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I am trying to diagonalize the Bogoliubov-de Gennes Hamiltonian. The problem is that the Hamiltonian contains a Laplacian. This could be solved by using a discretized Laplacian

If I use a simple Hamiltonian (just the Laplacian) and use eig(l1dd(...,...)) with l1dd the discretized Laplacian (2 and -1's) as in https://people.sc.fsu.edu/~jburkardt/m_src/laplacian/laplacian.html, Then I get:

image

I expect a linear or quadratic dispersion ($\omega \propto k or k^2$), but I get neither. What could be the problem? It seems that eig does a good job, so probably l1dd is the problem?

Adriaan
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MrQ
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  • If there is any comment on cross-posting, feel free to remove the other question on sci-comp https://scicomp.stackexchange.com/questions/41051/numerical-diagonalization-of-hamiltonian (I cannot delete it apparently) – MrQ Feb 22 '22 at 18:04

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