Wavefunctions from energies: Applications in simple potentials

A remarkable mathematical property-somehow hidden and recently rediscovered-allows obtaining the eigenvectors of a Hermitian matrix directly from their eigenvalues. This opens the possibility to get the wavefunctions from the spectrum, an elusive goal of many fields in physics. Here, the formula is assessed for simple potentials, recovering the theoretical wavefunctions within machine accuracy. A striking feature of this eigenvalue-eigenvector relation is that it does not require knowing any of the entries of the working matrix. However, it requires the knowledge of the eigenvalues of the minor matrices (in which a row and a column have been deleted from the original matrix). We found a pattern in these sub-matrix spectra, allowing us to get the eigenvectors analytically. The physical information hidden behind this pattern is analyzed.