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I have a Hermitian Matrix $H$ that evolves very slightly over ~50 iterations until convergence of a Schrödinger/Poisson system is achieved. The eigenvalues I'm interested in are $\lambda_1$ and the degenerate $\lambda_2 = \lambda_3$. The eigenvectors $v_n$ are all orthogonal. I'm interested in the evolution of the Eigenvectors for which I have very accurate starting guesses $v_{n}$. I'm calculating $\lambda_n$ using Rayleigh-coefficients and plug those into the inverse iteration $(H-\lambda_n I )^{-1} v_{n} = v_{n}'$.

My problem is, that there is no way of knowing which of the degenerate Eigenvectors I get when plugging in $v_2$ and $v_3$. I've tried using Lanczos-algorithm to calculate the first three eigenvectors and eigenvalues and then sort them manually using overlap integrals, however this approach runs into a myriad of problems in the long run which is why I've resorted to calculating the Eigenvectors myself.

My question now is: Is there any way to update degenerate eigenvectors e.g. $v_2$ to $v_2'$ and $v_3$ to $v_3'$ without manually sorting them afterwards? I'm more interested in updated eigenvectors similar to the old eigenvector than accuracy of eigenvectors or eigenvalues.

A similar question was asked here, however being an engineer and not a mathematician I was not able to do anything with the suggested approaches How to get the same eigenvectors of a degenerate eigenvalue as the matrix evolves slightly?

Apparizzle
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    Certainly, there are matrices $H$ that have degenerate eigenvalues, but it is most unusual that a perturbation $\Delta H$ produces a another matrix with $H+\Delta H$ with degenerate eigenvalues. Our expectation is that the eigenvalues of $H + \Delta H$ are distinct. You should tells us more about the origins of your matrix. Moreover, current choice of words suggests that you have a family of matrices, say, $t \rightarrow H(t)$ such that $H(t)$ has a two dimensional eigenspace $V(t)$ corresponding to the eigenvalue $\lambda(t)$. You seek to obtain basis for $V(t)$, correct? – Carl Christian Feb 16 '22 at 18:40
  • What are the properties that you are looking for in this map? – Carl Christian Feb 16 '22 at 18:40
  • The Matrix $H$ stems from finite difference discretization of the Schrödinger equation in a finite rectangular potential and is about 2000x2000 in size. For the simplest case, assume the box is square and the potential is slightly lowered or raised with each iteration. The eigenvalues are still degenerate with each iteration and $v_2$ and $v_3$ are the same but rotated by $\frac{\pi}{2}$. – Apparizzle Feb 17 '22 at 09:40
  • I hope I understand your wording correct but yes, I want to obtain basis for $V(t)$ as $H$ changes but do so for $v_2$ and $v_3$ seperately, as I need to deal with eigenvectors of degenerate eigenvalues seperate of each other.

    I'm well aware that my wording may not be mathematically correct and I'm not sure if what I'm trying to do is even possible in the first place.

     – Apparizzle Feb 17 '22 at 09:45
  • Your example is a wonderful one that illustrates the power of special perturbations! I don't think that it is possible to do what you want to do. Proving that it cannot be done is another matter entirely. However, do you really need this ability? Is it not enough that you can construct the orthogonal projector onto the eigenspace V(t) and that this projector is a continuous function of the parameter $t$? – Carl Christian Feb 17 '22 at 10:37
  • Somehow the idea of applying perturbation theory totally escaped my mind, thank you! I'll leave the question open for now and see if your suggested approach works in my case in the meantime. – Apparizzle Feb 17 '22 at 12:17
  • I hope this works out for you. I collect counter examples and I would dearly like to be able to replicate your problem. I don't want you to waste time typing writing it out. Do you have a link to a general description of the overall problem which is solved when your iteration converges. – Carl Christian Feb 18 '22 at 20:45
  • Sure, the approach presented in this paper is similar to mine Subband decomposition approach for the simulation of quantum electron transport in nanostructures. In short, a transport equation is solved along with Poisson's equation to determine current flow through nanoscale FETs. The problem arises when solving the Schrödinger equation for FETs with a larger channel size. The perturbation approach is looking very promising, as there are lot of resources for degenerate and nearly degenerate problems. – Apparizzle Feb 21 '22 at 08:54
  • Thank you for this information! – Carl Christian Feb 21 '22 at 20:26

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