Manuscript IV@1 · IV.tex · corollary · 351cb2d7

Corollary: Model 2 generator limit at fixed M

Model 2: spectral removal at fixed number cutoff › The fixed-M limit · lines 1932–1960 · proof included or linked

IV@1#cor:fixedM, IV@1#eq:deltaM

For every fixed finite $M \geq 2$ ,

B_{\infty, M} = Π_{M} B_{\infty} Π_{M}

is bounded self-adjoint, and

δ_{M} (R) := ∥ B_{R, M} - B_{\infty, M} ∥ \leq \frac{M C _{L}}{R} + \frac{2 M ( M - 1 ) C _{P}}{R} =: δ_{M} (R) .

Also, on $Dom N$ ,

∥ (B_{R} - B_{\infty}) Ψ ∥ \leq (C_{L} R^{- 1} + 2 C_{P} R^{- 1/2}) ∥ (N + 2) Ψ ∥ .

Proof. The off-diagonal block norms give $∥ T_{R} - T ∥ = ∥ L_{R} - L ∥$ , and the pair tensor difference has norm $2 ∥ P_{R} - P ∥_{HS}$ . Apply Lemma lem:quadratic2.

Displayed formulas in this passage

IV@1#eq:deltaM · line 1939
$δ_{M} (R) := ∥ B_{R, M} - B_{\infty, M} ∥ \leq \frac{M C _{L}}{R} + \frac{2 M ( M - 1 ) C _{P}}{R} =: δ_{M} (R) .$
unlabeled · line 1935
$B_{\infty, M} = Π_{M} B_{\infty} Π_{M}$
unlabeled · line 1948
$∥ (B_{R} - B_{\infty}) Ψ ∥ \leq (C_{L} R^{- 1} + 2 C_{P} R^{- 1/2}) ∥ (N + 2) Ψ ∥ .$

Corollary: Model 2 generator limit at fixed M

Displayed formulas in this passage

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