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I'm stuck with calculating the effective Hamiltonian of a model system. The energy-dependence of the effective Hamiltonian is confusing me a lot. My question is how to treat this dependence theoretically and numerically.

In principle, the model system I'm investigating is large but I'm even stuck with the easy three-level system. Hence, I will sketch you shortly what I have derived so far for this easy model. It would be very kind if you could help me with the problem.

My calculations are based on https://cec.mpg.de/fileadmin/media/Presse/Medien/Neese_Effective_Hamiltonian_Theory-MMER2014.pdf and Effective Hamiltonian / Perturbation theory for non-degenerate case and Finding the effective Hamiltonian in a certain subspace.

The Hamiltonian reads $H=\begin{pmatrix} \epsilon_1 & 0 & t \\ 0 & \epsilon_2 & t \\ t & t & \epsilon_3 \end{pmatrix}$. I would like to calculate the effective Hamiltonian by considering the levels with energy $\epsilon_2$ and $\epsilon_3$ as part 'b'. Then the effective Hamiltonian reads $H_{eff}(E)= \epsilon_1 - \begin{pmatrix} 0 & t \end{pmatrix} \big( \begin{pmatrix} \epsilon_2 & t \\ t & \epsilon_3 \end{pmatrix} - \begin{pmatrix} E & 0 \\ 0 & E \end{pmatrix} \big)^{-1} \begin{pmatrix} 0 \\ t \end{pmatrix}$. Moreover, the effective Hamiltonian has to fulfill the condition $H_{eff}(E) = E$. I'm not sure how to proceed with calculating the effective Hamiltonian. Of course, I'm able to invert a two-by-two matrix. However, the model system I want to treat is larger than the three-level system and hence, I cannot explicitly perform the matrix subtraction.

Question:

In general, the equation $H_{eff}(E) = E$ is very complex to solve. Is it possible to calculate all three eigenvalues $E_1$, $E_2$, $E_3$ of the whole Hamiltonian $H$ and just plug it into this equation? Is the effective Hamiltonian then the sum as $H_{eff}=H_{eff}(E_1) + H_{eff}(E_2) + H_{eff}(E_3)$ or does just the action of $H_{eff}$ depend on the energy of the ket state, e.g. $H_{eff} |E_1> = H_{eff}(E_1)|E_1>$ where $|E_1>$ eigenstate with energy $E_1$? Is there an iterative approach that can tackle the problem of solving $H_{eff}(E) = E$ for any arbitrary model system?

If I should provide you more calculations/information, please tell me. Thanks in advance for your help.

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  • $\begingroup$ If you can solve the Hamiltonian exactly, then there is no point to use an effective Hamiltonian. I suppose that for your large system such a solution is impossible, so obtaining an effective Hamiltonian involves an approximation. E.g., if it is the effective Hamiltonian for level 1, you may be justified setting $E=\epsilon_1$. Often it boils down to second order perturbation corrections. $\endgroup$
    – Roger V.
    Commented Apr 14, 2020 at 5:38
  • $\begingroup$ Thank you for your comment. Yes, it is impossible for the large system. However, how does the effective Hamiltonian read for the three-level system without any approximations? Is it $H(E_1)$ or a linear combination as $H(E_1) + H(E_2)$? $\endgroup$
    – user553052
    Commented Apr 14, 2020 at 8:06
  • $\begingroup$ It is not very clear from the question what kind of an effective Hamiltonian you are trying to derive: only for level 1 or something else? $\endgroup$
    – Roger V.
    Commented Apr 14, 2020 at 8:09
  • $\begingroup$ Thanks for the quick response. Yes, I'm trying to calculate the effective Hamiltonian for level 1 in the three-level system. $\endgroup$
    – user553052
    Commented Apr 14, 2020 at 8:24

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