Unique continuation and new Hohenberg-Kohn theorems Louis Garrigue Cirm, October 24, 2019 Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Hohenberg-Kohn theorem N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 � R d ( N − 1) | Ψ | 2 ( x , x 2 , . . . , x N ) d x 1 · · · d x N ρ Ψ ( x ) := N Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Hohenberg-Kohn theorem N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 � R d ( N − 1) | Ψ | 2 ( x , x 2 , . . . , x N ) d x 1 · · · d x N ρ Ψ ( x ) := N Theorem (Hohenberg-Kohn) Let w , v 1 , v 2 ∈ ? . If there are two ground states Ψ 1 and Ψ 2 of H N ( v 1 ) and H N ( v 2 ) , such that ρ Ψ 1 = ρ Ψ 2 , then v 1 = v 2 + E 1 − E 2 . N Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Hohenberg-Kohn theorem N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 � R d ( N − 1) | Ψ | 2 ( x , x 2 , . . . , x N ) d x 1 · · · d x N ρ Ψ ( x ) := N Theorem (Hohenberg-Kohn) Let w , v 1 , v 2 ∈ ? . If there are two ground states Ψ 1 and Ψ 2 of H N ( v 1 ) and H N ( v 2 ) , such that � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0 , then v 1 = v 2 + E 1 − E 2 . N Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Hohenberg-Kohn theorem N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 � R d ( N − 1) | Ψ | 2 ( x , x 2 , . . . , x N ) d x 1 · · · d x N ρ Ψ ( x ) := N Theorem (Hohenberg-Kohn) Let w , v 1 , v 2 ∈ ? . If there are two ground states Ψ 1 and Ψ 2 of H N ( v 1 ) and H N ( v 2 ) , such that � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0 , then v 1 = v 2 + E 1 − E 2 . N Works for bosons and fermions, in any dimension d . Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Hohenberg-Kohn theorem N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 � R d ( N − 1) | Ψ | 2 ( x , x 2 , . . . , x N ) d x 1 · · · d x N ρ Ψ ( x ) := N Theorem (Hohenberg-Kohn) Let w , v 1 , v 2 ∈ ? . If there are two ground states Ψ 1 and Ψ 2 of H N ( v 1 ) and H N ( v 2 ) , such that � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0 , then v 1 = v 2 + E 1 − E 2 . N Works for bosons and fermions, in any dimension d . Lieb remarked this relies on a strong unique continuation d 2 ( R d ) + L ∞ ( R d ) property (1983). He conjectured ? = L Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Hohenberg-Kohn theorem N N � � � H N ( v ) := − ∆ i + w ( x i − x j ) + v ( x i ) i =1 1 � i < j � N i =1 � R d ( N − 1) | Ψ | 2 ( x , x 2 , . . . , x N ) d x 1 · · · d x N ρ Ψ ( x ) := N Theorem (Hohenberg-Kohn) Let w , v 1 , v 2 ∈ ? . If there are two ground states Ψ 1 and Ψ 2 of H N ( v 1 ) and H N ( v 2 ) , such that � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0 , then v 1 = v 2 + E 1 − E 2 . N Works for bosons and fermions, in any dimension d . Lieb remarked this relies on a strong unique continuation d 2 ( R d ) + L ∞ ( R d ) property (1983). He conjectured ? = L d N 2 ( R d ) + L ∞ ( R d ) by Jerison-Kenig (1985) We can take ? = L Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Proof of the Hohenberg-Kohn theorem � �� N � � � Ψ , i =1 v ( x i ) Ψ = R d v ρ Ψ 1 Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Proof of the Hohenberg-Kohn theorem � �� N � � � Ψ , i =1 v ( x i ) Ψ = R d v ρ Ψ 1 2 E 1 � � Ψ 2 , H N ( v 1 )Ψ 2 � � = E 2 + R d ρ Ψ 2 ( v 1 − v 2 ) Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Proof of the Hohenberg-Kohn theorem � �� N � � � Ψ , i =1 v ( x i ) Ψ = R d v ρ Ψ 1 2 E 1 � � Ψ 2 , H N ( v 1 )Ψ 2 � � = E 2 + R d ρ Ψ 2 ( v 1 − v 2 ) 3 Exchanging 1 ↔ 2 gives E 1 − E 2 � � R d ρ Ψ 1 ( v 1 − v 2 ) Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Proof of the Hohenberg-Kohn theorem � �� N � � � Ψ , i =1 v ( x i ) Ψ = R d v ρ Ψ 1 2 E 1 � � Ψ 2 , H N ( v 1 )Ψ 2 � � = E 2 + R d ρ Ψ 2 ( v 1 − v 2 ) 3 Exchanging 1 ↔ 2 gives E 1 − E 2 � � R d ρ Ψ 1 ( v 1 − v 2 ) 4 Using � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0, the � ’s above are =, Ψ 2 , H N ( v 1 )Ψ 2 � � hence = E 1 , that is Ψ 2 is a ground state for H N ( v 1 ), so H N ( v 1 )Ψ 2 = E 1 Ψ 2 Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Proof of the Hohenberg-Kohn theorem � �� N � � � Ψ , i =1 v ( x i ) Ψ = R d v ρ Ψ 1 2 E 1 � � Ψ 2 , H N ( v 1 )Ψ 2 � � = E 2 + R d ρ Ψ 2 ( v 1 − v 2 ) 3 Exchanging 1 ↔ 2 gives E 1 − E 2 � � R d ρ Ψ 1 ( v 1 − v 2 ) 4 Using � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0, the � ’s above are =, Ψ 2 , H N ( v 1 )Ψ 2 � � hence = E 1 , that is Ψ 2 is a ground state for H N ( v 1 ), so H N ( v 1 )Ψ 2 = E 1 Ψ 2 5 Substracting it with H N ( v 2 )Ψ 2 = E 2 Ψ 2 , we get � N � � E 1 − E 2 + ( v 2 − v 1 )( x i ) Ψ 2 = 0 i =1 Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Proof of the Hohenberg-Kohn theorem � �� N � � � Ψ , i =1 v ( x i ) Ψ = R d v ρ Ψ 1 2 E 1 � � Ψ 2 , H N ( v 1 )Ψ 2 � � = E 2 + R d ρ Ψ 2 ( v 1 − v 2 ) 3 Exchanging 1 ↔ 2 gives E 1 − E 2 � � R d ρ Ψ 1 ( v 1 − v 2 ) 4 Using � R d ( v 1 − v 2 )( ρ Ψ 1 − ρ Ψ 2 ) = 0, the � ’s above are =, Ψ 2 , H N ( v 1 )Ψ 2 � � hence = E 1 , that is Ψ 2 is a ground state for H N ( v 1 ), so H N ( v 1 )Ψ 2 = E 1 Ψ 2 5 Substracting it with H N ( v 2 )Ψ 2 = E 2 Ψ 2 , we get � N � � E 1 − E 2 + ( v 2 − v 1 )( x i ) Ψ 2 = 0 i =1 6 By strong unique continuation, |{ Ψ 2 ( X ) = 0 }| = 0, thus E 1 − E 2 + � N i =1 ( v 2 − v 1 )( x i ) = 0 and integrating on [0 , L ] d ( N − 1) , we obtain v 1 = v 2 + ( E 1 − E 2 ) / N Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Strong UCP Theorem (Strong UCP for many-body Schr¨ odinger operators) Assume that the potentials satisfy v , w ∈ L p loc ( R d ) with p > max (2 d / 3 , 2) . If Ψ ∈ H 2 loc ( R dN ) is a non zero solution to H N ( v )Ψ = E Ψ , then |{ Ψ( X ) = 0 }| = 0 . L. Garrigue , Unique continuation for many-body Schr¨ odinger operators and the Hohenberg-Kohn theorem , • Math. Phys. Anal. Geom., 21 (2018), p. 27. L. Garrigue , Unique continuation for many-body Schr¨ odinger operators and the Hohenberg-Kohn theorem. • II. The Pauli Hamiltonian , (2019), arXiv:1901.03207. Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Strong UCP Theorem (Strong UCP for many-body Schr¨ odinger operators) Assume that the potentials satisfy v , w ∈ L p loc ( R d ) with p > max (2 d / 3 , 2) . If Ψ ∈ H 2 loc ( R dN ) is a non zero solution to H N ( v )Ψ = E Ψ , then |{ Ψ( X ) = 0 }| = 0 . In 3 D , we can take ? = L p > 2 ( R 3 ) + L ∞ ( R 3 ). Covers Coulomb-like singularities L. Garrigue , Unique continuation for many-body Schr¨ odinger operators and the Hohenberg-Kohn theorem , • Math. Phys. Anal. Geom., 21 (2018), p. 27. L. Garrigue , Unique continuation for many-body Schr¨ odinger operators and the Hohenberg-Kohn theorem. • II. The Pauli Hamiltonian , (2019), arXiv:1901.03207. Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Strong UCP Theorem (Strong UCP for many-body Schr¨ odinger operators) Assume that the potentials satisfy v , w ∈ L p loc ( R d ) with p > max (2 d / 3 , 2) . If Ψ ∈ H 2 loc ( R dN ) is a non zero solution to H N ( v )Ψ = E Ψ , then |{ Ψ( X ) = 0 }| = 0 . In 3 D , we can take ? = L p > 2 ( R 3 ) + L ∞ ( R 3 ). Covers Coulomb-like singularities Works for excited states L. Garrigue , Unique continuation for many-body Schr¨ odinger operators and the Hohenberg-Kohn theorem , • Math. Phys. Anal. Geom., 21 (2018), p. 27. L. Garrigue , Unique continuation for many-body Schr¨ odinger operators and the Hohenberg-Kohn theorem. • II. The Pauli Hamiltonian , (2019), arXiv:1901.03207. Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
Magnetic case, the Pauli Hamiltonian N � ( σ j · ( − i ∇ j + A ( x j ))) 2 + v ( x j ) � H N ( v , A ) := � � + w ( x i − x j ) j =1 1 � i < j � N Theorem (Strong UCP for the many-body Pauli operator) Assume that the potentials satisfy div A = 0 and A ∈ L q loc ( R d ) with q > 2 d , curl A , v , w ∈ L p loc ( R d ) with p > max (2 d / 3 , 2) . If Ψ ∈ H 2 loc ( R dN ) is a non zero solution to H N ( v , A )Ψ = E Ψ , then |{ Ψ( X ) = 0 }| = 0 . Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
History of related UCP results Weak Number of Hypothesis Date Magnetic ? or Strong particles on v (loc) L ∞ Carleman 39 W 1 (and N ) No L 2 d / 3 H¨ ormander 63 W 1 No L 2 d / 3 Georgescu 80 W N No L d Schechter-Simon 80 W No N L d / 2 Jerison-Kenig 85 S 1 No Kurata 97 S 1 Many Yes L d / 2 Koch-Tataru 01 S 1 Yes 18 S N Many Yes Laestadius-Benedicks-Penz L p > 2 d / 3 Garrigue 19 S N Yes Louis Garrigue Unique continuation and new Hohenberg-Kohn theorems
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