dimensionÑ . Since the part of f (r, θ, ϕ) depending on r consists of polynomials (Laguerre polynomials multiplied by a power of r ) and the dimension of D r is N , one would expect a set of N times (K + 1) 2 /4 eigenvectors f s coinciding exactly with the first eigenfunctions of the hydrogen atom evaluated at the nodes (provided M ≥ K ).…”

“…The way in which the matrix (1) is obtained is similar to the way in which the finite-dimensional matrix representation D x of d/dx, which has been used to solve two-point boundary value problems [2,5,8], was obtained. Such a matrix is given by…”