diagonalization of bosonic quadratic hamiltonians€¦ · i many-body qm (e ective theories like...
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Diagonalization of bosonic quadratic Hamiltoniansby Bogoliubov transformations1
P. T. Nam1 M. Napiorkowski 1 J. P. Solovej 2
1Institute of Science and Technology Austria
2Department of Mathematics, University of Copenhagen
GDR Dynqua meeting, Grenoble
1J. Funct. Anal. (in press)
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Bosonic quadratic Hamiltonians on Fock space
General form of quadratic Hamiltonian:
H = dΓ(h) +1
2
∑m,n≥1
(〈J∗kfm, fn〉a(fm)a(fn) + 〈J∗kfm, fn〉a∗(fm)a∗(fn)
)Here:
I a∗/a - bosonic creation/annihilation operators (CCR);
I h > 0 and dΓ(h) =∑
m,n≥1〈fm, hfn〉a∗(fm)a(fn)
I k : h→ h∗ is an (unbounded) linear operator withD(h) ⊂ D(k) (called pairing operator), k∗ = J∗kJ∗;
I J : h→ h∗ is the anti-unitary operator defined by
J(f)(g) = 〈f, g〉, ∀f, g ∈ h.
Operators of that type are important in physics!
I QFT (eg. scalar field with position dependent mass);
I many-body QM (effective theories like Bogoliubov or BCS).
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Bosonic quadratic Hamiltonians on Fock space
General form of quadratic Hamiltonian:
H = dΓ(h) +1
2
∑m,n≥1
(〈J∗kfm, fn〉a(fm)a(fn) + 〈J∗kfm, fn〉a∗(fm)a∗(fn)
)Here:
I a∗/a - bosonic creation/annihilation operators (CCR);
I h > 0 and dΓ(h) =∑
m,n≥1〈fm, hfn〉a∗(fm)a(fn)
I k : h→ h∗ is an (unbounded) linear operator withD(h) ⊂ D(k) (called pairing operator), k∗ = J∗kJ∗;
I J : h→ h∗ is the anti-unitary operator defined by
J(f)(g) = 〈f, g〉, ∀f, g ∈ h.
Operators of that type are important in physics!
I QFT (eg. scalar field with position dependent mass);
I many-body QM (effective theories like Bogoliubov or BCS).
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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The problem
Our goal:find (prove existence) a unitary transformation U on the Fockspace, such that
UHU∗ = E + dΓ(ξ).
Why?
I interpretation in terms of a non-interacting theory;
I access to spectral properties of H;
I . . .
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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The problem
Our goal:find (prove existence) a unitary transformation U on the Fockspace, such that
UHU∗ = E + dΓ(ξ).
Why?
I interpretation in terms of a non-interacting theory;
I access to spectral properties of H;
I . . .
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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H = dΓ(h) +1
2
∑m,n≥1
(〈J∗kfm, fn〉a(fm)a(fn) + 〈J∗kfm, fn〉a∗(fm)a∗(fn)
)
I Remark:The above definition is formal! If k is not Hilbert-Schmidt,then it is difficult to show that the domain is dense.
I More general approach: definition through quadraticforms!
I One-particle density matrices: γΨ : h→ h and αΨ : h→ h∗
〈f, γΨg〉 = 〈Ψ, a∗(g)a(f)Ψ〉 , 〈Jf, αΨg〉 = 〈Ψ, a∗(g)a∗(f)Ψ〉 , ∀f, g ∈ h.
I A formal calculation leads to the expression
〈Ψ,HΨ〉 = Tr(h1/2γΨh1/2) + <Tr(k∗αΨ).
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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H = dΓ(h) +1
2
∑m,n≥1
(〈J∗kfm, fn〉a(fm)a(fn) + 〈J∗kfm, fn〉a∗(fm)a∗(fn)
)
I Remark:The above definition is formal! If k is not Hilbert-Schmidt,then it is difficult to show that the domain is dense.
I More general approach: definition through quadraticforms!
I One-particle density matrices: γΨ : h→ h and αΨ : h→ h∗
〈f, γΨg〉 = 〈Ψ, a∗(g)a(f)Ψ〉 , 〈Jf, αΨg〉 = 〈Ψ, a∗(g)a∗(f)Ψ〉 , ∀f, g ∈ h.
I A formal calculation leads to the expression
〈Ψ,HΨ〉 = Tr(h1/2γΨh1/2) + <Tr(k∗αΨ).
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Unitary implementability
I Generalized creation and annihilation operators
A(f⊕Jg) = a(f)+a∗(g), A∗(f⊕Jg) = a∗(f)+a(g), ∀f, g ∈ h;
I Let V : h⊕ h∗ → h⊕ h∗, bounded;
Definition
A bounded operator V on h⊕ h∗ is unitarily implemented by aunitary operator UV on Fock space if
UVA(F )U∗V = A(VF ), ∀F ∈ h⊕ h∗.
I Our goal: Find UV such that UVHU∗V = E + dΓ(ξ).
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Unitary implementability
I Generalized creation and annihilation operators
A(f⊕Jg) = a(f)+a∗(g), A∗(f⊕Jg) = a∗(f)+a(g), ∀f, g ∈ h;
I Let V : h⊕ h∗ → h⊕ h∗, bounded;
Definition
A bounded operator V on h⊕ h∗ is unitarily implemented by aunitary operator UV on Fock space if
UVA(F )U∗V = A(VF ), ∀F ∈ h⊕ h∗.
I Our goal: Find UV such that UVHU∗V = E + dΓ(ξ).
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Quadratic Hamiltonians as quantizations of block operators
Let
A :=
(h k∗
k JhJ∗
)
and
HA :=1
2
∑m,n≥1
〈Fm,AFn〉A∗(Fm)A(Fn).
Then a formal calculation gives
H = HA −1
2Tr(h).
Thus, formally, H can be seen as ”quantization” of A.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Quadratic Hamiltonians as quantizations of block operators
Let
A :=
(h k∗
k JhJ∗
)and
HA :=1
2
∑m,n≥1
〈Fm,AFn〉A∗(Fm)A(Fn).
Then a formal calculation gives
H = HA −1
2Tr(h).
Thus, formally, H can be seen as ”quantization” of A.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Quadratic Hamiltonians as quantizations of block operators
Let
A :=
(h k∗
k JhJ∗
)and
HA :=1
2
∑m,n≥1
〈Fm,AFn〉A∗(Fm)A(Fn).
Then a formal calculation gives
H = HA −1
2Tr(h).
Thus, formally, H can be seen as ”quantization” of A.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Diagonalization
If UVA(F )U∗V = A(VF ), then
UVHAU∗V = HVAV∗ .
Thus, if V diagonalizes A:
VAV∗ =
(ξ 00 JξJ∗
)for some operator ξ : h→ h, then
UVHU∗V = UV(HA −
1
2Tr(h)
)U∗V = dΓ(ξ) +
1
2Tr(ξ − h).
These formal arguments suggest it is enough to consider thediagonalization of block operators.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Diagonalization
If UVA(F )U∗V = A(VF ), then
UVHAU∗V = HVAV∗ .
Thus, if V diagonalizes A:
VAV∗ =
(ξ 00 JξJ∗
)for some operator ξ : h→ h, then
UVHU∗V = UV(HA −
1
2Tr(h)
)U∗V = dΓ(ξ) +
1
2Tr(ξ − h).
These formal arguments suggest it is enough to consider thediagonalization of block operators.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Diagonalization
If UVA(F )U∗V = A(VF ), then
UVHAU∗V = HVAV∗ .
Thus, if V diagonalizes A:
VAV∗ =
(ξ 00 JξJ∗
)for some operator ξ : h→ h, then
UVHU∗V = UV(HA −
1
2Tr(h)
)U∗V = dΓ(ξ) +
1
2Tr(ξ − h).
These formal arguments suggest it is enough to consider thediagonalization of block operators.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Questions
Question 1:
what are the conditions on V so that UVA(F )U∗V = A(VF )?
Question 2:
what are the conditions on A so that there exists a V thatdiagonalizes A?
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Questions
Question 1:
what are the conditions on V so that UVA(F )U∗V = A(VF )?
Question 2:
what are the conditions on A so that there exists a V thatdiagonalizes A?
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Questions
Question 1:
what are the conditions on V so that UVA(F )U∗V = A(VF )?
Question 2:
what are the conditions on A so that there exists a V thatdiagonalizes A?
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Questions
Question 1:
what are the conditions on V so that UVA(F )U∗V = A(VF )?
Question 2:
what are the conditions on A so that there exists a V thatdiagonalizes A?
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Question 1 - symplectic transformations
Recall A(f ⊕ Jg) = a(f) + a∗(g) and UVA(F )U∗V = A(VF ).
I Conjugate and canonical commutation relations:
A∗(F1) = A(JF1), [A(F1), A∗(F2)] = (F1,SF2), ∀F1, F2 ∈ h⊕h∗
where
S =
(1 0
0 − 1
), J =
(0 J∗
J 0
).
I S = S−1 = S∗ is unitary, J = J −1 = J ∗ is anti-unitary.
I Compatibility (wrt implementability) conditions
JVJ = V, V∗SV = S = VSV∗. (1)
I Any bounded operator V on h⊕ h∗ satisfying (1) is called asymplectic transformation.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Question 1 - symplectic transformations
Recall A(f ⊕ Jg) = a(f) + a∗(g) and UVA(F )U∗V = A(VF ).
I Conjugate and canonical commutation relations:
A∗(F1) = A(JF1), [A(F1), A∗(F2)] = (F1,SF2), ∀F1, F2 ∈ h⊕h∗
where
S =
(1 0
0 − 1
), J =
(0 J∗
J 0
).
I S = S−1 = S∗ is unitary, J = J −1 = J ∗ is anti-unitary.
I Compatibility (wrt implementability) conditions
JVJ = V, V∗SV = S = VSV∗. (1)
I Any bounded operator V on h⊕ h∗ satisfying (1) is called asymplectic transformation.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Question 1 - implementability
I Symplecticity of V implies
JVJ = V ⇒ V =
(U J∗V J∗
V JUJ∗
)
Fundamental result:
Shale’s theorem (’62)
A symplectic transformation V is unitarily implementable (i.e.UVA(F )U∗V = A(VF )), if and only if
‖V ‖2HS = Tr(V ∗V ) <∞.
UV , a unitary implementer on the Fock space of a symplectictransformation V, is called a Bogoliubov transformation.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Question 1 - implementability
I Symplecticity of V implies
JVJ = V ⇒ V =
(U J∗V J∗
V JUJ∗
)Fundamental result:
Shale’s theorem (’62)
A symplectic transformation V is unitarily implementable (i.e.UVA(F )U∗V = A(VF )), if and only if
‖V ‖2HS = Tr(V ∗V ) <∞.
UV , a unitary implementer on the Fock space of a symplectictransformation V, is called a Bogoliubov transformation.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Question 1 - implementability
I Symplecticity of V implies
JVJ = V ⇒ V =
(U J∗V J∗
V JUJ∗
)Fundamental result:
Shale’s theorem (’62)
A symplectic transformation V is unitarily implementable (i.e.UVA(F )U∗V = A(VF )), if and only if
‖V ‖2HS = Tr(V ∗V ) <∞.
UV , a unitary implementer on the Fock space of a symplectictransformation V, is called a Bogoliubov transformation.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
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Question 2 - example: commuting operators in ∞ dim
I h > 0 and k = k∗ be commuting operators on h = L2(Ω,C)
I
A :=
(h kk h
)> 0 on h⊕ h∗.
if and only if G < 1 with G := |k|h−1.I A is diagonalized by the linear operator
V :=
√1
2+
1
2√
1−G2
(1 −G
1+√
1−G2
−G1+√
1−G21
)in the sense that
VAV∗ =
(ξ 00 ξ
)with ξ := h
√1−G2 =
√h2 − k2 > 0.
I V satisfies the compatibility conditions and is bounded (andhence a symplectic transformation) iff ‖G‖ = ‖kh−1‖ < 1
I V is unitarily implementable iff kh−1 is Hilbert-Schmidt.
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Historical remarks
I For dim h <∞ this follows from Williamson’s Theorem (’36);
I Friedrichs (’50s) and Berezin (’60s): h ≥ µ > 0 bounded withgap and k Hilbert-Schmidt;
I Grech-Seiringer (’13): h > 0 with compact resolvent, kHilbert-Schmidt;
I Lewin-Nam-Serfaty-Solovej (’15): h ≥ µ > 0 unbounded, kHilbert-Schmidt;
I Bach-Bru (’16): h > 0, ‖kh−1‖ < 1 and kh−s isHilbert-Schmidt for all s ∈ [0, 1 + ε] for some ε > 0.
I Our result: essentially optimal conditions
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Theorem [Diagonalization of block operators]
(i) (Existence). Let h : h→ h and k : h→ h∗ be (unbounded)linear operators satisfying h = h∗ > 0, k∗ = J∗kJ∗ andD(h) ⊂ D(k). Assume that the operator G := h−1/2J∗kh−1/2 isdensely defined and extends to a bounded operator satisfying‖G‖ < 1. Then we can define the self-adjoint operator
A :=
(h k∗
k JhJ∗
)> 0 on h⊕ h∗
by Friedrichs’ extension. This operator can be diagonalized by asymplectic transformation V on h⊕ h∗ in the sense that
VAV∗ =
(ξ 00 JξJ∗
)for a self-adjoint operator ξ > 0 on h. Moreover, we have
‖V‖ ≤(
1 + ‖G‖1− ‖G‖
)1/4
.
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Theorem [Diagonalization of block operators]
(ii) (Implementability). Assume further that G isHilbert-Schmidt. Then V is unitarily implementable and
‖V ‖HS ≤2
1− ‖G‖‖G‖HS.
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Theorem [Diagonalization of quadratic Hamiltonians]
Recall G := h−1/2J∗kh−1/2. Assume, as before, that ‖G‖ < 1 andG is Hilbert-Schmidt. Assume further that kh−1/2 isHilbert-Schmidt. Then the quadratic Hamiltonian H, definedbefore as a quadratic form, is bounded from below and closable,and hence its closure defines a self-adjoint operator which we stilldenote by H. Moreover, if UV is the unitary operator on Fockspace implementing the symplectic transformation V, then
UVHU∗V = dΓ(ξ) + inf σ(H)
and
inf σ(H) ≥ −1
2‖kh−1/2‖2HS.
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Sketch of proof
Step 1. - fermionic case. If B is a self-adjoint and such thatJBJ = −B, then there exists a unitary operator U on h⊕ h∗
such that JUJ = U and
UBU∗ =
(ξ 00 −JξJ∗
).
Step 2. Apply Step 1 to B = A1/2SA1/2.
Step 3. Explicit construction of the symplectic transformation V:
V := U|B|1/2A−1/2.
Step 4. A detailed study of V∗V = A−1/2|B|A−1/2.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
![Page 31: Diagonalization of bosonic quadratic Hamiltonians€¦ · I many-body QM (e ective theories like Bogoliubov or BCS). Marcin Napi orkowski Quadratic Hamiltonians and Bogoliubov transformations](https://reader030.vdocuments.mx/reader030/viewer/2022041118/5f2eea3a80ded4432766294f/html5/thumbnails/31.jpg)
Sketch of proof
Step 1. - fermionic case. If B is a self-adjoint and such thatJBJ = −B, then there exists a unitary operator U on h⊕ h∗
such that JUJ = U and
UBU∗ =
(ξ 00 −JξJ∗
).
Step 2. Apply Step 1 to B = A1/2SA1/2.
Step 3. Explicit construction of the symplectic transformation V:
V := U|B|1/2A−1/2.
Step 4. A detailed study of V∗V = A−1/2|B|A−1/2.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
![Page 32: Diagonalization of bosonic quadratic Hamiltonians€¦ · I many-body QM (e ective theories like Bogoliubov or BCS). Marcin Napi orkowski Quadratic Hamiltonians and Bogoliubov transformations](https://reader030.vdocuments.mx/reader030/viewer/2022041118/5f2eea3a80ded4432766294f/html5/thumbnails/32.jpg)
Sketch of proof
Step 1. - fermionic case. If B is a self-adjoint and such thatJBJ = −B, then there exists a unitary operator U on h⊕ h∗
such that JUJ = U and
UBU∗ =
(ξ 00 −JξJ∗
).
Step 2. Apply Step 1 to B = A1/2SA1/2.
Step 3. Explicit construction of the symplectic transformation V:
V := U|B|1/2A−1/2.
Step 4. A detailed study of V∗V = A−1/2|B|A−1/2.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
![Page 33: Diagonalization of bosonic quadratic Hamiltonians€¦ · I many-body QM (e ective theories like Bogoliubov or BCS). Marcin Napi orkowski Quadratic Hamiltonians and Bogoliubov transformations](https://reader030.vdocuments.mx/reader030/viewer/2022041118/5f2eea3a80ded4432766294f/html5/thumbnails/33.jpg)
Sketch of proof
Step 1. - fermionic case. If B is a self-adjoint and such thatJBJ = −B, then there exists a unitary operator U on h⊕ h∗
such that JUJ = U and
UBU∗ =
(ξ 00 −JξJ∗
).
Step 2. Apply Step 1 to B = A1/2SA1/2.
Step 3. Explicit construction of the symplectic transformation V:
V := U|B|1/2A−1/2.
Step 4. A detailed study of V∗V = A−1/2|B|A−1/2.
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations
![Page 34: Diagonalization of bosonic quadratic Hamiltonians€¦ · I many-body QM (e ective theories like Bogoliubov or BCS). Marcin Napi orkowski Quadratic Hamiltonians and Bogoliubov transformations](https://reader030.vdocuments.mx/reader030/viewer/2022041118/5f2eea3a80ded4432766294f/html5/thumbnails/34.jpg)
Thank you for your attention!
Marcin Napiorkowski Quadratic Hamiltonians and Bogoliubov transformations