Quantum Coulomb systems at the Brink Dirk Hundertmark (based on - PowerPoint PPT Presentation

Quantum Coulomb systems at the Brink Dirk Hundertmark (based on joint work with Markus Lange (UBC) and Michal Jex (KIT)) CIRM: The Analysis of Complex Quantum Systems: Large Coulomb Systems and Related Matters. Karlsruhe Institute of Technology – Institute for Analysis // CRC 1173 Wave Phenomena: Analysis and Numerics supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

Outline The approach Outline � Some results. � Decay of eigenstates at thresholds: Repulsion is your friend � The physical heuristic. � Making the heuristic into a proof: One body operators. � Helium. 1 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach An atom The operator for an atom with nuclear charge Z is � � � � N e 2 1 j − Ze 2 mP 2 H = + | x j | | x j − x k | j = 1 1 ≤ j < k ≤ N P j = − i � ∇ j = − i � ∇ x j , the kinetic energy of particle j , m electron mass, Z nuclear charge. Real atoms: Z = ze , z ∈ N , helium: Z = 2 e , but in theory any Z > 0 allowed � . � Acts on L 2 ( R 3 N ) (either with or without symmetry constraints) � No spin, but this can be easily handled. 2 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Scaling Let ( U λ ψ )( x ) ≔ λ 3 N ψ ( λ x ) , x = ( x 1 , · , x N ) ∈ R 3 N . Then U λ : L 2 ( R 3 N ) → L 2 ( R 3 N ) is unitary and � � � � � � N λ 2 � 2 λ e 2 (− ∆ j ψ )( λ x ) − Ze λ H ( U λ ψ )( λ x ) = λ 3 N | λ x j | ψ ( λ x ) | λ x j − λ x k | ψ ( λ x ) + 2 m j = 1 1 ≤ j < k ≤ N � N � � � � � = λ 3 N λ 2 � 2 2 me 2 2 mZ (− ∆ 2 j ψ )( λ x ) − � 2 λ | λ x j | ψ ( λ x ) � 2 λ | λ x j − λ x k | ψ ( λ x ) + 2 m j = 1 1 ≤ j < k ≤ N Choose λ = 2 mZ / � 2 , then � � = λ 3 N 2 mZ 2 2 mZ 2 � 2 [ H U ψ ] ( λ x ) = U λ � 2 H U ψ ( x ) / / with � � � � N j − 1 U P 2 H U = | x j − x k | , U = 1 / Z , P k = − i ∇ k . + | x j | 1 ≤ j < k ≤ N j = 1 3 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Letś concentrate on helium, N = 2. So we will consider the operator � � � 2 j − 1 U P 2 H U = | x 1 − x 2 | , + | x j | j = 1 with U > 0 from now on. 4 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Letś concentrate on helium, N = 2. So we will consider the operator � � � 2 j − 1 U P 2 H U = | x 1 − x 2 | , + | x j | j = 1 with U > 0 from now on. Known results: � Essential spectrum: σ ess ( H U ) = σ ( P 2 − | x | ) = [− 1 1 4 , ∞) , HVZ Theorem (Hunziker, van Winter, Zhislin). 4 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Letś concentrate on helium, N = 2. So we will consider the operator � � � 2 j − 1 U P 2 H U = | x 1 − x 2 | , + | x j | j = 1 with U > 0 from now on. Known results: � Essential spectrum: σ ess ( H U ) = σ ( P 2 − | x | ) = [− 1 1 4 , ∞) , HVZ Theorem (Hunziker, van Winter, Zhislin). For Physicists: You reach the continuum spectrum once you kicked one electron to infinity! 4 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Letś concentrate on helium, N = 2. So we will consider the operator � � � 2 j − 1 U P 2 H U = | x 1 − x 2 | , + | x j | j = 1 with U > 0 from now on. Known results: � Essential spectrum: σ ess ( H U ) = σ ( P 2 − | x | ) = [− 1 1 4 , ∞) , HVZ Theorem (Hunziker, van Winter, Zhislin). For Physicists: You reach the continuum spectrum once you kicked one electron to infinity! � If U = 1 there are infinitely many bound states below − 1 / 4. (Kato: More than 40.000, see e.g., Zhislin for full result ). 4 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach What about U > 1 ? Let E U = ground state energy of H U . Easy to see: 0 < U �→ E U is strictly increasing. So by convexity, it must ‘really increase’. 5 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach What about U > 1 ? Let E U = ground state energy of H U . Easy to see: 0 < U �→ E U is strictly increasing. So by convexity, it must ‘really increase’. � Bethe (1929): For some U > 1 one still has E U < − 1 / 4. � ⇒ there exists a critical U c > 1 with E U c = − 1 / 4. � ⇒ for any 1 < U < U c , the operator H U has a ground state ψ U with ground states energy E U < − 1 / 4. 5 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach What about U > 1 ? Let E U = ground state energy of H U . Easy to see: 0 < U �→ E U is strictly increasing. So by convexity, it must ‘really increase’. � Bethe (1929): For some U > 1 one still has E U < − 1 / 4. � ⇒ there exists a critical U c > 1 with E U c = − 1 / 4. � ⇒ for any 1 < U < U c , the operator H U has a ground state ψ U with ground states energy E U < − 1 / 4. � What happens with these ground states as U ր U c ? 5 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach What about U > 1 ? Let E U = ground state energy of H U . Easy to see: 0 < U �→ E U is strictly increasing. So by convexity, it must ‘really increase’. � Bethe (1929): For some U > 1 one still has E U < − 1 / 4. � ⇒ there exists a critical U c > 1 with E U c = − 1 / 4. � ⇒ for any 1 < U < U c , the operator H U has a ground state ψ U with ground states energy E U < − 1 / 4. � What happens with these ground states as U ր U c ? Theorem (Maria and Thomas Hoffmann-Ostenhof - Barry Simon (H 2 O 2 -S) 1984) There is binding for Helium at U c , that is, H U c has a ground state eigenfunction ψ U c ∈ L 2 ( R 6 ) and H U c ψ U c = − 1 4 ψ U c . 5 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach The result says nothing about how fast Ψ U c decays. 6 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach The result says nothing about how fast Ψ U c decays. Recall that with Agmon’s method one can show that as long as − 1 4 > E U = − ε U − 1 4 then | ψ U ( x 1 , x 2 )| � exp (−( 1 − ε ) ρ ( x )) for all 0 < ε < 1, ρ the Agmon distance ρ ( x ) = √ ε U � x ∞ � + 1 2 � x 0 � with � x ∞ � = max (� x 1 � , � x 2 �) (distance of the outer particle to the nucleus) and � x 0 � = min (� x 1 � , � x 2 �) (distance of the inner particle to the nucleus). This says nothing when U = U c , since then ε U c = 0. 6 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Frank–Lieb–Seiringer found a different proof from the one by H 2 O 2 -S, which also yields a kind of localization result: Theorem (Frank-Lieb-Seiringer 2012) For any δ > 0 , there is a constant C δ > 0 such that for all 1 + δ ≤ U < U c and for all normalized ground states ψ U of H U one has � ψ U , � x ∞ � − 1 ψ U � ≥ C δ 7 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Frank–Lieb–Seiringer found a different proof from the one by H 2 O 2 -S, which also yields a kind of localization result: Theorem (Frank-Lieb-Seiringer 2012) For any δ > 0 , there is a constant C δ > 0 such that for all 1 + δ ≤ U < U c and for all normalized ground states ψ U of H U one has � ψ U , � x ∞ � − 1 ψ U � ≥ C δ � By a compactness argument, this shows that a ground state at critical repulsion U = U c exists. � They have a similar result for bi-polarons and multi-polarons (for near energy minimizers). � Helium case got extended to atoms by Bellazzini–Frank–Lieb–Seiringer (2014). � Polaron case: no existence of bound states (in zero total momentum channel) Atoms: They need infinite nucleus mass approximation. 7 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Our result for Helium Theorem (51 e , Markus Lange, Michal Jex 2019) There exist constants c 1 , c 2 > 0 such that the ground state at critical coupling obeys � � � ψ U c ( x ) � exp − c 1 � x ∞ � + c 2 log � x ∞ � 8 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173

Outline The approach Our result for Helium Theorem (51 e , Markus Lange, Michal Jex 2019) There exist constants c 1 , c 2 > 0 such that the ground state at critical coupling obeys � � � ψ U c ( x ) � exp − c 1 � x ∞ � + c 2 log � x ∞ � � These constants are quantitative (have a ‘physical meaning’....) � Similar result for atoms. � We do not need the infinite nuclear mass aproximation! � Similar result for multipolaron systems at critical coupling in the zero total momentum channel. 8 September 12, 2019 Dirk Hundertmark – The Brink KIT – Institute for Analysis // CRC 1173