OAI/MathematicalPhysics/DefocusingNLS/Linear/HomogeneousFiniteRankCriterion.lean

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3 declarations, 0 contain sorry, 0 declared as axioms

Invertibility reduces to the actual finite-dimensional remainder

This is the finite-dimensional Schur complement of the checked resolvent formula. It will be applied to a contraction plus the actual finite-rank remainder of the linearized time step.

  1. theorem OAI.DefocusingNLS.finiteRank_isUnit_iff line 26
  2. theorem OAI.DefocusingNLS.finiteRank_isUnit_iff_det_ne_zero line 63

    Source documentation

    The only obstruction is the determinant on the finite-dimensional range.

    theorem finiteRank_isUnit_iff_det_ne_zero (A : E →L[ℂ] E) (U : F →L[ℂ] E)
    (V : E →L[ℂ] F) (hA : IsUnit A) :
    IsUnit (A - U.comp V) ↔
      (ContinuousLinearMap.id ℂ F - V.comp ((Ring.inverse A).comp U)).det ≠ 0 := by
…
  3. theorem OAI.DefocusingNLS.finiteRank_spectrum_iff_det_zero line 71

    Source documentation

    Outside the spectrum of the contracting part, the spectral problem is finite-dimensional.

    theorem finiteRank_spectrum_iff_det_zero (B : E →L[ℂ] E) (U : F →L[ℂ] E)
    (V : E →L[ℂ] F) (z : ℂ) (hz : z ∈ resolventSet ℂ B) :
    z ∈ spectrum ℂ (B + U.comp V) ↔
      (ContinuousLinearMap.id ℂ F - V.comp ((resolvent B z).comp U)).det = 0 := by
…