Green s functions in N-body quantum mechanics A mathematical perspective

Transcription

1 Green s functions in N-body quantum mechanics A mathematical perspective Eric CANCES Ecole des Ponts and INRIA, Paris, France Aussois, June 19th, 2015

2 Outline of the course 2 1 Linear operators 2 Electronic Hamiltonians 3 One-body Green s function and self-energy 4 The dynamically screened Coulomb operator W 5 Hedin s equations and the GW approximation A1 Fourier transform A2 Causal functions, Hilbert transform and Kramers-Kronig relations

3 1 - Linear operators References: EB Davies, Linear operators and their spectra, Cambridge University Press 2007 B Helffer, Spectral theory and its applications, Cambridge University Press 2013 M Reed and B Simon, Modern methods in mathematical physics, Vol 1, 2nd edition, Academic Press 1980 Notation: in this section, H denotes a separable complex Hilbert space, its scalar product, and the associated norm

4 1 - Linear operators 4 The finite dimensional case (H = C d ) The spectrum of a matrix A C d d is the finite set σ(a) = { z C (z A) C d d non-invertible }

5 1 - Linear operators 4 The finite dimensional case (H = C d ) The spectrum of a matrix A C d d is the finite set σ(a) = { z C (z A) C d d non-invertible } As C d is finite dimensional, (z A) non-invertible (z A) non-injective: σ(a) = { z C x C d \ {0} st Ax = zx } = {eigenvalues of A}

6 1 - Linear operators 4 The finite dimensional case (H = C d ) The spectrum of a matrix A C d d is the finite set σ(a) = { z C (z A) C d d non-invertible } As C d is finite dimensional, (z A) non-invertible (z A) non-injective: σ(a) = { z C x C d \ {0} st Ax = zx } = {eigenvalues of A} A matrix A C d d is called hermitian if A = A (ie A ij = A ji, 1 i, j d)

7 1 - Linear operators 4 The finite dimensional case (H = C d ) The spectrum of a matrix A C d d is the finite set σ(a) = { z C (z A) C d d non-invertible } As C d is finite dimensional, (z A) non-invertible (z A) non-injective: σ(a) = { z C x C d \ {0} st Ax = zx } = {eigenvalues of A} A matrix A C d d is called hermitian if A = A (ie A ij = A ji, 1 i, j d) Key properties of hermitian matrices: the spectrum of a hermitian matrix is real: σ(a) R;

8 1 - Linear operators 4 The finite dimensional case (H = C d ) The spectrum of a matrix A C d d is the finite set σ(a) = { z C (z A) C d d non-invertible } As C d is finite dimensional, (z A) non-invertible (z A) non-injective: σ(a) = { z C x C d \ {0} st Ax = zx } = {eigenvalues of A} A matrix A C d d is called hermitian if A = A (ie A ij = A ji, 1 i, j d) Key properties of hermitian matrices: the spectrum of a hermitian matrix is real: σ(a) R; any hermitian matrix A can be diagonalized in an orthonormal basis: d A = λ i x i x i, λ i σ(a) R, x i R d, x i x j = δ ij, Ax i = λ i x i ; i=1

9 1 - Linear operators 4 The finite dimensional case (H = C d ) The spectrum of a matrix A C d d is the finite set σ(a) = { z C (z A) C d d non-invertible } As C d is finite dimensional, (z A) non-invertible (z A) non-injective: σ(a) = { z C x C d \ {0} st Ax = zx } = {eigenvalues of A} A matrix A C d d is called hermitian if A = A (ie A ij = A ji, 1 i, j d) Key properties of hermitian matrices: the spectrum of a hermitian matrix is real: σ(a) R; any hermitian matrix A can be diagonalized in an orthonormal basis: d A = λ i x i x i, λ i σ(a) R, x i R d, x i x j = δ ij, Ax i = λ i x i ; i=1 there exists a functional calculus for hermitian matrices

10 1 - Linear operators 5 Functional calculus for hermitian matrices Let A be a hermitian matrix of C d d such that d A = λ i x i x i, λ i σ(a) R, x i C d, x i x j = δ ij, Ax i = λ i x i i=1

11 1 - Linear operators 5 Functional calculus for hermitian matrices Let A be a hermitian matrix of C d d such that d A = λ i x i x i, λ i σ(a) R, x i C d, x i x j = δ ij, Ax i = λ i x i i=1 For any f : R C, the matrix d f(a) := f(λ i )x i x i is independent of the choice of the spectral decomposition of A i=1

12 1 - Linear operators 5 Functional calculus for hermitian matrices Let A be a hermitian matrix of C d d such that d A = λ i x i x i, λ i σ(a) R, x i C d, x i x j = δ ij, Ax i = λ i x i i=1 For any f : R C, the matrix d f(a) := f(λ i )x i x i i=1 is independent of the choice of the spectral decomposition of A Functional calculus can be extended to self-adjoint operators in Hilbert spaces Functional calculus is extremely useful in quantum physics, eg to define the propagator e ith associated with a Hamiltonian H; 1 the density matrix of a fermionic system at temperature 1 + e (H ε F)/(k B T ) T and chemical potential (Fermi level) ε F

13 1 - Linear operators 6 Bounded linear operators on Hilbert spaces Definition-Theorem (bounded linear operator) A bounded operator on H is a linear map A : H H such that Au A := u < sup u H\{0}

14 1 - Linear operators 6 Bounded linear operators on Hilbert spaces Definition-Theorem (bounded linear operator) A bounded operator on H is a linear map A : H H such that Au A := u < sup u H\{0} The set B(H) of the bounded operators on H is a non-commutative algebra and is a norm on B(H)

15 1 - Linear operators 6 Bounded linear operators on Hilbert spaces Definition-Theorem (bounded linear operator) A bounded operator on H is a linear map A : H H such that Au A := u < sup u H\{0} The set B(H) of the bounded operators on H is a non-commutative algebra and is a norm on B(H) Remark A bounded linear operator is uniquely defined by the values of the sesquilinear form H H (u, v) u Av C

16 1 - Linear operators 6 Bounded linear operators on Hilbert spaces Definition-Theorem (bounded linear operator) A bounded operator on H is a linear map A : H H such that Au A := u < sup u H\{0} The set B(H) of the bounded operators on H is a non-commutative algebra and is a norm on B(H) Remark A bounded linear operator is uniquely defined by the values of the sesquilinear form H H (u, v) u Av C Definition-Theorem (adjoint of a bounded linear operator) Let A B(H) The operator A B(H) defined by is called the adjoint of A (u, v) H H, u A v = Au v,

17 1 - Linear operators 6 Bounded linear operators on Hilbert spaces Definition-Theorem (bounded linear operator) A bounded operator on H is a linear map A : H H such that Au A := u < sup u H\{0} The set B(H) of the bounded operators on H is a non-commutative algebra and is a norm on B(H) Remark A bounded linear operator is uniquely defined by the values of the sesquilinear form H H (u, v) u Av C Definition-Theorem (adjoint of a bounded linear operator) Let A B(H) The operator A B(H) defined by (u, v) H H, u A v = Au v, is called the adjoint of A The operator A is called self-adjoint if A = A

18 1 - Linear operators 6 Bounded linear operators on Hilbert spaces Definition-Theorem (bounded linear operator) A bounded operator on H is a linear map A : H H such that Au A := u < sup u H\{0} The set B(H) of the bounded operators on H is a non-commutative algebra and is a norm on B(H) Remark A bounded linear operator is uniquely defined by the values of the sesquilinear form H H (u, v) u Av C Definition-Theorem (adjoint of a bounded linear operator) Let A B(H) The operator A B(H) defined by (u, v) H H, u A v = Au v, is called the adjoint of A The operator A is called self-adjoint if A = A Endowed with its norm and the operation, B(H) is a C -algebra

19 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces

20 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces Definition (linear operator) A linear operator on H is a linear map A : D(A) H, where D(A) is a subspace of H called the domain of A

21 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces Definition (linear operator) A linear operator on H is a linear map A : D(A) H, where D(A) is a subspace of H called the domain of A Note that bounded linear operators are particular linear operators

22 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces Definition (linear operator) A linear operator on H is a linear map A : D(A) H, where D(A) is a subspace of H called the domain of A Note that bounded linear operators are particular linear operators Definition (extensions of operators) Let A 1 and A 2 be operators on H A 2 is called an extension of A 1 if D(A 1 ) D(A 2 ) and if u D(A 1 ), A 2 u = A 1 u

23 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces Definition (linear operator) A linear operator on H is a linear map A : D(A) H, where D(A) is a subspace of H called the domain of A Note that bounded linear operators are particular linear operators Definition (extensions of operators) Let A 1 and A 2 be operators on H A 2 is called an extension of A 1 if D(A 1 ) D(A 2 ) and if u D(A 1 ), A 2 u = A 1 u Definition (unbounded linear operator) An operator A on H which does not possess a bounded extension is called an unbounded operator on H

24 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces Definition (linear operator) A linear operator on H is a linear map A : D(A) H, where D(A) is a subspace of H called the domain of A Note that bounded linear operators are particular linear operators Definition (extensions of operators) Let A 1 and A 2 be operators on H A 2 is called an extension of A 1 if D(A 1 ) D(A 2 ) and if u D(A 1 ), A 2 u = A 1 u Definition (unbounded linear operator) An operator A on H which does not possess a bounded extension is called an unbounded operator on H Definition (symmetric operator) A linear operator A on H with dense domain D(A) is called symmetric if (u, v) D(A) D(A), Au v = u Av

25 1 - Linear operators 7 (Non necessarily bounded) linear operators on Hilbert spaces Definition (linear operator) A linear operator on H is a linear map A : D(A) H, where D(A) is a subspace of H called the domain of A Note that bounded linear operators are particular linear operators Definition (extensions of operators) Let A 1 and A 2 be operators on H A 2 is called an extension of A 1 if D(A 1 ) D(A 2 ) and if u D(A 1 ), A 2 u = A 1 u Definition (unbounded linear operator) An operator A on H which does not possess a bounded extension is called an unbounded operator on H Definition (symmetric operator) A linear operator A on H with dense domain D(A) is called symmetric if (u, v) D(A) D(A), Au v = u Av Symmetric operators are not very interesting Only self-adjoint operators represent physical observables and have nice mathematical properties: real spectrum; spectral decomposition and functional calculus

26 1 - Linear operators 8 Definition (adjoint of a linear operator with dense domain) Let A be a linear operator on H with dense domain D(A), and D(A ) the vector space defined as D(A ) = {v H w v H st u D(A), Au v = u w v } The linear operator A on H, with domain D(A ), defined by v D(A ), A v = w v, (if w v exists, it is unique since D(A) is dense) is called the adjoint of A

27 1 - Linear operators 8 Definition (adjoint of a linear operator with dense domain) Let A be a linear operator on H with dense domain D(A), and D(A ) the vector space defined as D(A ) = {v H w v H st u D(A), Au v = u w v } The linear operator A on H, with domain D(A ), defined by v D(A ), A v = w v, (if w v exists, it is unique since D(A) is dense) is called the adjoint of A (This definition agrees with the one on Slide 6 for bounded operators)

28 1 - Linear operators 8 Definition (adjoint of a linear operator with dense domain) Let A be a linear operator on H with dense domain D(A), and D(A ) the vector space defined as D(A ) = {v H w v H st u D(A), Au v = u w v } The linear operator A on H, with domain D(A ), defined by v D(A ), A v = w v, (if w v exists, it is unique since D(A) is dense) is called the adjoint of A (This definition agrees with the one on Slide 6 for bounded operators) Definition (self-adjoint operator) A linear operator A with dense domain is called self-adjoint if A = A (that is if A symmetric and D(A ) = D(A))

29 1 - Linear operators 8 Definition (adjoint of a linear operator with dense domain) Let A be a linear operator on H with dense domain D(A), and D(A ) the vector space defined as D(A ) = {v H w v H st u D(A), Au v = u w v } The linear operator A on H, with domain D(A ), defined by v D(A ), A v = w v, (if w v exists, it is unique since D(A) is dense) is called the adjoint of A (This definition agrees with the one on Slide 6 for bounded operators) Definition (self-adjoint operator) A linear operator A with dense domain is called self-adjoint if A = A (that is if A symmetric and D(A ) = D(A)) Case of bounded operators: symmetric self-adjoint

30 1 - Linear operators 8 Definition (adjoint of a linear operator with dense domain) Let A be a linear operator on H with dense domain D(A), and D(A ) the vector space defined as D(A ) = {v H w v H st u D(A), Au v = u w v } The linear operator A on H, with domain D(A ), defined by v D(A ), A v = w v, (if w v exists, it is unique since D(A) is dense) is called the adjoint of A (This definition agrees with the one on Slide 6 for bounded operators) Definition (self-adjoint operator) A linear operator A with dense domain is called self-adjoint if A = A (that is if A symmetric and D(A ) = D(A)) Case of bounded operators: symmetric self-adjoint Case of unbounded operators: symmetric (easy to check) self-adjoint (sometimes difficult to check)

31 1 - Linear operators 9 Some unbounded self-adjoint operators arising in quantum mechanics position operator along the j axis: H = L 2 (R d ), D( r j ) = { u L 2 (R d ) r j u L 2 (R d ) }, ( r j φ)(r) = r j φ(r); momentum operator along the j axis: H = L 2 (R d ), D( p j ) = { u L 2 (R d ) rj u L 2 (R d ) }, ( p j φ)(r) = i rj φ(r); kinetic energy operator: H = L 2 (R d ), D(T ) = H 2 (R d ) := { u L 2 (R d ) u L 2 (R d ) }, T = 1 2 ; Schrödinger operators in 3D: let V L 2 unif (R3, R) (V (r) = Z r OK) H = L 2 (R 3 ), D(H) = H 2 (R 3 ), H = V

32 1 - Linear operators 10 Linear operators and Green s functions

33 1 - Linear operators 10 Linear operators and Green s functions Kernel of a linear operator on L 2 (R d ) Let A be a linear operator on L 2 (R d ) with domain D(A) The kernel of A, if it exists, is the distribution A(x, x ) such that φ D(A), (Aφ)(x) = " A(x, x ) φ(x ) dx " R d

34 1 - Linear operators 10 Linear operators and Green s functions Kernel of a linear operator on L 2 (R d ) Let A be a linear operator on L 2 (R d ) with domain D(A) The kernel of A, if it exists, is the distribution A(x, x ) such that φ D(A), (Aφ)(x) = " A(x, x ) φ(x ) dx " R d Schwartz kernel theorem 66: all "well-behaved" operators have kernels

35 1 - Linear operators 10 Linear operators and Green s functions Kernel of a linear operator on L 2 (R d ) Let A be a linear operator on L 2 (R d ) with domain D(A) The kernel of A, if it exists, is the distribution A(x, x ) such that φ D(A), (Aφ)(x) = " A(x, x ) φ(x ) dx " R d Schwartz kernel theorem 66: all "well-behaved" operators have kernels Green s function of a linear operator on L 2 (R d ) If A is invertible, the kernel G(x, x ) of A 1, if it exists, is called the Green s function of A The solution u to the equation Au = f then is u(x) = " G(x, x ) f(x ) dx " for aa x R d R d

36 1 - Linear operators 10 Linear operators and Green s functions Kernel of a linear operator on L 2 (R d ) Let A be a linear operator on L 2 (R d ) with domain D(A) The kernel of A, if it exists, is the distribution A(x, x ) such that φ D(A), (Aφ)(x) = " A(x, x ) φ(x ) dx " R d Schwartz kernel theorem 66: all "well-behaved" operators have kernels Green s function of a linear operator on L 2 (R d ) If A is invertible, the kernel G(x, x ) of A 1, if it exists, is called the Green s function of A The solution u to the equation Au = f then is u(x) = " G(x, x ) f(x ) dx " for aa x R d R d Remark: the Green s functions used in many-body perturbation theory are related to, but are not exactly, this kind of Green s functions

37 1 - Linear operators 11 Definition-Theorem (spectrum of a linear operator) Let A be a closed 1 linear operator on H The open set ρ(a) = {z C (z A) : D(A) H invertible} is called the resolvent set of A 1 The operator A is called closed if its graph Γ(A) := {(u, Au), u D(A)} is a closed subspace of H H

38 1 - Linear operators 11 Definition-Theorem (spectrum of a linear operator) Let A be a closed 1 linear operator on H The open set ρ(a) = {z C (z A) : D(A) H invertible} is called the resolvent set of A The analytic function ρ(a) z R z (A) := (z A) 1 B(H) is called the resolvent of A It holds R z (A) R z (A) = (z z)r z (A)R z (A) 1 The operator A is called closed if its graph Γ(A) := {(u, Au), u D(A)} is a closed subspace of H H

39 1 - Linear operators 11 Definition-Theorem (spectrum of a linear operator) Let A be a closed 1 linear operator on H The open set ρ(a) = {z C (z A) : D(A) H invertible} is called the resolvent set of A The analytic function ρ(a) z R z (A) := (z A) 1 B(H) is called the resolvent of A It holds R z (A) R z (A) = (z z)r z (A)R z (A) The closed set σ(a) = C \ ρ(a) is called the spectrum of A 1 The operator A is called closed if its graph Γ(A) := {(u, Au), u D(A)} is a closed subspace of H H

40 1 - Linear operators 11 Definition-Theorem (spectrum of a linear operator) Let A be a closed 1 linear operator on H The open set ρ(a) = {z C (z A) : D(A) H invertible} is called the resolvent set of A The analytic function ρ(a) z R z (A) := (z A) 1 B(H) is called the resolvent of A It holds R z (A) R z (A) = (z z)r z (A)R z (A) The closed set σ(a) = C \ ρ(a) is called the spectrum of A If A is self-adjoint, then σ(a) R 1 The operator A is called closed if its graph Γ(A) := {(u, Au), u D(A)} is a closed subspace of H H

41 1 - Linear operators 11 Definition-Theorem (spectrum of a linear operator) Let A be a closed 1 linear operator on H The open set ρ(a) = {z C (z A) : D(A) H invertible} is called the resolvent set of A The analytic function ρ(a) z R z (A) := (z A) 1 B(H) is called the resolvent of A It holds R z (A) R z (A) = (z z)r z (A)R z (A) The closed set σ(a) = C \ ρ(a) is called the spectrum of A If A is self-adjoint, then σ(a) R and it holds σ(a) = σ p (A) σ c (A), where σ p (A) and σ c (A) are respectively the point spectrum and the continuous spectrum of A defined as σ p (A) = {z C (z A) : D(A) H non-injective} = {eigenvalues of A} σ c (A) = {z C (z A) : D(A) H injective but non surjective} 1 The operator A is called closed if its graph Γ(A) := {(u, Au), u D(A)} is a closed subspace of H H

42 1 - Linear operators 12 On the physical meaning of point and continuous spectra

43 1 - Linear operators 12 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78) Let H be a locally compact self-adjoint operator on L 2 (R d ) [Ex: the Hamiltonian of the hydrogen atom satisfies these assumptions]

44 1 - Linear operators 12 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78) Let H be a locally compact self-adjoint operator on L 2 (R d ) [Ex: the Hamiltonian of the hydrogen atom satisfies these assumptions] Let H p = Span {eigenvectors of H} and H c = H p [Ex: for the Hamiltonian of the hydrogen atom, dim(h p ) = dim(h c ) = ]

45 1 - Linear operators 12 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78) Let H be a locally compact self-adjoint operator on L 2 (R d ) [Ex: the Hamiltonian of the hydrogen atom satisfies these assumptions] Let H p = Span {eigenvectors of H} and H c = H p [Ex: for the Hamiltonian of the hydrogen atom, dim(h p ) = dim(h c ) = ] Let χ BR be the characteristic function of the ball B R = { r R d r < R }

46 1 - Linear operators 12 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78) Let H be a locally compact self-adjoint operator on L 2 (R d ) [Ex: the Hamiltonian of the hydrogen atom satisfies these assumptions] Let H p = Span {eigenvectors of H} and H c = H p [Ex: for the Hamiltonian of the hydrogen atom, dim(h p ) = dim(h c ) = ] Let χ BR be the characteristic function of the ball B R = { r R d r < R } Then (φ 0 H p ) ε > 0, R > 0, t 0, (1 χbr )e ith φ 0 2 L 2 ε; (φ 0 H c ) R > 0, lim T + 1 T T 0 χ BR e ith φ 0 2 L 2 dt = 0

47 1 - Linear operators 12 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78) Let H be a locally compact self-adjoint operator on L 2 (R d ) [Ex: the Hamiltonian of the hydrogen atom satisfies these assumptions] Let H p = Span {eigenvectors of H} and H c = H p [Ex: for the Hamiltonian of the hydrogen atom, dim(h p ) = dim(h c ) = ] Let χ BR be the characteristic function of the ball B R = { r R d r < R } Then (φ 0 H p ) ε > 0, R > 0, t 0, (1 χbr )e ith φ 0 2 L 2 ε; (φ 0 H c ) R > 0, lim T + H p : set of bound states, 1 T T 0 χ BR e ith φ 0 2 L 2 dt = 0 H c : set of diffusive states

48 1 - Linear operators 13 Diagonalizable self-adjoint operators and Dirac s bra-ket notation Let A be a self-adjoint operator that can be diagonalized in an orthonormal basis (e n ) n N (this is not the case for many useful self-adjoint operators!) Dirac s bra-ket notation: A = n N λ n e n e n, λ n R, e m e n = δ mn

49 1 - Linear operators 13 Diagonalizable self-adjoint operators and Dirac s bra-ket notation Let A be a self-adjoint operator that can be diagonalized in an orthonormal basis (e n ) n N (this is not the case for many useful self-adjoint operators!) Dirac s bra-ket notation: A = n N λ n e n e n, λ n R, e m e n = δ mn Then, the operator A is bounded if and only if A = sup n λ n < ;

50 1 - Linear operators 13 Diagonalizable self-adjoint operators and Dirac s bra-ket notation Let A be a self-adjoint operator that can be diagonalized in an orthonormal basis (e n ) n N (this is not the case for many useful self-adjoint operators!) Dirac s bra-ket notation: A = n N λ n e n e n, λ n R, e m e n = δ mn Then, the operator A is bounded if and only if A = sup n λ n < ; D(A) = { u = n N u n e n n N (1 + λ n 2 ) u n 2 < } ;

51 1 - Linear operators 13 Diagonalizable self-adjoint operators and Dirac s bra-ket notation Let A be a self-adjoint operator that can be diagonalized in an orthonormal basis (e n ) n N (this is not the case for many useful self-adjoint operators!) Dirac s bra-ket notation: A = n N λ n e n e n, λ n R, e m e n = δ mn Then, the operator A is bounded if and only if A = sup n λ n < ; D(A) = { u = n N u n e n n N (1 + λ n 2 ) u n 2 < } ; σ p (A) = {λ n } n N and σ c (A) = { accumulation points of {λ n } n N } \σp (A);

52 1 - Linear operators 13 Diagonalizable self-adjoint operators and Dirac s bra-ket notation Let A be a self-adjoint operator that can be diagonalized in an orthonormal basis (e n ) n N (this is not the case for many useful self-adjoint operators!) Dirac s bra-ket notation: A = n N λ n e n e n, λ n R, e m e n = δ mn Then, the operator A is bounded if and only if A = sup n λ n < ; D(A) = { u = n N u n e n n N (1 + λ n 2 ) u n 2 < } ; σ p (A) = {λ n } n N and σ c (A) = { accumulation points of {λ n } n N } \σp (A); H p = H and H c = {0} (no diffusive states);

53 1 - Linear operators 13 Diagonalizable self-adjoint operators and Dirac s bra-ket notation Let A be a self-adjoint operator that can be diagonalized in an orthonormal basis (e n ) n N (this is not the case for many useful self-adjoint operators!) Dirac s bra-ket notation: A = n N λ n e n e n, λ n R, e m e n = δ mn Then, the operator A is bounded if and only if A = sup n λ n < ; D(A) = { u = n N u n e n n N (1 + λ n 2 ) u n 2 < } ; σ p (A) = {λ n } n N and σ c (A) = { accumulation points of {λ n } n N } \σp (A); H p = H and H c = {0} (no diffusive states); functional calculus for diagonalizable self-adjoint operators: for all f : R C, the operator f(a) defined by { } D(f(A)) = u = u n e n + f(λ n ) n N n N(1 2 ) u n 2 <, f(a) = f(λ n ) e n e n n N is independent of the choice of the spectral decomposition of A

54 1 - Linear operators 14 Theorem (functional calculus for bounded functions) Let B(R, C) be the -algebra of bounded C-valued Borel functions on R and let A be a selfadjoint operator on H Then there exists a unique map satisfies the following properties: Φ A : B(R, C) f f(a) B(H) 1 Φ A is a homomorphism of -algebras: (αf + βg)(a) = αf(a) + βg(a), (fg)(a) = f(a)g(a), f(a) = f(a) ; 2 f(a) sup f(x) ; x R 3 if f n (x) x pointwise and f n (x) x for all n and all x R, then u D(A), f n (A)u Au in H; 4 if f n (x) f(x) pointwise and sup n sup x R f n (x) <, then u H, f n (A)u f(a)u in H; In addition, if u H is such that Au = λu, then f(a)u = f(λ)u

55 1 - Linear operators 15 Theorem (spectral projections and functional calculus - general case -) Let A be a self-adjoint operator on H For all λ R, the bounded operator P A λ := 1 ],λ](a), where 1 ],λ] ( ) is the characteristic function of ], λ], is an orthogonal projection Spectral decomposition of A: for all u D(A) and v H, it holds v Au = λ d v Pλ A u, which we denote by A = λ dpλ A R Functional calculus: let f be a (not necessarily bounded) C-valued Borel function on R The operator f(a) can be defined by { } D(f(A)) := u H f(λ) 2 d u Pλ A u < and (u, v) D(f(A)) H, v f(a)u := R R R f(λ) v P A dλu

56 2 - Electronic Hamiltonians

57 2 - Electronic Hamiltonians 17 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 v ext (r) = M k=1 z k r R k 1 2 N ri + i=1 N v ext (r i ) + i=1 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2, p S N, Ψ(r p(1),, r p(n) ) = ε(p)ψ(r 1,, r N ), (Pauli principle)

58 2 - Electronic Hamiltonians 17 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 v ext (r) = M k=1 z k r R k 1 2 N ri + i=1 N v ext (r i ) + i=1 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2, Ψ H N = N H1, H 1 = L 2 (R 3, C)

59 2 - Electronic Hamiltonians 17 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 v ext (r) = M k=1 z k r R k 1 2 N ri + i=1 N v ext (r i ) + i=1 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2, Ψ H N = N H1, H 1 = L 2 (R 3, C 2 ) with spin

60 2 - Electronic Hamiltonians 17 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 v ext (r) = M k=1 z k r R k 1 2 N ri + i=1 N v ext (r i ) + i=1 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2, N Ψ H N = H1, H 1 = L 2 (R 3, C) Theorem (Kato 51) The operator H N := 1 N N ri + v ext (r i )+ 2 i=1 i=1 with domain D(H N ) := H N H 2 (R 3N ) is self-adjoint on H N 1 i<j N 1 r i r j

61 2 - Electronic Hamiltonians 18 Theorem (spectrum of H N ) 1 HVZ theorem (Hunziger 66, van Winten 60, Zhislin 60) σ c (H N ) = [Σ N, + ) with Σ N = min σ(h N 1 ) 0 and Σ N < 0 iff N 2 2 Bound states of neutral molecules and positive ions (Zhislin 61) M If N Z := z k, then H N has an infinite number of bound states k=1 Ground state Excited states Ε N 0 Σ N Continuous spectrum 3 Bound states of negative ions (Yafaev 72) If N Z + 1, then H N has at most a finite number of bound states

62 2 - Electronic Hamiltonians 19 Assumptions 1 Non-degeneracy of the N-particle ground state EN 0 is a simple eigenvalue of H N, H N Ψ 0 N = ENΨ 0 0 N, Ψ 0 N = 1 2 Stability of the N-particle system 2EN 0 < EN EN 1 0

63 2 - Electronic Hamiltonians 20 Photoemission spectroscopy (PES) hν E kin E k N 1 E0 N = hν E kin System with N electrons System with N 1 electrons Ε 0 N Σ N σ(h N ) Ε k N 1 Σ N 1 σ(h N 1 )

64 2 - Electronic Hamiltonians 21 Inverse photoemission spectroscopy (IPES) E kin hν E 0 N Ek N+1 = hν E kin System with N electrons System with N + 1 electrons 0 Ε N Σ N σ(h N ) k Ε N+1 Σ N+1 σ(h N+1 )

65 2 - Electronic Hamiltonians 22 Goal: compute the excitation energies E k N+1 E0 N and Ek N 1 E0 N Wavefunction methods: scales from N 6 b (CISD) to N b! (full CI) Time-dependent density functional theory (TDDFT): lots of problems (especially for extended systems)

66 2 - Electronic Hamiltonians 22 Goal: compute the excitation energies E k N+1 E0 N and Ek N 1 E0 N Wavefunction methods: scales from N 6 b (CISD) to N b! (full CI) Time-dependent density functional theory (TDDFT): lots of problems (especially for extended systems) GW: decent to very good results (especially for extended systems) Electronic excitations for perfect crystals (N + )

67 2 - Electronic Hamiltonians 23 Electronic ground state density ρ 0 N(r) = N Ψ 0 N(r, r 2,, r N ) 2 dr 2 dr N R 3(N 1) One-body electronic ground state density matrix γn(r, 0 r ) = N Ψ 0 N(r, r 2,, r N ) Ψ 0 N(r, r 2,, r N ) dr 2 dr N R 3(N 1) One-body Green s function G(r, r, t t ) = i Ψ N 0 T (Ψ H (r, t)ψ H (r, t )) Ψ N 0

68 2 - Electronic Hamiltonians 23 Electronic ground state density ρ 0 N(r) = N Ψ 0 N(r, r 2,, r N ) 2 dr 2 dr N R 3(N 1) One-body electronic ground state density matrix γn(r, 0 r ) = N Ψ 0 N(r, r 2,, r N ) Ψ 0 N(r, r 2,, r N ) dr 2 dr N R 3(N 1) One-body Green s function G(r, r, t t ) = i Ψ N 0 T (Ψ H (r, t)ψ H (r, t )) Ψ N 0 No concept of wavefunction for infinite systems (such as perfect crystals)

69 2 - Electronic Hamiltonians 23 Electronic ground state density ρ 0 N(r) = N Ψ 0 N(r, r 2,, r N ) 2 dr 2 dr N R 3(N 1) One-body electronic ground state density matrix γn(r, 0 r ) = N Ψ 0 N(r, r 2,, r N ) Ψ 0 N(r, r 2,, r N ) dr 2 dr N R 3(N 1) One-body Green s function G(r, r, t t ) = i Ψ N 0 T (Ψ H (r, t)ψ H (r, t )) Ψ N 0 No concept of wavefunction for infinite systems (such as perfect crystals) Thermodynamic limit problem (for periodic crystals): L E 0 ZL 3? L 3 L E0 per, ρ 0 ZL 3 (r)? L ρ0 per(r) γ 0 ZL 3 (r, r )? L γ0 per(r, r ), G(r, r?, t) G per(r, r, t) L

70 3 - One-body Green s function and self-energy Let X be a Banach space (typically X = B(H 1 )) Fourier transform: let f L 1 (R t, X) ω R, [Ff](ω) = f(ω) = + f(t) e iωt dt

71 3 - One-body Green s function and self-energy Let X be a Banach space (typically X = B(H 1 )) Fourier transform: let f L 1 (R t, X) ω R, [Ff](ω) = f(ω) = + f(t) e iωt dt Laplace transform of causal functions: let f L (R t, X) st f(t) = 0 for t < 0 z U = {z C I(z) > 0}, [Lf](z) = + f(t) e izt dt = + 0 f(t) e izt dt

72 3 - One-body Green s function and self-energy Let X be a Banach space (typically X = B(H 1 )) Fourier transform: let f L 1 (R t, X) ω R, [Ff](ω) = f(ω) = + f(t) e iωt dt Laplace transform of causal functions: let f L (R t, X) st f(t) = 0 for t < 0 z U = {z C I(z) > 0}, [Lf](z) = + f(t) e izt dt = + 0 f(t) e izt dt Laplace transform of anti-causal functions: let f L (R t, X) st f(t) = 0 for t > 0 z = {z C I(z) < 0}, [Lf](z) = + f(t) e izt dt = 0 f(t) e izt dt

73 3 - One-body Green s function and self-energy Let X be a Banach space (typically X = B(H 1 )) Fourier transform: let f L 1 (R t, X) ω R, [Ff](ω) = f(ω) = + f(t) e iωt dt Laplace transform of causal functions: let f L (R t, X) st f(t) = 0 for t < 0 z U = {z C I(z) > 0}, [Lf](z) = + f(t) e izt dt = + 0 f(t) e izt dt Laplace transform of anti-causal functions: let f L (R t, X) st f(t) = 0 for t > 0 z = {z C I(z) < 0}, [Lf](z) = + f(t) e izt dt = 0 f(t) e izt dt The Fourier and Laplace transforms can be extended to some distribution spaces (extension of the Fourier transform to the space of tempered distributions)

74 3 - One-body Green s function and self-energy 25 Second quantization formalism (for fermions) Fock space F := + N=0 H N, H 0 = C, H 1 = L 2 (R 3, C), H N = Creation and annihilation operators N H1 a A(H 1, B(F)), a B(H 1, B(F)), a(φ) = a(φ) = φ, φ H 1, a(φ) HN : H N H N 1, a (φ) HN : H N H N+1, a (φ) = (a(φ)), Ψ N H N, (a(φ)ψ N )(r 1,, r N 1 ) = N φ(r) Ψ N (r, r 1,, r N 1 ) dr R 3 Canonical commutation relations (CCR) φ, ψ H 1, a(φ)a(ψ) + a(ψ) a(φ) = φ ψ Id F

75 3 - One-body Green s function and self-energy 26 Particle Green s function Time representation: G p L (R t, B(H 1 )) defined by t R, (f, g) H 1 H 1, g G p (t) f = iθ(t) Ψ N 0 a(g)e it(h N+1 E N 0 ) a (f) Ψ N 0 E kin hν System with N electrons System with N + 1 electrons

76 3 - One-body Green s function and self-energy 26 Particle Green s function Time representation: G p L (R t, B(H 1 )) defined by t R, (f, g) H 1 H 1, g G p (t) f = iθ(t) Ψ N 0 a(g)e it(h N+1 E0 N) a (f) Ψ N 0 Frequency representation (Fourier transform) Ĝ p (ω) = (FG p )(ω), Ĝ p H 1 (R ω, B(H 1 ))

77 3 - One-body Green s function and self-energy 26 Particle Green s function Time representation: G p L (R t, B(H 1 )) defined by t R, (f, g) H 1 H 1, g G p (t) f = iθ(t) Ψ N 0 a(g)e it(h N+1 E0 N) a (f) Ψ N 0 Frequency representation (Fourier transform) Ĝ p (ω) = (FG p )(ω), Ĝ p H 1 (R ω, B(H 1 )) Complex plane representation (analytic continuation of the Laplace transform) G p (z) = A + (z (H N+1 E 0 N)) 1 A + where A + : H 1 H N+1 f a (f) Ψ 0 N The singularities of z G p (z) are contained in σ(h N+1 E 0 N ) 0 0 N+1 N E E

78 3 - One-body Green s function and self-energy 27 Hole Green s function Time representation: G h L (R t, B(H 1 )) defined by t R, (f, g) H 1 H 1, g G h (t) f = iθ( t) Ψ N 0 a (g)e it(h N 1 E N 0 ) a(f) Ψ N 0 hν E kin System with N electrons System with N 1 electrons

79 3 - One-body Green s function and self-energy 27 Hole Green s function Time representation: G h L (R t, B(H 1 )) defined by t R, (f, g) H 1 H 1, g G h (t) f = iθ( t) Ψ N 0 a (g)e it(h N 1 E0 N) a(f) Ψ N 0 Frequency representation (Fourier transform) Ĝ h (ω) = (FG h )(ω), Ĝ h H 1 (R ω, B(H 1 )) Complex plane representation (analytic continuation of the Laplace transform) G h (z) = A (z (E 0 N H N 1 )) 1 A where A : H 1 H N 1 f a(f) Ψ 0 N The singularities of z G h (z) are contained in σ(e 0 N H N 1) 0 N E E 0 N 1

80 3 - One-body Green s function and self-energy 28 Properties of the particle and hole Green s functions Spectral functions (operator-valued measures on R ω ) b B(R ω ), A p (b) = π 1 IĜp(b) A h (b) = +π 1 IĜh(b) A(b) = A p (b) + A h (b)

81 3 - One-body Green s function and self-energy 28 Properties of the particle and hole Green s functions Spectral functions (operator-valued measures on R ω ) b B(R ω ), A p (b) = π 1 IĜp(b) = A + 1 b (H N+1 E 0 N)A + 0, A h (b) = +π 1 IĜh(b) = A 1 b (E 0 N H N 1 )A 0, A(b) = A p (b) + A h (b) 0

82 3 - One-body Green s function and self-energy 28 Properties of the particle and hole Green s functions Spectral functions (operator-valued measures on R ω ) b B(R ω ), A p (b) = π 1 IĜp(b) = A + 1 b (H N+1 E 0 N)A + 0, A h (b) = +π 1 IĜh(b) = A 1 b (E 0 N H N 1 )A 0, A(b) = A p (b) + A h (b) 0 Sum rule: Supp(A h ) σ(e 0 N H N 1 ), Supp(A p ) σ(h N+1 E 0 N) A p (R) = A + A + = 1 γ 0 N, A h (R) = A A = γ 0 N, A(R) = 1

83 3 - One-body Green s function and self-energy 28 Properties of the particle and hole Green s functions Spectral functions (operator-valued measures on R ω ) b B(R ω ), A p (b) = π 1 IĜp(b) = A + 1 b (H N+1 E 0 N)A + 0, A h (b) = +π 1 IĜh(b) = A 1 b (E 0 N H N 1 )A 0, A(b) = A p (b) + A h (b) 0 Supp(A h ) σ(en 0 H N 1 ), Supp(A p ) σ(h N+1 EN) 0 Sum rule: A p (R) = A + A + = 1 γn, 0 A h (R) = A A = γn, 0 A(R) = 1 It holds γn 0 = ig h(0 )

84 3 - One-body Green s function and self-energy 28 Properties of the particle and hole Green s functions Spectral functions (operator-valued measures on R ω ) b B(R ω ), A p (b) = π 1 IĜp(b) = A + 1 b (H N+1 E 0 N)A + 0, A h (b) = +π 1 IĜh(b) = A 1 b (E 0 N H N 1 )A 0, A(b) = A p (b) + A h (b) 0 Sum rule: Supp(A h ) σ(e 0 N H N 1 ), Supp(A p ) σ(h N+1 E 0 N) A p (R) = A + A + = 1 γ 0 N, A h (R) = A A = γ 0 N, A(R) = 1 It holds γ 0 N = ig h(0 ) Galitskii-Migdal formula E 0 N = 1 2 Tr H 1 (( d dτ i ( 1 )) 2 + v ext G h (τ) τ=0 )

85 3 - One-body Green s function and self-energy 29 Green s functions of non-interacting systems System of non-interacting electrons subjected to an effective potential V N ( H 0,N = 1 ) 2 r i + V (r i ) on H N, h 1 = V on H 1 i=1

86 3 - One-body Green s function and self-energy 29 Green s functions of non-interacting systems System of non-interacting electrons subjected to an effective potential V N ( H 0,N = 1 ) 2 r i + V (r i ) on H N, h 1 = V on H 1 i=1 Ground state of non-interacting systems Φ 0 N = φ 1 φ N, γ 0 0,N = 1 ],µ0 ](h 1 ) = N φ i φ i i=1

87 3 - One-body Green s function and self-energy 29 Green s functions of non-interacting systems System of non-interacting electrons subjected to an effective potential V N ( H 0,N = 1 ) 2 r i + V (r i ) on H N, h 1 = V on H 1 i=1 Ground state of non-interacting systems Φ 0 N = φ 1 φ N, γ 0 0,N = 1 ],µ0 ](h 1 ) = N φ i φ i i=1 Particle and hole Green s functions G 0,p (z) = (1 γ 0 0,N)(z h 1 ) 1 (1 γ 0 0,N), G0,h (z) = γ 0 0,N(z h 1 ) 1 γ 0 0,N

88 3 - One-body Green s function and self-energy 29 Green s functions of non-interacting systems System of non-interacting electrons subjected to an effective potential V N ( H 0,N = 1 ) 2 r i + V (r i ) on H N, h 1 = V on H 1 i=1 Ground state of non-interacting systems Φ 0 N = φ 1 φ N, γ 0 0,N = 1 ],µ0 ](h 1 ) = N φ i φ i i=1 Particle and hole Green s functions G 0,p (z) = (1 γ 0 0,N)(z h 1 ) 1 (1 γ 0 0,N), G0,h (z) = γ 0 0,N(z h 1 ) 1 γ 0 0,N Time-ordered Green s function for interacting and non-interacting systems G = G p + G h, G 0 = G 0,p + G 0,h

89 3 - One-body Green s function and self-energy 29 Green s functions of non-interacting systems System of non-interacting electrons subjected to an effective potential V N ( H 0,N = 1 ) 2 r i + V (r i ) on H N, h 1 = V on H 1 i=1 Ground state of non-interacting systems Φ 0 N = φ 1 φ N, γ 0 0,N = 1 ],µ0 ](h 1 ) = N φ i φ i i=1 Particle and hole Green s functions G 0,p (z) = (1 γ 0 0,N)(z h 1 ) 1 (1 γ 0 0,N), G0,h (z) = γ 0 0,N(z h 1 ) 1 γ 0 0,N, Time-ordered Green s function for interacting and non-interacting systems G = G p + G h, G 0 = G 0,p + G 0,h G0 (z) = (z h 1 ) 1 (resolvent of h 1 at z)

90 3 - One-body Green s function and self-energy 30 Dynamical Hamiltonian Non-interacting systems: G 0 (z) = (z h 1 ) 1 Interacting systems: G(z) = (z H(z)) 1, H(z) : dynamical Hamiltonian

91 3 - One-body Green s function and self-energy 30 Dynamical Hamiltonian Non-interacting systems: G 0 (z) = (z h 1 ) 1 Interacting systems: G(z) = (z H(z)) 1, H(z) : dynamical Hamiltonian Proposition Let z C \ R The dynamical Hamiltonian is a well-defined closed unbounded operator on H 1 with dense domain D(z) H 2 (R 3 )

92 3 - One-body Green s function and self-energy 30 Dynamical Hamiltonian Non-interacting systems: G 0 (z) = (z h 1 ) 1 Interacting systems: G(z) = (z H(z)) 1, H(z) : dynamical Hamiltonian Proposition Let z C \ R The dynamical Hamiltonian is a well-defined closed unbounded operator on H 1 with dense domain D(z) H 2 (R 3 ) Self-energy operator z C\R, Σ(z) := ( G0 (z)) 1 1 ( G(z))

93 3 - One-body Green s function and self-energy 30 Dynamical Hamiltonian Non-interacting systems: G 0 (z) = (z h 1 ) 1 Interacting systems: G(z) = (z H(z)) 1, H(z) : dynamical Hamiltonian Proposition Let z C \ R The dynamical Hamiltonian is a well-defined closed unbounded operator on H 1 with dense domain D(z) H 2 (R 3 ) Self-energy operator z C\R, Σ(z) := ( G0 (z)) 1 ( G(z)) 1 G(z) = G 0 (z)+ G 0 (z) Σ(z) G(z) (Dyson equation)

94 3 - One-body Green s function and self-energy 30 Dynamical Hamiltonian Non-interacting systems: G 0 (z) = (z h 1 ) 1 Interacting systems: G(z) = (z H(z)) 1, H(z) : dynamical Hamiltonian Proposition Let z C \ R The dynamical Hamiltonian is a well-defined closed unbounded operator on H 1 with dense domain D(z) H 2 (R 3 ) Self-energy operator z C\R, Σ(z) := ( G0 (z)) 1 ( G(z)) 1 H(z) = h 1 + Σ(z)

95 3 - One-body Green s function and self-energy 30 Dynamical Hamiltonian Non-interacting systems: G 0 (z) = (z h 1 ) 1 Interacting systems: G(z) = (z H(z)) 1, H(z) : dynamical Hamiltonian Proposition Let z C \ R The dynamical Hamiltonian is a well-defined closed unbounded operator on H 1 with dense domain D(z) H 2 (R 3 ) Self-energy operator z C\R, Σ(z) := ( G0 (z)) 1 ( G(z)) 1 H(z) = h 1 + Σ(z) Road map: 1 construct a non-interacting Green s function G 0 (using eg the Kohn-Sham LDA Hamiltonian); 2 construct an approximation Σ app (z) of the self-energy operator; 3 seek the singularities of G app (z) := (z (h 1 + Σ app (z))) 1

96 4 - The dynamically screened Coulomb operator W

97 4 - The dynamically screened Coulomb operator W 32 The (bare) Coulomb operator v c In the vacuum and neglecting relativistic effects, the electrostatic potential created by a time-dependent charge distribution ρ at point r and time t is 1 [V ρ](r, t) = R 3 r r ρ(r, t) dr V (τ) = v c δ 0 (τ), v c ρ(k) = 4π k 2 ρ(k)

98 4 - The dynamically screened Coulomb operator W 32 The (bare) Coulomb operator v c In the vacuum and neglecting relativistic effects, the electrostatic potential created by a time-dependent charge distribution ρ at point r and time t is 1 [V ρ](r, t) = R 3 r r ρ(r, t) dr V (τ) = v c δ 0 (τ), v c ρ(k) = 4π k 2 ρ(k) Screening: in the presence of the molecular system, the perturbation of the electrostatic potential created at point r and time t by a time-dependent external charge distribution δρ, is given, in the linear response regime, by δv (r, t) = t W + (r, r, t t ) δρ(r, t ) dt

99 4 - The dynamically screened Coulomb operator W 32 The (bare) Coulomb operator v c In the vacuum and neglecting relativistic effects, the electrostatic potential created by a time-dependent charge distribution ρ at point r and time t is 1 [V ρ](r, t) = R 3 r r ρ(r, t) dr V (τ) = v c δ 0 (τ), v c ρ(k) = 4π k 2 ρ(k) Screening: in the presence of the molecular system, the perturbation of the electrostatic potential created at point r and time t by a time-dependent external charge distribution δρ, is given, in the linear response regime, by δv (r, t) = t W + (r, r, t t ) δρ(r, t ) dt The dynamically screened Coulomb operator is defined by τ R, W (τ) = Θ(τ)W + (τ) + Θ( τ)w + ( τ) = v 1/2 c (δ(τ) χ sym (τ))v 1/2 c

100 4 - The dynamically screened Coulomb operator W 32 The (bare) Coulomb operator v c In the vacuum and neglecting relativistic effects, the electrostatic potential created by a time-dependent charge distribution ρ at point r and time t is 1 [V ρ](r, t) = R 3 r r ρ(r, t) dr V (τ) = v c δ 0 (τ), v c ρ(k) = 4π k 2 ρ(k) Screening: in the presence of the molecular system, the perturbation of the electrostatic potential created at point r and time t by a time-dependent external charge distribution δρ, is given, in the linear response regime, by δv (r, t) = t W + (r, r, t t ) δρ(r, t ) dt The dynamically screened Coulomb operator is defined by τ R, W (τ) = Θ(τ)W + (τ) + Θ( τ)w + ( τ) = vc 1/2 (δ(τ) χ }{{} sym (τ))v1/2 c L (R, B(L 2 (R 3 )))

101 5 - Hedin s equations and the GW approximation

102 5 - Hedin s equations and the GW approximation Notation Kernel of a space-time operator A A((r 1, t 1 ), (r 2, t 2 )) A(12) If A is a time-translation invariant space-time operator A(12 + ) = A((r 1, t 1 ), (r 2, t + 2 )) = lim t t + 2 A((r 1, t 1 ), (r 2, t)) = [A((t 1 t 2 ) )](r 1, r 2 )

103 5 - Hedin s equations and the GW approximation 34 Hedin s equations (Hedin 65) Dyson equation G(12) = G 0 (12) + d(34)g 0 (13)Σ(34)G(42) The Hedin-Lundqvist Equations The Dyson eq: G (12) = G (0) (12) + d (34) G (0) (13) Σ (34) G (42) [H I] Self-energy Σ(12) = i d(34)g(13)w (41 + )Γ(32; 4) Self-energy: Σ (12) = i d (34) W (1 + 3) G(14)Γ (42; 3) [H II] Screened interaction W (12) = v c (12) + d(34)v c (13)P (34)W (42) Irreducible polarization P (12) = i d(34)g(13)g(41 + )Γ(34; 2) Screened interaction: W (12) = v(12) + d (34) W (13) P (34) v(42) Irred Polarisation: P (12) = i d (34) G(23)G(42)Γ (34; 1) [H IV] [H III] Vertex function: Γ (12; 3) = δ (12) δ (13) + d (4567) δσ(12) G(46)G(75)Γ (67; 3) [H V] δg(45) Vertex function Γ(12; 3) = δ(12)δ(13) + d(4567) δσ(12) G(46)G(75)Γ(67; 3) δg(45) 51

104 5 - Hedin s equations and the GW approximation 35 GW approximation (Hedin 65) Dyson equation G(12) = G 0 (12) + d(34)g 0 (13)Σ(34)G(42) The Hedin-Lundqvist Equations The Dyson eq: G (12) = G (0) (12) + d (34) G (0) (13) Σ (34) G (42) [H I] Self-energy Σ(12) = i d(34)g(13)w (41 + )Γ(32; 4) Self-energy: Σ (12) = i d (34) W (1 + 3) G(14)Γ (42; 3) [H II] Screened interaction W (12) = v c (12) + d(34)v c (13)P (34)W (42) Irreducible polarization P (12) = i d(34)g(13)g(41 + )Γ(34; 2) Screened interaction: W (12) = v(12) + d (34) W (13) P (34) v(42) Irred Polarisation: P (12) = i d (34) G(23)G(42)Γ (34; 1) [H IV] [H III] Vertex function: Γ (12; 3) = δ (12) δ (13) + d (4567) δσ(12) G(46)G(75)Γ (67; 3) [H V] δg(45) Vertex function Γ(12; 3) = δ(12)δ(13) + d(4567) δσ(12) G(46)G(75)Γ(67; 3) δg(45) }{{} Neglect this term 51

105 5 - Hedin s equations and the GW approximation 36 GW equations (Hedin 65) Dyson equation G app (12) = G 0 (12) + Self-energy d(34)g 0 (13)Σ app (34)G app (42) Σ app (12) = ig app (12)W app (21 + ) Screened interaction W app (12) = v c (12) + d(34)v c (13)P app (34)W app (42) Irreducible polarization P app (12) = ig app (12)G app (21)

106 5 - Hedin s equations and the GW approximation 37 GW equations in a mixed time-frequency representation Dyson equation Ĝ app (ω) = Ĝ0(ω) + Ĝ0(ω) Σ app (ω)ĝapp (ω) Self-energy Σ app (τ) = ig app (0 ) v c δ 0 (τ) + G app (τ) Wc app ( τ) Screened interaction [ ( ] 1 Ŵc app (ω) = 1 v c P (ω)) app 1 Irreducible polarization P app (τ) = ig app (τ) G app ( τ) v c Hadamard product of two operators: (A B)(r 1, r 2 ) = A(r 1, r 2 )B(r 2, r 1 )

107 5 - Hedin s equations and the GW approximation 38 GW equations on imaginary axes Dyson equation G app (µ + iω) = G 0 (µ + iω) + G 0 (µ + iω) Σ app (µ + iω) G app (µ + iω) Self-energy Σ app (µ+iω) = γ 0,LDA N v c 1 2π Screened interaction W app c (iω) = + [ ( 1 v c P app (iω)) 1 1 ] G app (µ+i(ω ω )) Irreducible polarization P app (iω) = 1 + G app (µ + iω ) 2π G app (µ + i(ω ω)) dω v c W app c (iω ) dω µ 0 0 E N N 1 0 E N+1 N 0 E E G(z) E 1 E 0 N N W (z)

108 5 - Hedin s equations and the GW approximation 39 GW flowchart Kohn Sham LDA G 0 0,LDA γ N G app Σ app P app W app The existence of a solution to these equations is an open problem

109 5 - Hedin s equations and the GW approximation 40 G 0 W 0 method Kohn Sham LDA G 0 0,LDA γ N G app 00 RPA Σ 0 P 0 W 0 Theorem (EC, Gontier, Stoltz 15) The G 0 W 0 method is well defined

110 5 - Hedin s equations and the GW approximation 41 Self-consistent GW 0 method Kohn Sham LDA G 0 0,LDA γ N G app RPA Σ app P 0 W 0

111 5 - Hedin s equations and the GW approximation 41 Self-consistent GW 0 method Kohn Sham LDA G 0 0,LDA γ N G app RPA Σ app xλ P 0 W 0 Theorem (EC, Gontier, Stoltz 15) The self-consistent GW 0 method is well defined in the perturbative regime (λ small enough)

112 5 - Hedin s equations and the GW approximation 42 Summary of the current mathematical results EC, D Gontier and G Stoltz, A mathematical analysis of the GW method for computing electronic excited state energies of molecules, arxiv The fundamental objects (G, G 0, Σ, P, W ) involved in the GW formalism are mathematically well-defined Some of their properties (sum rules, signs, Galitskii-Migdal formula) have been rigorously established The G 0 W 0 version of the GW approach is well defined The self-consistent GW 0 method is well-defined in the perturbation regime Work in progress Analysis of the partially self-consistent GW method (self-consistency on the eigenvalues only) Analysis of the fully self-consistent GW method Infinite systems (periodic crystals, disordered materials) Numerical algorithms

113 A1 - Fourier transform

114 A1 - Fourier transform 44 Definition (Schwartz space) A function φ : R C of class C is called rapidly decreasing if for all p N, N p (φ) := max max sup φ 0 k p 0 l p tkdl t R dt (t) l < The vector space of all C rapidly decreasing functions from R to C is denoted by S(R) and is called the Schwartz space Gaussian functions and gaussian-polynomial functions are in S(R) Definition (convergence in S(R)) A sequence (φ n ) n N of functions of S(R) converges in S(R) to φ S(R) if p N, N p (φ n φ) n + 0

115 A1 - Fourier transform 45 Definition (Fourier transform in S(R)) The Fourier transform of a function φ S(R) is the function denoted by φ or Fφ and defined by + ω R, φ(ω) = Fφ(ω) := φ(t) e iωt dt Remark Other sign and normalization conventions are also commonly used in the physics and mathematical literatures Theorem (some properties of S(R)) The Schwartz space S(R) is stable by 1 translation, scaling, complex conjugation; 2 derivation and multiplication by polynomials; 3 Fourier transform ( p N, C p R + st φ S(R), N p ( φ) C p N p+2 (φ)) Besides, the Fourier transform defines a sequentially bicontinuous linear map from S(R) onto itself, with inverse F 1 defined by ψ S(R), t R, [F 1 ψ](t) = 1 2π + ψ(ω) e iωt dω

116 A1 - Fourier transform 46 Definition (tempered distributions) We denote by S (R) the vector space of the linear forms u : S(R) C satisfying the following continuity property : there exists p N and C R + such that φ S(R), u, φ CN p (φ) (1) Theorem (canonical embedding of L p (R) in S (R)) Let 1 p + and f L p (R) Then, the linear form u f : S(R) C defined by φ S(R), u f, φ := + f(t)φ(t) dt is a tempered distribution, and if f 1 and f 2 are both in L p (R) and such that u f1 = u f2, then f 1 = f 2 We can therefore, with no ambiguity, denote f instead of u f : f L p (R) S (R) and φ S(R), f, φ := + f(t)φ(t) dt

117 A1 - Fourier transform 47 Definition-Theorem (basic operations on S (R)) 1 Let u S (R) The derivative of u is the element of S (R) denote by du dt and defined by φ S(R), du dφ, φ = u, dt dt 2 Let u S (R) and p : R C a polynomial function The product pu is the element of S (R) defined by φ S(R), pu, φ = u, pφ 3 Let u S (R) The Fourier transform of u is the element of S (R) denoted by û or Fu and defined by φ S(R), û, φ = u, φ Crucial point: the above definitions are consistent with the usual definitions for "nice" functions (ex: û(ω) = + u(t)e iωt dt for all u L 1 (R)) Exercise: define the translation, scaling, and complex conjugation operations on S (R)

118 A1 - Fourier transform 48 Definition (convergence in S (R)) A sequence (u n ) n N of elements of S (R) converges in S (R) to u S (R) if and only if φ S(R), u n, φ n + u, φ Theorem (some properties of the Fourier transform on S (R)) 1 Let u S (R) Then ( ) du F = iωû(ω) dt and F (tu(t)) = i dû dω (ω) 2 The Fourier transform is a sequentially bicontinuous linear map from S (R) onto itself, with inverse F 1 defined by φ S(R), F 1 u, φ S,S = u, F 1 φ S,S Exercise: compute the Fourier transform of a translated tempered distribution, of a scaled tempered distribution, and of the complex conjugate of a tempered distribution