What is it about?

A remarkable mathematical property—somehow hidden and recently rediscovered—allows obtaining the eigenvectors of a Hermitian matrix directly from their eigenvalues. The eigenvalue–eigenvector identity expresses a curious and surprising relationship between the eigenvector of a matrix and the eigenvalues of its minors. A striking feature of this 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. Also, the physical information hidden behind this pattern is analyzed.

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Why is it important?

The identity opens the possibility to get the wavefunctions from the spectrum, an elusive goal of many fields in physics. It can be very useful from the numerical point of view, encouraging the study of new techniques to solve large eigenproblems. In general, it is much simpler and faster to calculate the eigenvalues than the eigenvectors. Moreover, for very large matrices, the memory storage requirement for the standard eigenvector solver can lead to serious problems, which could be overcome by this approach. A critical element to strengthening the use of the identity is the ability to avoid the calculation of the eigenvalues of all the minor matrices, as we did here.

Perspectives

The generalization to other potentials in 3 dimensions and for real scenarios can be a matter of further interesting development.

Dario Mitnik
Instituto de Astronomia y Fisica del Espacio - CONICET-UBA Argentina

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This page is a summary of: Wavefunctions from energies: Applications in simple potentials, Journal of Mathematical Physics, June 2020, American Institute of Physics, DOI: 10.1063/5.0011115.
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