Eigenvalues and eigenvectors of the second derivative

Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and discrete cases. In the discrete case, the standard central difference approximation of the second derivative is used on a uniform grid. These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension. The index j represents the jth eigenvalue or eigenvector and runs from 1 to ∞ {displaystyle infty } . Assuming the equation is defined on the domain x ∈ [ 0 , L ] {displaystyle xin } , the following are the eigenvalues and normalized eigenvectors. The eigenvalues are ordered in descending order. (That is: 0 {displaystyle 0} is a simple eigenvalue and all further eigenvalues are given by j 2 π 2 L 2 {displaystyle {frac {j^{2}pi ^{2}}{L^{2}}}} , j = 1 , 2 , … {displaystyle j=1,2,ldots } , each with multiplicity 2).