Trails in Quantum Mechanics and Surroundings
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1 A Remark on the Hamiltonian for Three Identical Bosons with Zero-Range Interactions Giulia Basti Dipartimento di Matematica Università degli studi di Roma La Sapienza 09 July 2015 Trails in Quantum Mechanics and Surroundings Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
2 Outline 1 Introduction 2 Explicit Construction 3 Fermionic Case 4 Bosonic Case Main References [FM] L. Faddeev, R.A. Minlos, Soviet Phys. Dokl., 6 (1962). [MeM] A.M. Melnikov, R.A. Minlos, Adv. Soviet Math., 5 (1991). [CDFMT] M. Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D. Finco, A. Teta, Europhysics Letters, (2015 ). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
![Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D.](/docs-images/44/15228296/images/page_2.jpg "Finco, A. Teta, Europhysics Letters, (2015 ).")
3 Outline 1 Introduction 2 Explicit Construction 3 Fermionic Case 4 Bosonic Case Main References [FM] L. Faddeev, R.A. Minlos, Soviet Phys. Dokl., 6 (1962). [MeM] A.M. Melnikov, R.A. Minlos, Adv. Soviet Math., 5 (1991). [CDFMT] M. Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D. Finco, A. Teta, Europhysics Letters, (2015 ). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
![Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D.](/docs-images/44/15228296/images/page_3.jpg "Finco, A. Teta, Europhysics Letters, (2015 ).")
4 Outline 1 Introduction 2 Explicit Construction 3 Fermionic Case 4 Bosonic Case Main References [FM] L. Faddeev, R.A. Minlos, Soviet Phys. Dokl., 6 (1962). [MeM] A.M. Melnikov, R.A. Minlos, Adv. Soviet Math., 5 (1991). [CDFMT] M. Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D. Finco, A. Teta, Europhysics Letters, (2015 ). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
![Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D.](/docs-images/44/15228296/images/page_4.jpg "Finco, A. Teta, Europhysics Letters, (2015 ).")
5 Outline 1 Introduction 2 Explicit Construction 3 Fermionic Case 4 Bosonic Case Main References [FM] L. Faddeev, R.A. Minlos, Soviet Phys. Dokl., 6 (1962). [MeM] A.M. Melnikov, R.A. Minlos, Adv. Soviet Math., 5 (1991). [CDFMT] M. Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D. Finco, A. Teta, Europhysics Letters, (2015 ). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
![Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D.](/docs-images/44/15228296/images/page_5.jpg "Finco, A. Teta, Europhysics Letters, (2015 ).")
6 Outline 1 Introduction 2 Explicit Construction 3 Fermionic Case 4 Bosonic Case Main References [FM] L. Faddeev, R.A. Minlos, Soviet Phys. Dokl., 6 (1962). [MeM] A.M. Melnikov, R.A. Minlos, Adv. Soviet Math., 5 (1991). [CDFMT] M. Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D. Finco, A. Teta, Europhysics Letters, (2015 ). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
![Correggi, G. Dell Antonio, D. Finco, A. Michelangeli, A. Teta, Rev. Math. Phys., 24 (2012). [CFT] M. Correggi, D.](/docs-images/44/15228296/images/page_6.jpg "Finco, A. Teta, Europhysics Letters, (2015 ).")
7 Zero-Range Interaction System of three particles interacting via zero-range two-body interactions. Formally 3ÿ 1 H = x 2m i + ÿ ij (x i x j ) i=1 i i<j where x i œ R 3, X =(x 1, x 2, x 3 ) œ R 9 and ij œ R. Motivations Nuclear Physics Ultra-Cold Quantum Gases We look for a rigorous definition: self-adjoint operator (crucial to define the dynamics), possibly bounded from below (in order to ensure the stability of the system). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
8 Zero-Range Interaction System of three particles interacting via zero-range two-body interactions. Formally 3ÿ 1 H = x 2m i + ÿ ij (x i x j ) i=1 i i<j where x i œ R 3, X =(x 1, x 2, x 3 ) œ R 9 and ij œ R. Motivations Nuclear Physics Ultra-Cold Quantum Gases We look for a rigorous definition: self-adjoint operator (crucial to define the dynamics), possibly bounded from below (in order to ensure the stability of the system). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
9 Zero-Range Interaction System of three particles interacting via zero-range two-body interactions. Formally 3ÿ 1 H = x 2m i + ÿ ij (x i x j ) i=1 i i<j where x i œ R 3, X =(x 1, x 2, x 3 ) œ R 9 and ij œ R. Motivations Nuclear Physics Ultra-Cold Quantum Gases We look for a rigorous definition: self-adjoint operator (crucial to define the dynamics), possibly bounded from below (in order to ensure the stability of the system). Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
10 Rigorous Definition Note that So we consider H = H 0  if  vanishes on each hyperplane {x i = x j } ÊH 0 = 1 2m i 3ÿ i=1 x i D( Ê H 0 )=C Œ 0 1 R 9 \ i<j{x 2 i = x j } symmetric but not self-adjoint. = by definition any self-adjoint extension of Ê H 0, di erent from H 0, is a Hamiltonian for a system of three particles in R 3 with zero-range interactions. The problem is to give an explicit construction. = We proceed by analogy with the well known two-body case. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
11 Rigorous Definition Note that So we consider H = H 0  if  vanishes on each hyperplane {x i = x j } ÊH 0 = 1 2m i 3ÿ i=1 x i D( Ê H 0 )=C Œ 0 1 R 9 \ i<j{x 2 i = x j } symmetric but not self-adjoint. = by definition any self-adjoint extension of Ê H 0, di erent from H 0, is a Hamiltonian for a system of three particles in R 3 with zero-range interactions. The problem is to give an explicit construction. = We proceed by analogy with the well known two-body case. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
12 Rigorous Definition Note that So we consider H = H 0  if  vanishes on each hyperplane {x i = x j } ÊH 0 = 1 2m i 3ÿ i=1 x i D( Ê H 0 )=C Œ 0 1 R 9 \ i<j{x 2 i = x j } symmetric but not self-adjoint. = by definition any self-adjoint extension of Ê H 0, di erent from H 0, is a Hamiltonian for a system of three particles in R 3 with zero-range interactions. The problem is to give an explicit construction. = We proceed by analogy with the well known two-body case. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
13 Rigorous Definition Note that So we consider H = H 0  if  vanishes on each hyperplane {x i = x j } ÊH 0 = 1 2m i 3ÿ i=1 x i D( Ê H 0 )=C Œ 0 1 R 9 \ i<j{x 2 i = x j } symmetric but not self-adjoint. = by definition any self-adjoint extension of Ê H 0, di erent from H 0, is a Hamiltonian for a system of three particles in R 3 with zero-range interactions. The problem is to give an explicit construction. = We proceed by analogy with the well known two-body case. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
14 STM Extensions STM extensions H a of  H 0 : H a  = H 0  if x i = x j and  œd(h a ) satisfies the boundary conditions A 1  ƒ x i x j æ0 x i x j 1 B Q ij (r ij, x k )+o(1), a r ij = m ix i + m j x j m i + m j. In general H a is neither self-adjoint nor bounded from below and its s.a. extensions are in general unbounded from below (Thomas e ect [FM], [MeM]). We show s.a. and boundness from below for two systems using the quadratic form (a = Œ): F[Â] =(Â, HÂ) for any  œd(h) = If F is closed and bounded from below then H is self-adjoint and bounded from below. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
15 STM Extensions STM extensions H a of  H 0 : H a  = H 0  if x i = x j and  œd(h a ) satisfies the boundary conditions A 1  ƒ x i x j æ0 x i x j 1 B Q ij (r ij, x k )+o(1), a r ij = m ix i + m j x j m i + m j. In general H a is neither self-adjoint nor bounded from below and its s.a. extensions are in general unbounded from below (Thomas e ect [FM], [MeM]). We show s.a. and boundness from below for two systems using the quadratic form (a = Œ): F[Â] =(Â, HÂ) for any  œd(h) = If F is closed and bounded from below then H is self-adjoint and bounded from below. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
16 STM Extensions STM extensions H a of  H 0 : H a  = H 0  if x i = x j and  œd(h a ) satisfies the boundary conditions A 1  ƒ x i x j æ0 x i x j 1 B Q ij (r ij, x k )+o(1), a r ij = m ix i + m j x j m i + m j. In general H a is neither self-adjoint nor bounded from below and its s.a. extensions are in general unbounded from below (Thomas e ect [FM], [MeM]). We show s.a. and boundness from below for two systems using the quadratic form (a = Œ): F[Â] =(Â, HÂ) for any  œd(h) = If F is closed and bounded from below then H is self-adjoint and bounded from below. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
17 STM Extensions STM extensions H a of  H 0 : H a  = H 0  if x i = x j and  œd(h a ) satisfies the boundary conditions A 1  ƒ x i x j æ0 x i x j 1 B Q ij (r ij, x k )+o(1), a r ij = m ix i + m j x j m i + m j. In general H a is neither self-adjoint nor bounded from below and its s.a. extensions are in general unbounded from below (Thomas e ect [FM], [MeM]). We show s.a. and boundness from below for two systems using the quadratic form (a = Œ): F[Â] =(Â, HÂ) for any  œd(h) = If F is closed and bounded from below then H is self-adjoint and bounded from below. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
18 The Potential Produced by Q ij Let us introduce the potential produced by the charges Q ij : (GQ)(X) = ÿ 1 (2fi) i<j 5 dpe ˆQ ix P ij (p i + p j, p k ) µ ij h 0 (P) where h 0 (P) = p2 1 2m 1 + p2 2 2m 2 + p2 3 2m 3 Properties H 0 (GQ)(X) =2fi q ij and µ ij = m i m j m i +m j. 1 µ ij Q ij (r ij, x k ) (x i x j ) (GQ)(X) ƒ Q ij(r ij,x k ) x i x j ( Q) ij (r ij, x k )+o(1) if x i x j æ0 Writing  = w + GQ the boundary conditions become w smooth lim ij(r ij, x k ). x i x j æ0 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
19 The Potential Produced by Q ij Let us introduce the potential produced by the charges Q ij : (GQ)(X) = ÿ 1 (2fi) i<j 5 dpe ˆQ ix P ij (p i + p j, p k ) µ ij h 0 (P) where h 0 (P) = p2 1 2m 1 + p2 2 2m 2 + p2 3 2m 3 Properties H 0 (GQ)(X) =2fi q ij and µ ij = m i m j m i +m j. 1 µ ij Q ij (r ij, x k ) (x i x j ) (GQ)(X) ƒ Q ij(r ij,x k ) x i x j ( Q) ij (r ij, x k )+o(1) if x i x j æ0 Writing  = w + GQ the boundary conditions become w smooth lim ij(r ij, x k ). x i x j æ0 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
20 The Potential Produced by Q ij Let us introduce the potential produced by the charges Q ij : (GQ)(X) = ÿ 1 (2fi) i<j 5 dpe ˆQ ix P ij (p i + p j, p k ) µ ij h 0 (P) where h 0 (P) = p2 1 2m 1 + p2 2 2m 2 + p2 3 2m 3 Properties H 0 (GQ)(X) =2fi q ij and µ ij = m i m j m i +m j. 1 µ ij Q ij (r ij, x k ) (x i x j ) (GQ)(X) ƒ Q ij(r ij,x k ) x i x j ( Q) ij (r ij, x k )+o(1) if x i x j æ0 Writing  = w + GQ the boundary conditions become w smooth lim ij(r ij, x k ). x i x j æ0 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
21 Quadratic Form Defining D Á = {X œ R 9 : x i x j >Á} we have (Â, HÂ) =lim dx ÂH 0  Áæ0 D Á =(w, H 0 w)+ lim dx GQ H 0 w Áæ0 D Á =(w, H 0 w)+ ÿ 2fi µ i<j ij Q 1 c ÿ ijk 2Ÿ a 2fiµ ij µ jk dpdk ij (k, p) 2 Û µij M m k (m i + m j ) k2 + µ ij M p2 ij (k jk, p) jk (k ij, p) dpdk ij dk jk kij 2 2µ ij + k2 jk 2µ jk + k ij k jk m j where ij (k, p) = ˆQ ij ( m i +m j M p k, m k M p + k) with M = q i m i. + p2 2M R d b Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
22 Quadratic Form Defining D Á = {X œ R 9 : x i x j >Á} we have (Â, HÂ) =lim dx ÂH 0  Áæ0 D Á =(w, H 0 w)+ lim dx GQ H 0 w Áæ0 D Á =(w, H 0 w)+ ÿ 2fi µ i<j ij Q 1 c ÿ ijk 2Ÿ a 2fiµ ij µ jk dpdk ij (k, p) 2 Û µij M m k (m i + m j ) k2 + µ ij M p2 ij (k jk, p) jk (k ij, p) dpdk ij dk jk kij 2 2µ ij + k2 jk 2µ jk + k ij k jk m j where ij (k, p) = ˆQ ij ( m i +m j M p k, m k M p + k) with M = q i m i. + p2 2M R d b Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
23 2 Fermions + 1 Let us consider two identical fermions (e.g. 1 and 3) of mass m interacting with a di erent particle (e.g. 2) of mass m 0. Let = m m 0 the mass ratio. Hilbert space: L 2 asym(r 9 ) 13 = 0 and 12 = 23 = because of antisymmetry Energy form D[F] ={ œ L 2 asym(r 9 ): = w + G,w œ H 1 asym(r 9 ), œ H 1/2 (R 6 )} F[Â] =F 0 [w]+ 2(1 + ) mfi where F 0 [w] =(w, H 0 w) and [ ] = s dp p [ ] = s dp( p diag [ ]+ p o [ ]) is a form acting on the charge. [ ] Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
24 2 Fermions + 1 Let us consider two identical fermions (e.g. 1 and 3) of mass m interacting with a di erent particle (e.g. 2) of mass m 0. Let = m m 0 the mass ratio. Hilbert space: L 2 asym(r 9 ) 13 = 0 and 12 = 23 = because of antisymmetry Energy form D[F] ={ œ L 2 asym(r 9 ): = w + G,w œ H 1 asym(r 9 ), œ H 1/2 (R 6 )} F[Â] =F 0 [w]+ 2(1 + ) mfi where F 0 [w] =(w, H 0 w) and [ ] = s dp p [ ] = s dp( p diag [ ]+ p o [ ]) is a form acting on the charge. [ ] Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
25 2 Fermions + 1 Let us consider two identical fermions (e.g. 1 and 3) of mass m interacting with a di erent particle (e.g. 2) of mass m 0. Let = m m 0 the mass ratio. Hilbert space: L 2 asym(r 9 ) 13 = 0 and 12 = 23 = because of antisymmetry Energy form D[F] ={ œ L 2 asym(r 9 ): = w + G,w œ H 1 asym(r 9 ), œ H 1/2 (R 6 )} F[Â] =F 0 [w]+ 2(1 + ) mfi where F 0 [w] =(w, H 0 w) and [ ] = s dp p [ ] = s dp( p diag [ ]+ p o [ ]) is a form acting on the charge. [ ] Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
26 Stability The following theorem has been proved [CDFMT]: Theorem The energy form is positive and closed if and only if Æ c where c is the solution of the equation ( ) = Ô arcsin = 1 fi This implies that H is self-adjoint and bounded from below. Remark The result is optimal. For the analogous N + 1 problem there is only a partial result. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
27 Stability The following theorem has been proved [CDFMT]: Theorem The energy form is positive and closed if and only if Æ c where c is the solution of the equation ( ) = Ô arcsin = 1 fi This implies that H is self-adjoint and bounded from below. Remark The result is optimal. For the analogous N + 1 problem there is only a partial result. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
28 3 Bosons Let us now consider a system composed by three identical bosons of mass 1. Hilbert space: L 2 sym(r 9 ) ij = for all i, j. Energy form: D[F] ={ œ L 2 sym(r 9 ): = w + G,w œ H 1 sym (R 9 ), œ H 1/2 (R 6 )} F[Â] =(w, H 0 w)+ 6 fi [ ] where [ ] = s dp p [ ] = s dp( p diag [ ]+ p o [ ]) with p diag [ ] =2fi 2 s Ò dk (k, p) k2 + p2 6 p o [ ] = 2 s (k 1, p) (k 2, p) dk 1 dk 2 k1 2 + k2 2 + k 1 k 2 + p2 6 Such a form is well known to be unbounded. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
29 3 Bosons Let us now consider a system composed by three identical bosons of mass 1. Hilbert space: L 2 sym(r 9 ) ij = for all i, j. Energy form: D[F] ={ œ L 2 sym(r 9 ): = w + G,w œ H 1 sym (R 9 ), œ H 1/2 (R 6 )} F[Â] =(w, H 0 w)+ 6 fi [ ] where [ ] = s dp p [ ] = s dp( p diag [ ]+ p o [ ]) with p diag [ ] =2fi 2 s Ò dk (k, p) k2 + p2 6 p o [ ] = 2 s (k 1, p) (k 2, p) dk 1 dk 2 k1 2 + k2 2 + k 1 k 2 + p2 6 Such a form is well known to be unbounded. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
30 3 Bosons Let us now consider a system composed by three identical bosons of mass 1. Hilbert space: L 2 sym(r 9 ) ij = for all i, j. Energy form: D[F] ={ œ L 2 sym(r 9 ): = w + G,w œ H 1 sym (R 9 ), œ H 1/2 (R 6 )} F[Â] =(w, H 0 w)+ 6 fi [ ] where [ ] = s dp p [ ] = s dp( p diag [ ]+ p o [ ]) with p diag [ ] =2fi 2 s Ò dk (k, p) k2 + p2 6 p o [ ] = 2 s (k 1, p) (k 2, p) dk 1 dk 2 k1 2 + k2 2 + k 1 k 2 + p2 6 Such a form is well known to be unbounded. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
31 Main Result Theorem [ ] is positive for (k, p) = ÿ lm (k, p)y lm (, ). m,l l>0 Note: p diag [ ] positive p o [ ] not definite in sign Crucial point: p o [ ] Ø diag p [ ] with < 1. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
32 Main Result Theorem [ ] is positive for (k, p) = ÿ lm (k, p)y lm (, ). m,l l>0 Note: p diag [ ] positive p o [ ] not definite in sign Crucial point: p o [ ] Ø diag p [ ] with < 1. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
33 Main Result Theorem [ ] is positive for (k, p) = ÿ lm (k, p)y lm (, ). m,l l>0 Note: p diag [ ] positive p o [ ] not definite in sign Crucial point: p o [ ] Ø diag p [ ] with < 1. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
34 Main Result Theorem [ ] is positive for (k, p) = ÿ lm (k, p)y lm (, ). m,l l>0 Note: p diag [ ] positive p o [ ] not definite in sign Crucial point: p o [ ] Ø diag p [ ] with < 1. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
35 Sketch of the Proof p o [ ] Ø p diag [ ] [I]: p o [ ] = 2 dk 1 dk 2 (k 1, p) (k 2, p) k k2 2 + k 1 k 2 + p2 6 [exp. in spherical harmonics] = 4fi ÿ Œ Œ dk 1 dk 2 k1 2 lm (k 1, p)k2 2 lm (k 2, p) lm,l = P l (y) dy 1 k1 2 + k2 2 + k 1k 2 y + p2 6 [terms with l odd are positive, terms with l even are decreasing in p] Ø 4fi ÿ Œ Œ dk 1 dk 2 k1 2 lm (k 1, p)k2 2 lm (k 2, p) lm,l =0 0 0 l even 1 P l (y) dy 1 k1 2 + k2 2 + k 1k 2 y Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
36 Sketch of the Proof p o [ ] Ø p diag [ ] [I]: p o [ ] = 2 dk 1 dk 2 (k 1, p) (k 2, p) k k2 2 + k 1 k 2 + p2 6 [exp. in spherical harmonics] = 4fi ÿ Œ Œ dk 1 dk 2 k1 2 lm (k 1, p)k2 2 lm (k 2, p) lm,l = P l (y) dy 1 k1 2 + k2 2 + k 1k 2 y + p2 6 [terms with l odd are positive, terms with l even are decreasing in p] Ø 4fi ÿ Œ Œ dk 1 dk 2 k1 2 lm (k 1, p)k2 2 lm (k 2, p) lm,l =0 0 0 l even 1 P l (y) dy 1 k1 2 + k2 2 + k 1k 2 y Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
37 Sketch of the Proof p o [ ] Ø p diag [ ] [I]: p o [ ] = 2 dk 1 dk 2 (k 1, p) (k 2, p) k k2 2 + k 1 k 2 + p2 6 [exp. in spherical harmonics] = 4fi ÿ Œ Œ dk 1 dk 2 k1 2 lm (k 1, p)k2 2 lm (k 2, p) lm,l = P l (y) dy 1 k1 2 + k2 2 + k 1k 2 y + p2 6 [terms with l odd are positive, terms with l even are decreasing in p] Ø 4fi ÿ Œ Œ dk 1 dk 2 k1 2 lm (k 1, p)k2 2 lm (k 2, p) lm,l =0 0 0 l even 1 P l (y) dy 1 k1 2 + k2 2 + k 1k 2 y Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
38 Sketch of the Proof p o [ ] Ø p diag [ ] [II]: Ë 1 diagonalization using: lm (q, p) = 2fi = ÿ lm,l =0 l even dqs l (q) lm (q, p) 2 È dxe ikx e 2x lm (e x, p) Ë S l (q) Æ S 2 (q) Æ for l > 0with =2fi fi 3Ô 3 where b = Ø ÿ lm,l =0 l even fi/6 dt(12 sin 2 t 1), t 0 = arcsin(1/ Ô È 12) t 0 dq lm (q, p) fi 2 b 9 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
39 Sketch of the Proof p o [ ] Ø p diag [ ] [II]: Ë 1 diagonalization using: lm (q, p) = 2fi = ÿ lm,l =0 l even dqs l (q) lm (q, p) 2 È dxe ikx e 2x lm (e x, p) Ë S l (q) Æ S 2 (q) Æ for l > 0with =2fi fi 3Ô 3 where b = Ø ÿ lm,l =0 l even fi/6 dt(12 sin 2 t 1), t 0 = arcsin(1/ Ô È 12) t 0 dq lm (q, p) fi 2 b 9 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
40 Sketch of the Proof p o [ ] Ø p diag [ ] [III]: [inverse transform] ÿ Œ = dkk 3 lm (k, p) 2 lm,l =0 0 l even [ =Ô 3fi 2 < 1andaddp2 6 ] = ÿ 2fi 2 Œ lm,l =0 l even [add terms for l odd ] = p diag [ ]. 0 dkk 2 Û 3 4 k2 + p2 6 lm(k, p) 2 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
41 Sketch of the Proof p o [ ] Ø p diag [ ] [III]: [inverse transform] ÿ Œ = dkk 3 lm (k, p) 2 lm,l =0 0 l even [ =Ô 3fi 2 < 1andaddp2 6 ] = ÿ 2fi 2 Œ lm,l =0 l even [add terms for l odd ] = p diag [ ]. 0 dkk 2 Û 3 4 k2 + p2 6 lm(k, p) 2 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
42 Sketch of the Proof p o [ ] Ø p diag [ ] [III]: [inverse transform] ÿ Œ = dkk 3 lm (k, p) 2 lm,l =0 0 l even [ =Ô 3fi 2 < 1andaddp2 6 ] = ÿ 2fi 2 Œ lm,l =0 l even [add terms for l odd ] = p diag [ ]. 0 dkk 2 Û 3 4 k2 + p2 6 lm(k, p) 2 Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
43 Thank you for the attention! Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
44 Appendix S 2 (q) Æ To obtain : So 1 S 2 (q) =fi 2 dy(3y 2 cosh(q arcsin y 2 1) ) cos(arcsin y 2 ) cosh(q fi 2 ) 1 A B 1 S 2 (0) =fi 2 dy(3y ) 1 cos(arcsin( y = 2fi2 2 )) 3 fi 3Ô 3 Thus it s enough to show S 2 (q) S 2 (0) Æ 4fi 2 b 9, q Ø 0 (even function). 1 S 2 (q) S 2 (0) = fi 2 dy (3y 2 A 1) cos(arcsin y 2 ) 1 cosh(q arcsin y 2 ) B cosh(q fi 2 ) = 4fi 2 I 1 with I = s A B fi/6 0 dt(12 sin 2 t 1) 1 cosh(qt) cosh(q fi 2 ) Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
45 Appendix S 2 (q) Æ To obtain : So 1 S 2 (q) =fi 2 dy(3y 2 cosh(q arcsin y 2 1) ) cos(arcsin y 2 ) cosh(q fi 2 ) 1 A B 1 S 2 (0) =fi 2 dy(3y ) 1 cos(arcsin( y = 2fi2 2 )) 3 fi 3Ô 3 Thus it s enough to show S 2 (q) S 2 (0) Æ 4fi 2 b 9, q Ø 0 (even function). 1 S 2 (q) S 2 (0) = fi 2 dy (3y 2 A 1) cos(arcsin y 2 ) 1 cosh(q arcsin y 2 ) B cosh(q fi 2 ) = 4fi 2 I 1 with I = s A B fi/6 0 dt(12 sin 2 t 1) 1 cosh(qt) cosh(q fi 2 ) Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
46 Appendix S 2 (q) Æ To obtain : So 1 S 2 (q) =fi 2 dy(3y 2 cosh(q arcsin y 2 1) ) cos(arcsin y 2 ) cosh(q fi 2 ) 1 A B 1 S 2 (0) =fi 2 dy(3y ) 1 cos(arcsin( y = 2fi2 2 )) 3 fi 3Ô 3 Thus it s enough to show S 2 (q) S 2 (0) Æ 4fi 2 b 9, q Ø 0 (even function). 1 S 2 (q) S 2 (0) = fi 2 dy (3y 2 A 1) cos(arcsin y 2 ) 1 cosh(q arcsin y 2 ) B cosh(q fi 2 ) = 4fi 2 I 1 with I = s A B fi/6 0 dt(12 sin 2 t 1) 1 cosh(qt) cosh(q fi 2 ) Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
47 Appendix S 2 (q) Æ We want I Æ b 9 1 Ë 1 I Æ cosh(q fi 2 ) a +(b a) cosh where a = s t 0 0 dt 12 sin2 t 1. Introduce then I Æ cosh(q fi 2 ) + b 9. q fi b cosh q fi 2È g(q) =a + 9 b a cosh q fi 2 1 b cosh q fi g(q) g(0) = b 9 > 0 g Õ (q) Ø 0 g(q) Ø 0 and I Æ b 9. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
48 Appendix S 2 (q) Æ We want I Æ b 9 1 Ë 1 I Æ cosh(q fi 2 ) a +(b a) cosh where a = s t 0 0 dt 12 sin2 t 1. Introduce then I Æ cosh(q fi 2 ) + b 9. q fi b cosh q fi 2È g(q) =a + 9 b a cosh q fi 2 1 b cosh q fi g(q) g(0) = b 9 > 0 g Õ (q) Ø 0 g(q) Ø 0 and I Æ b 9. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
49 Appendix S 2 (q) Æ We want I Æ b 9 1 Ë 1 I Æ cosh(q fi 2 ) a +(b a) cosh where a = s t 0 0 dt 12 sin2 t 1. Introduce then I Æ cosh(q fi 2 ) + b 9. q fi b cosh q fi 2È g(q) =a + 9 b a cosh q fi 2 1 b cosh q fi g(q) g(0) = b 9 > 0 g Õ (q) Ø 0 g(q) Ø 0 and I Æ b 9. Giulia Basti (La Sapienza) 3-body systems with interactions TQMS / 15
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