# Apply GEVP methods to find energies of oscillators in lattice theory

1. May 11, 2014

### magnetic flux

I am writing an undergrad (Bachelor) thesis covering the harmonic and anharmonic oscillator using path integrals on discrete time lattice. With the material from [CF80], I have created a program that creates quantum trajectories on the lattice and calculates $|\psi(x)|^2$ and $E_0$ from that.

Now I have been given [Blo+09] as well to calculate higher energy eigenvalues. From what I understand so far, the energies are given by
$$E_n^\text{eff}(t, t_0) = - \frac1a \left( \log \lambda_n(t+a, t_0) - \log \lambda_n(t, t_0) \right),$$
where the eigenvalues $\lambda_n$ are determined by the GEVP (generalized eigenvalue problem)
$$C(t) v_n(t, t_0) = \lambda_n(t, t_0) C(t_0) v_n(t, t_0).$$
The matrices $C$ are “matrices of Euclidean space correlation function”, see (2.1) in the paper. Those contain operators/fields $O_i$ that are “some interpolating fields”.

I see that the energy values are what I need, but I am not sure where I can put my data to use. Since I have just one harmonic oscillator, I am not sure what my “fields” are supposed to be.

Do you see how I can extend my Metropolis Monte Carlo algorithm to compute higher energies with the contents of that paper? If so, could you please point me into the right direction?

Last edited by a moderator: May 6, 2017