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This method is well known from the experts. I thank Michael Barnsley for mentioning the Schur complement, and Pierre Duclos for the reference to Feschback. Some informations can be found in: P. LOWDIN, “.The calculation of upper and lower bounds of energy eigenvalues in perturbation theory by mean of partitioning technics” In Int. Summer School on “ Quantum theory of polyatomic molecules”, Menton. (1965).
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Bellissard, J. (1986). K-theory of C*—Algebras in solid state physics. In: Dorlas, T.C., Hugenholtz, N.M., Winnink, M. (eds) Statistical Mechanics and Field Theory: Mathematical Aspects. Lecture Notes in Physics, vol 257. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16777-3_74
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