lean/OAI/Analysis/HardSphere/CondensationDefinitions.lean

import Mathlib

/-!
# Ground-state condensation in the dilute hard-sphere gas

The main result covers all pure ground states and all density operators supported on
 the ground space. In particular it covers the normalized ground-space projection.
Only the definitions needed to state this result are retained below.
-/

noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators InnerProductSpace ComplexOrder NNReal

namespace OAI.HardSphere

abbrev Length := {L : ℝ // 0 < L}

instance lengthPositive (L : Length) : Fact (0 < L.val) := ⟨L.property⟩

abbrev Torus (L : Length) := Fin 3 → AddCircle L.val

abbrev Configuration (N : ℕ) (L : Length) := Fin N → Torus L

abbrev LiftedConfiguration (N : ℕ) := Fin N → Fin 3 → ℝ

abbrev Direction (N : ℕ) := Fin N × Fin 3

def torusDistance (L : Length) (x y : Torus L) : ℝ :=
  Real.sqrt (∑ j : Fin 3, dist (x j) (y j) ^ 2)

def allowed (N : ℕ) (L : Length) (a : ℝ) : Set (Configuration N L) :=
  {X | ∀ i j : Fin N, i < j → a < torusDistance L (X i) (X j)}

def project (N : ℕ) (L : Length) (X : LiftedConfiguration N) : Configuration N L :=
  fun i j ↦ (X i j : AddCircle L.val)

def representative (N : ℕ) (L : Length) (X : Configuration N L) : LiftedConfiguration N :=
  fun i j ↦ (AddCircle.equivIco L.val 0 (X i j)).val

def coordDerivative (N : ℕ) (L : Length) (f : Configuration N L → ℂ)
    (p : Direction N) (X : Configuration N L) : ℂ :=
  fderiv ℝ (fun Y : LiftedConfiguration N ↦ f (project N L Y))
    (representative N L X) (Pi.single p.1 (Pi.single p.2 1))

abbrev ScalarL2 (N : ℕ) (L : Length) : Type :=
  Lp ℂ 2 (volume : Measure (Configuration N L))

abbrev Jet (N : ℕ) (L : Length) :=
  ScalarL2 N L × (Direction N → ScalarL2 N L)

structure SmoothTest (N : ℕ) (L : Length) (a : ℝ) where
  toFun : Configuration N L → ℂ
  smooth : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun X : LiftedConfiguration N ↦ toFun (project N L X))
  compact : HasCompactSupport toFun
  support_allowed : tsupport toFun ⊆ allowed N L a
  memLp : MemLp toFun (2 : ℕ) (volume : Measure (Configuration N L))
  partial_memLp : ∀ p : Direction N, MemLp (coordDerivative N L toFun p) (2 : ℕ) volume

def SmoothTest.jet {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a) : Jet N L :=
  (f.memLp.toLp f.toFun, fun p ↦ (f.partial_memLp p).toLp (coordDerivative N L f.toFun p))

def dirichletDomain (N : ℕ) (L : Length) (a : ℝ) : Set (Jet N L) :=
  closure (Set.range (SmoothTest.jet (N := N) (L := L) (a := a)))

def energy {N : ℕ} {L : Length} (u : Jet N L) : ℝ :=
  ∑ p : Direction N, ‖u.2 p‖ ^ 2

def bosonic {N : ℕ} {L : Length} (ψ : ScalarL2 N L) : Prop :=
  ∀ σ : Equiv.Perm (Fin N),
    (fun X : Configuration N L ↦ ψ (fun i ↦ X (σ i))) =ᵐ[volume] ψ

def IsGroundVector (N : ℕ) (L : Length) (a : ℝ) (u : Jet N L) : Prop :=
  u ∈ dirichletDomain N L a ∧ bosonic u.1 ∧ ‖u.1‖ = 1 ∧
    ∀ v ∈ dirichletDomain N L a, bosonic v.1 → ‖v.1‖ = 1 → energy u ≤ energy v

def insertFirst {N : ℕ} (hN : 0 < N) {L : Length}
    (x : Torus L) (Y : Configuration (N - 1) L) : Configuration N L :=
  fun i ↦ Fin.cons (α := fun _ ↦ Torus L) x Y (Fin.cast (Nat.sub_add_cancel (Nat.succ_le_of_lt hN)).symm i)

def occupation {N : ℕ} (hN : 0 < N) {L : Length} (ψ : ScalarL2 N L) : ℝ :=
  (L.val ^ 3)⁻¹ * ∫ Y : Configuration (N - 1) L,
    ‖∫ x : Torus L, ψ (insertFirst hN x Y)‖ ^ 2

namespace Density
variable (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

def basisSet : Set H := (exists_hilbertBasis (𝕜:=ℂ) (E:=H)).choose

def basis : HilbertBasis (basisSet H) ℂ H :=
  (exists_hilbertBasis (𝕜:=ℂ) (E:=H)).choose_spec.choose

structure Operator where
  toCLM : H →L[ℂ] H
  positive : 0≤toCLM
  trace_one : HasSum (fun i : basisSet H ↦ (inner ℂ (basis H i) (toCLM (basis H i))).re) 1

variable {H K : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
  [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]

def Operator.compressionTrace (T : Operator H) (A : H →L[ℂ] K) : ℝ :=
  let adjoint := DFunLike.coe (F := (H →L[ℂ] K) ≃ₗᵢ⋆[ℂ] K →L[ℂ] H)
    (β := fun _ ↦ K →L[ℂ] H) ContinuousLinearMap.adjoint A
  ∑' j : basisSet K,(inner ℂ (adjoint (basis K j)) (T.toCLM (adjoint (basis K j)))).re

end Density

namespace Orbital

def sliceIntegral {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) (Y : Configuration (N-1) L) : ℂ :=
  (Real.sqrt (L.val^3))⁻¹ • ∫x : Torus L,ψ (insertFirst hN x Y)

def map {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : ScalarL2 (N-1) L := by
  have integral_mul_le_sqrt {X : Type} [MeasurableSpace X] {ν : Measure X} (f g : X → ℝ) (hf : MemLp f 2 ν) (hg : MemLp g 2 ν)
      (hf0 : 0 ≤ᵐ[ν] f) (hg0 : 0 ≤ᵐ[ν] g) :
      ∫ x, f x * g x ∂ν ≤ Real.sqrt (∫ x, f x ^ 2 ∂ν) *
        Real.sqrt (∫ x, g x ^ 2 ∂ν) := by
    have h := integral_mul_le_Lp_mul_Lq_of_nonneg
      Real.HolderConjugate.two_two hf0 hg0 (by simpa using hf) (by simpa using hg)
    simpa only [Real.rpow_two, ← Real.sqrt_eq_rpow] using h

  have integral_norm_sq_le {X : Type} [MeasurableSpace X] {ν : Measure X} [IsFiniteMeasure ν] {f : X → ℂ} (hf : MemLp f 2 ν) :
      (∫ x, ‖f x‖ ∂ν)^2 ≤ ν.real univ * ∫ x, ‖f x‖ ^ 2 ∂ν := by
    have h := integral_mul_le_sqrt (fun _ : X ↦ 1) (fun x ↦ ‖f x‖)
      (memLp_const (1 : ℝ)) hf.norm (Eventually.of_forall fun _ ↦ zero_le_one)
      (Eventually.of_forall fun _ ↦ norm_nonneg _)
    simp only [one_mul, one_pow, integral_const, smul_eq_mul, mul_one] at h
    have ha : 0 ≤ ∫ x, ‖f x‖ ∂ν := integral_nonneg fun _ ↦ norm_nonneg _
    have hm : 0 ≤ ∫ x, ‖f x‖^2 ∂ν := integral_nonneg fun _ ↦ sq_nonneg _
    have hs := (sq_le_sq₀ ha (by positivity)).2 h
    simpa only [mul_pow, Real.sq_sqrt (measureReal_nonneg), Real.sq_sqrt hm] using hs

  have memLp_slices {X Y : Type} [MeasurableSpace X] [MeasurableSpace Y] {ν : Measure X} {μ : Measure Y} [IsFiniteMeasure ν] [IsFiniteMeasure μ] {f : Y × X → ℂ} (hf : MemLp f 2 (μ.prod ν)) :
      ∀ᵐ y ∂μ, MemLp (fun x ↦ f (y,x)) 2 ν := by
    filter_upwards [hf.aestronglyMeasurable.prodMk_left,
      (hf.integrable_norm_pow (by norm_num : (2 : ℕ) ≠ 0)).prod_right_ae] with y hm hi
    exact (memLp_two_iff_integrable_sq_norm hm).2 hi

  have memLp_integral_norm {X Y : Type} [MeasurableSpace X] [MeasurableSpace Y] {ν : Measure X} {μ : Measure Y} [IsFiniteMeasure ν] [IsFiniteMeasure μ] {f : Y × X → ℂ} (hf : MemLp f 2 (μ.prod ν)) :
      MemLp (fun y ↦ ∫ x, ‖f (y,x)‖ ∂ν) 2 μ := by
    have hm := hf.aestronglyMeasurable.norm.integral_prod_right'
    apply (memLp_two_iff_integrable_sq hm).2
    have hi := (hf.integrable_norm_pow (by norm_num : (2 : ℕ) ≠ 0)).integral_prod_left
    apply (hi.const_mul (ν.real univ)).mono' (hm.pow 2)
    filter_upwards [memLp_slices hf] with y hy
    change ‖(∫ x, ‖f (y,x)‖ ∂ν)^2‖ ≤ _
    rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
    exact integral_norm_sq_le hy

  have cons_preserving {n : ℕ} (L : Length) :
      MeasurePreserving (fun p : Configuration n L × Torus L ↦ Fin.cons (α:=fun _ ↦ Torus L) p.2 p.1)
        (volume.prod volume) volume := by
    have h := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm
    have hh := h.comp (Measure.measurePreserving_swap (μ:= (volume : Measure (Configuration n L))) (ν:=volume))
    convert hh using 1; try rfl
    funext p i
    simp [MeasurableEquiv.piFinSuccAbove]

  have insertFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) :
      MeasurePreserving (fun p : Torus L×Configuration (N-1) L ↦ insertFirst hN p.1 p.2)
        ((volume : Measure (Torus L)).prod volume) volume := by
    cases N with
    | zero => omega
    | succ n =>
      unfold insertFirst
      have h := (cons_preserving (n:=n) L).comp
        (Measure.measurePreserving_swap (μ:=(volume : Measure (Torus L))) (ν:=volume))
      convert h using 1 <;> simp [Function.comp_def]

  have bathFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) :
      MeasurePreserving (fun p : Configuration (N-1) L×Torus L ↦ insertFirst hN p.2 p.1)
        ((volume : Measure (Configuration (N-1) L)).prod volume) volume :=
    (insertFirst_preserving hN L).comp (Measure.measurePreserving_swap)

  have ground_slice_memLp {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) :
      MemLp (fun p : Configuration (N-1) L×Torus L ↦ ψ (insertFirst hN p.2 p.1)) 2
        ((volume : Measure (Configuration (N-1) L)).prod volume) :=
    (Lp.memLp ψ).comp_measurePreserving (bathFirst_preserving hN L)

  have sliceIntegral_memLp {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : MemLp (sliceIntegral hN ψ) 2 volume := by
    have hf := ground_slice_memLp hN ψ
    have ha := memLp_integral_norm hf
    have hh : MemLp (fun Y ↦ ∫x : Torus L,ψ (insertFirst hN x Y)) 2 volume := by
      refine ha.mono' hf.aestronglyMeasurable.integral_prod_right' ?_
      exact Eventually.of_forall fun Y ↦ norm_integral_le_integral_norm _
    exact hh.const_smul ((Real.sqrt (L.val^3))⁻¹ : ℝ)

  exact (sliceIntegral_memLp hN ψ).toLp (sliceIntegral hN ψ)

def continuous {N : ℕ} (hN : 0<N) {L : Length} : ScalarL2 N L →L[ℂ] ScalarL2 (N-1) L := by
  have scalar_norm_sq_integral {N : ℕ} {L : Length} (u : ScalarL2 N L) :
      ‖u‖^2 = ∫ X, ‖u X‖^2 := by
    rw [← real_inner_self_eq_norm_sq (F := ScalarL2 N L), L2.inner_def]
    simp only [real_inner_self_eq_norm_sq]

  have volume_torus (L : Length) : (volume : Measure (Torus L)) Set.univ = ENNReal.ofReal (L.val^3) := by
    rw [volume_pi,Measure.pi_univ]
    simp only [AddCircle.measure_univ,Finset.prod_const,Finset.card_univ,Fintype.card_fin,
      ENNReal.ofReal_pow L.property.le]

  have integral_mul_le_sqrt {X : Type} [MeasurableSpace X] {ν : Measure X} (f g : X → ℝ) (hf : MemLp f 2 ν) (hg : MemLp g 2 ν)
      (hf0 : 0 ≤ᵐ[ν] f) (hg0 : 0 ≤ᵐ[ν] g) :
      ∫ x, f x * g x ∂ν ≤ Real.sqrt (∫ x, f x ^ 2 ∂ν) *
        Real.sqrt (∫ x, g x ^ 2 ∂ν) := by
    have h := integral_mul_le_Lp_mul_Lq_of_nonneg
      Real.HolderConjugate.two_two hf0 hg0 (by simpa using hf) (by simpa using hg)
    simpa only [Real.rpow_two, ← Real.sqrt_eq_rpow] using h

  have integral_norm_sq_le {X : Type} [MeasurableSpace X] {ν : Measure X} [IsFiniteMeasure ν] {f : X → ℂ} (hf : MemLp f 2 ν) :
      (∫ x, ‖f x‖ ∂ν)^2 ≤ ν.real univ * ∫ x, ‖f x‖ ^ 2 ∂ν := by
    have h := integral_mul_le_sqrt (fun _ : X ↦ 1) (fun x ↦ ‖f x‖)
      (memLp_const (1 : ℝ)) hf.norm (Eventually.of_forall fun _ ↦ zero_le_one)
      (Eventually.of_forall fun _ ↦ norm_nonneg _)
    simp only [one_mul, one_pow, integral_const, smul_eq_mul, mul_one] at h
    have ha : 0 ≤ ∫ x, ‖f x‖ ∂ν := integral_nonneg fun _ ↦ norm_nonneg _
    have hm : 0 ≤ ∫ x, ‖f x‖^2 ∂ν := integral_nonneg fun _ ↦ sq_nonneg _
    have hs := (sq_le_sq₀ ha (by positivity)).2 h
    simpa only [mul_pow, Real.sq_sqrt (measureReal_nonneg), Real.sq_sqrt hm] using hs

  have memLp_slices {X Y : Type} [MeasurableSpace X] [MeasurableSpace Y] {ν : Measure X} {μ : Measure Y} [IsFiniteMeasure ν] [IsFiniteMeasure μ] {f : Y × X → ℂ} (hf : MemLp f 2 (μ.prod ν)) :
      ∀ᵐ y ∂μ, MemLp (fun x ↦ f (y,x)) 2 ν := by
    filter_upwards [hf.aestronglyMeasurable.prodMk_left,
      (hf.integrable_norm_pow (by norm_num : (2 : ℕ) ≠ 0)).prod_right_ae] with y hm hi
    exact (memLp_two_iff_integrable_sq_norm hm).2 hi

  have memLp_integral_norm {X Y : Type} [MeasurableSpace X] [MeasurableSpace Y] {ν : Measure X} {μ : Measure Y} [IsFiniteMeasure ν] [IsFiniteMeasure μ] {f : Y × X → ℂ} (hf : MemLp f 2 (μ.prod ν)) :
      MemLp (fun y ↦ ∫ x, ‖f (y,x)‖ ∂ν) 2 μ := by
    have hm := hf.aestronglyMeasurable.norm.integral_prod_right'
    apply (memLp_two_iff_integrable_sq hm).2
    have hi := (hf.integrable_norm_pow (by norm_num : (2 : ℕ) ≠ 0)).integral_prod_left
    apply (hi.const_mul (ν.real univ)).mono' (hm.pow 2)
    filter_upwards [memLp_slices hf] with y hy
    change ‖(∫ x, ‖f (y,x)‖ ∂ν)^2‖ ≤ _
    rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
    exact integral_norm_sq_le hy

  have cons_preserving {n : ℕ} (L : Length) :
      MeasurePreserving (fun p : Configuration n L × Torus L ↦ Fin.cons (α:=fun _ ↦ Torus L) p.2 p.1)
        (volume.prod volume) volume := by
    have h := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm
    have hh := h.comp (Measure.measurePreserving_swap (μ:= (volume : Measure (Configuration n L))) (ν:=volume))
    convert hh using 1; try rfl
    funext p i
    simp [MeasurableEquiv.piFinSuccAbove]

  have insertFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) :
      MeasurePreserving (fun p : Torus L×Configuration (N-1) L ↦ insertFirst hN p.1 p.2)
        ((volume : Measure (Torus L)).prod volume) volume := by
    cases N with
    | zero => omega
    | succ n =>
      unfold insertFirst
      have h := (cons_preserving (n:=n) L).comp
        (Measure.measurePreserving_swap (μ:=(volume : Measure (Torus L))) (ν:=volume))
      convert h using 1 <;> simp [Function.comp_def]

  have bathFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) :
      MeasurePreserving (fun p : Configuration (N-1) L×Torus L ↦ insertFirst hN p.2 p.1)
        ((volume : Measure (Configuration (N-1) L)).prod volume) volume :=
    (insertFirst_preserving hN L).comp (Measure.measurePreserving_swap)

  have ground_slice_memLp {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) :
      MemLp (fun p : Configuration (N-1) L×Torus L ↦ ψ (insertFirst hN p.2 p.1)) 2
        ((volume : Measure (Configuration (N-1) L)).prod volume) :=
    (Lp.memLp ψ).comp_measurePreserving (bathFirst_preserving hN L)

  have occupation_le_norm_sq {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) :
      occupation hN ψ≤‖ψ‖^2 := by
    have hLp := L.property
    let f := fun p : Configuration (N-1) L×Torus L ↦ ψ (insertFirst hN p.2 p.1)
    have hf : MemLp f 2 (volume.prod volume) := ground_slice_memLp hN ψ
    have ha := memLp_integral_norm hf
    have hc : MemLp (fun Y ↦ ‖∫x,f (Y,x)‖) 2 volume := by
      refine ha.mono' hf.aestronglyMeasurable.integral_prod_right'.norm ?_
      exact Eventually.of_forall fun Y ↦ by
        rw [Real.norm_eq_abs,abs_of_nonneg (norm_nonneg _)]
        exact norm_integral_le_integral_norm _
    have hi := (hf.integrable_norm_pow (by norm_num : (2:ℕ)≠0)).integral_prod_left
    have hbound : (∫Y,‖∫x,f (Y,x)‖^2)≤(volume : Measure (Torus L)).real univ * ∫p,‖f p‖^2 ∂volume.prod volume := by
      calc
        _ ≤ ∫Y,(volume : Measure (Torus L)).real univ * ∫x,‖f (Y,x)‖^2 := by
          apply integral_mono_ae hc.integrable_sq (hi.const_mul _)
          filter_upwards [memLp_slices hf] with Y hY
          exact (pow_le_pow_left₀ (norm_nonneg _) (norm_integral_le_integral_norm _) 2).trans
            (integral_norm_sq_le hY)
        _ = _ := by rw [integral_const_mul,←integral_prod _ (hf.integrable_norm_pow (by norm_num : (2:ℕ)≠0))]
    have hm : (∫p,‖f p‖^2 ∂volume.prod volume)=‖ψ‖^2 := by
      rw [scalar_norm_sq_integral]
      exact (bathFirst_preserving hN L).integral_comp
        (by
          cases N with
          | zero => omega
          | succ n =>
            convert (MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm.measurableEmbedding.comp
              (MeasurableEquiv.prodComm (α:=Configuration n L) (β:=Torus L)).measurableEmbedding using 1 <;> try rfl
            funext p i
            simp [insertFirst,MeasurableEquiv.piFinSuccAbove,MeasurableEquiv.prodComm])
        (fun X ↦ ‖ψ X‖^2)
    rw [hm,Measure.real,volume_torus,ENNReal.toReal_ofReal (by positivity : 0≤L.val^3)] at hbound
    unfold occupation
    change (L.val^3)⁻¹*(∫Y,‖∫x,f (Y,x)‖^2)≤‖ψ‖^2
    calc
      _ ≤ (L.val^3)⁻¹*(L.val^3*‖ψ‖^2) := mul_le_mul_of_nonneg_left hbound (by positivity)
      _ = _ := by rw [←mul_assoc,inv_mul_cancel₀ (pow_ne_zero _ L.property.ne'),one_mul]

  have map_ae {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : (map hN ψ : Configuration (N-1) L→ℂ)=ᵐ[volume] sliceIntegral hN ψ := by
    unfold map
    exact MemLp.coeFn_toLp _

  have map_add {N : ℕ} (hN : 0<N) {L : Length} (f g : ScalarL2 N L) : map hN (f+g)=map hN f+map hN g := by
    apply Lp.ext
    have hp := (bathFirst_preserving hN L).quasiMeasurePreserving.tendsto_ae.eventually (Lp.coeFn_add f g)
    have hs := Measure.ae_ae_of_ae_prod hp
    filter_upwards [map_ae hN (f+g),map_ae hN f,map_ae hN g,Lp.coeFn_add (map hN f) (map hN g),hs,
      memLp_slices (ground_slice_memLp hN f),memLp_slices (ground_slice_memLp hN g)] with Y hh hf hg hsum hY hfi hgi
    rw [hh,hsum]
    change sliceIntegral hN (f+g) Y=(map hN f) Y+(map hN g) Y
    rw [hf,hg]
    unfold sliceIntegral
    rw [←smul_add,←integral_add (hfi.integrable (by norm_num)) (hgi.integrable (by norm_num))]
    congr 1
    apply integral_congr_ae
    exact hY

  have map_smul {N : ℕ} (hN : 0<N) {L : Length} (c : ℂ) (f : ScalarL2 N L) : map hN (c • f)=c • map hN f := by
    apply Lp.ext
    have hp := (bathFirst_preserving hN L).quasiMeasurePreserving.tendsto_ae.eventually (Lp.coeFn_smul c f)
    have hs := Measure.ae_ae_of_ae_prod hp
    filter_upwards [map_ae hN (c•f),map_ae hN f,Lp.coeFn_smul c (map hN f),hs] with Y hh hf hc hY
    rw [hh,hc]
    change sliceIntegral hN (c•f) Y=c • (map hN f) Y
    rw [hf]
    change (Real.sqrt (L.val^3))⁻¹ • (∫x : Torus L,(c•f) (insertFirst hN x Y))=
      c • ((Real.sqrt (L.val^3))⁻¹ • ∫x : Torus L,f (insertFirst hN x Y))
    rw [integral_congr_ae hY]
    simp only [Pi.smul_apply]
    rw [integral_smul]
    exact smul_comm _ _ _

  have map_norm_sq {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : ‖map hN ψ‖^2=occupation hN ψ := by
    have hLp := L.property
    rw [scalar_norm_sq_integral]
    have hsi := integral_congr_ae ((map_ae hN ψ).fun_comp fun x ↦ ‖x‖^2)
    simp only [Function.comp_def] at hsi
    rw [hsi]
    unfold sliceIntegral occupation
    simp only [norm_smul,Real.norm_eq_abs,mul_pow,sq_abs,inv_pow,
      integral_const_mul,Real.sq_sqrt (by positivity : 0≤L.val^3)]

  have map_norm_le {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : ‖map hN ψ‖≤‖ψ‖ := by
    have hh := (map_norm_sq hN ψ).trans_le (occupation_le_norm_sq hN ψ)
    nlinarith [norm_nonneg (map hN ψ),norm_nonneg ψ]

  exact ({ toFun := map hN, map_add' := map_add hN, map_smul' := map_smul hN } : ScalarL2 N L →ₗ[ℂ] ScalarL2 (N-1) L).mkContinuous 1 (fun ψ ↦ by
    change ‖map hN ψ‖ ≤ 1 * ‖ψ‖
    simpa only [one_mul] using map_norm_le hN ψ)

end Orbital

local instance groundScalarL2Complete (N : ℕ) (L : Length) : CompleteSpace (ScalarL2 N L) :=
  inferInstance

def GroundSupported {N : ℕ} {L : Length} (a : ℝ)
    (T : Density.Operator (ScalarL2 N L)) : Prop :=
  ∀ψ∈T.toCLM.kerᗮ,‖ψ‖=1 →
    ∃u : Jet N L,u.1=ψ ∧ IsGroundVector N L a u

def mixedOccupation {N : ℕ} (hN : 0<N) {L : Length}
    (T : Density.Operator (ScalarL2 N L)) : ℝ :=
  T.compressionTrace (Orbital.continuous hN)

end OAI.HardSphere