# rspert.m

Rayleigh-Schrodinger perturbation theory to arbitrary order, Eqs 2.21-2.23 from Stefan Stoll's PhD thesis, with the typo fixed in the numerator of Eq 2.21.

## Syntax

[E,V]=rspert(E0,H1,order)

## Arguments

E0 - eigenvalues of H0, a column vector of real numbers H1 - perturbation, written in the basis that di- agonalises H0 order - order of perturbation theory to be used, 6 is the sensible maximum

## Outputs

E - eigenvalues of H0+H1 to the specified order in perturbation theory, a vector of reals V - normalised eigenvectors of H0+H1 to the spe- cified order in perturbation theory, a squa- re unitary matrix with eigenvectors in cols in the same order as the eigenvalues in E

## Examples

Below is the output of examples/fundamentals/perturb_theory.m example file.

## Notes

- There must be no degeneracies in H0.
- H1 must be Hermitian.
- The theory only converges when H1 << H0 in 2-norm.
- Numerical artefacts appear beyond sixth order.
- Complexity is linear in the order and cubic in matrix dimension.

## See also

vvpert.m, Numerical infrastructure

*Version 2.6, authors: Ilya Kuprov*