| import Mathlib | |
| import OAI.Analysis.HardSphere.Operators.SemigroupLaplaceCommute | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| open scoped BigOperators ComplexConjugate | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Real Complex | |
| open scoped ENNReal ComplexConjugate | |
| open ProbabilityTheory | |
| namespace HardSphere | |
| open MeasureTheory Set Filter ContinuousLinearMap | |
| open scoped Topology | |
| lemma killedTransition_eq_formHeat (N : ℕ) (L : Length) (a : ℝ) {t : ℝ} (ht : 0<t) : | |
| killedTransition N L a t=formHeat N L a t := by | |
| rcases subsingleton_or_nontrivial (ScalarL2 N L) with h | h | |
| · exact Subsingleton.elim _ _ | |
| exact semigroup_eq_heat (killedTransition N L a) (killedTransition_add_nonneg N L a) | |
| (killedTransition_clamped_continuous N L a) | |
| (fun t _ ↦ killedTransition_apply_contraction N L a t) | |
| (killedTransition_selfAdjoint N L a) | |
| (formResolvent N L a) (formResolvent_isPositive N L a) | |
| (formResolvent_norm_le_one N L a) (killedLaplace_eq_formResolvent (N := N) (L := L) (a := a)) ht | |
| end HardSphere | |
| end | |
| end OAI | |