AN0022295669;i3f01nov.06;2019Feb27.14:25;v2.2.500
On the Generator of the Solution of a Quantum Stochastic Differential Equation.
If the Hamiltonian of the system is a bounded selfadjoint operator, we give a simple alternative characterization of the infinitesimal generator iC of the unitary group associated with a Quantum Stochastic Differential Equation (QSDE). By using the defect indices of the symmetric operator C, we give a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of isometries in the case when the Hamiltonian of the system is an unbounded symmetric operator.
Keywords: Defect indices; Fock space; Infinitesimal generator; Quantum stochastic differential equation; Strongly continuous unitary group; Symmetric operator; 60J25; 47D10; 81Q10
1. INTRODUCTION
A Quantum Stochastic Differential Equation (QSDE) of the Hudson–Parthasarathy type (Ref.[10]) has the form,
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where W, L, and H are bounded operators in a separable Hilbert space ℋ and dΛ<subs>t</subs>, dA<subs>t</subs>, are the basic stochastic differentials in the symmetric Fock space Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>+</subs>)). The solution {V<subs>t</subs>}<subs>t≥0</subs> of this QSDE is an adapted process of operators in Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>+</subs>))⊗ℋ describing the total evolution of a quantum system together with its environment.
There exists a strongly continuous unitary group {U<subs>t</subs>}<subs>t∈ℝ</subs> related with {V<subs>t</subs>}<subs>t≥0</subs>. The unitary group U<subs>t</subs> is constructed by introducing on Γ<sups>s</sups>(L<sups>2</sups>(ℝ)) the second quantization Θ<subs>t</subs>, see Ref.[10], of the shift operator <subs>t</subs> in L<sups>2</sups>(ℝ) and defining
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where Θ<subs>t</subs> is identified with its ampliation Θ<subs>t</subs>⊗ I to Γ<sups>s</sups>(L<sups>2</sups>(ℝ))⊗ℋ. The strong continuity of V<subs>t</subs>, Θ<subs>t</subs> and the cocycle property of V<subs>t</subs> assure the strong continuity of the unitary group U<subs>t</subs> in Γ<sups>s</sups>(L<subs>2</subs>(ℝ))⊗ℋ, see Refs.[1][5][9].
By Stone's Theorem U<subs>t</subs> is generated by a selfadjoint operator C, then for each t ∈ ℝ, we have formally
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It is surprising that the characterization of the operator C remained open for around fifteen years. The main problem is that C is a singular operator and its domain should be described by using boundary conditions; recently Chebotarev[3] found the explicit form of this operator, and proved that it can be obtained as the strong resolvent limit of a family C<sups>α</sups>, α > 0, of selfadjoint operators. A different approach can be found in Refs.[7][8] where the case of arbitrary multiplicity and bounded system operators was considered, one of the referees informed us that a gap in the proofs of the latter paper has been corrected by the author and should appear soon. In Section 2 we provide an alternative characterization of C by using a different approach that permit us easily determine a core for it. In Section 3 we give an example where the defect indices of the symmetric operator C can be explicitly computed and a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of contractions.
2. THE INFINITESIMAL GENERATOR OF U t
We shall consider the case when L = 0, and W is a unitary operator commuting with the bounded selfadjoint operator H, the Hamiltonian of the system. The unitary group associated with the QSDE is defined for every t ∈ ℝ by Ref.[3],
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where
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with ψ<subs>n</subs> continuous for every n ≥ 1, ψ<subs>0</subs> ∈ ℂ and ℝ<subs>*</subs> = ℝ∖{0}. For every n ≥ 1 we have,
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where
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here e<sups>−itH</sups> and W are identified with its ampliations I⊗ e<sups>−itH</sups> and I⊗ W, respectively, I<subs>A</subs> denotes the indicator of the subset A and ψ<subs>n</subs>(x<subs>1</subs>,... x<subs>n</subs>) = v(x<subs>1</subs>)⊗···⊗ v(x<subs>n</subs>)⊗ h, h ∈ ℋ. The definition of U<subs>t,n</subs> is extended by density to the whole for every n ∈ ℕ.
Our main idea is to work first with the n-particles subspace , for every n ≥ 1. Let us consider the dense subspace of Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>*</subs>))⊗ℋ given by
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Lemma 2.1
Let ��<subs>2,1</subs>(ℝ<subs>*</subs>) be the Sobolev space of absolutely continuous functions with first derivative in L<sups>2</sups>(ℝ<subs>*</subs>). Let v ∈ ��<subs>2,1</subs>(ℝ<subs>*</subs>). Then the right and left limits at zero v(0<sups>±</sups>) there exist and moreover
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where ‖·‖<subs>2,1</subs> denotes the norm in ��<subs>2,1</subs>(ℝ<subs>*</subs>).
Proof
We prove the estimate for |v(0<sups>−</sups>)|, the estimate for |v(0<sups>+</sups>)| is similar. For c, x ∈ (− ∞, 0), x < c, we have that , therefore lim<subs>x→−∞</subs>|v(x)| = 0 since v ∈ ��<subs>2,1</subs>(ℝ<subs>*</subs>). Now using Schwarz inequality we obtain .
For v<subs>j</subs> ∈ ��<subs>2,1</subs>(ℝ<subs>*</subs>), j = 1,..., n, we define
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where the means that the corresponding factor is omitted.
It follows from the lemma that
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therefore can be extended to bounded operator from ��<subs>2,1</subs>(ℝ<subs>*</subs>)<sups>⊗ n</sups> into and . □
Lemma 2.2
For any ∈ ��<subs>2,1</subs>(ℝ<subs>*</subs>)<sups>⊗ n</sups> we have that for 1 ≤ j ≤ n.
Proof
The conclusion obviously holds for any finite linear combination of simple tensors. For a general ∈ ��<subs>2,1</subs>(ℝ<subs>*</subs>)<sups>⊗ n</sups> there exists a sequence of finite linear combinations of simple tensors (<subs>k</subs>)<subs>k≥1</subs> that converges to in ��<subs>2,1</subs>(ℝ<subs>*</subs>)<sups>⊗ n</sups>. By taking subsequences we can assure that (<subs>k</subs>)<subs>k≥1</subs> converges almost everywhere in ℝ<subs>*</subs> and that (∂<subs>j</subs><subs>k</subs>)<subs>k≥1</subs> also converges almost everywhere. The absolute continuity of and <subs>k</subs> for every k ≥ 0, imply that for every x > 0.
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almost everywhere. Taking again a subsequence if necessary we obtain that
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pointwise. This proves the result for . A similar argument works for .
The space ��<subs>2,1</subs>(ℝ<subs>*</subs>)<sups>⊗ n</sups> is densely and continuously embedded in , therefore the above definition can be extended to obtain a bounded operator from into and the result of Lemma 2.2 also holds for any .
We shall use the natural identifications and . Let C<subs>n</subs>, n ∈ ℕ, be the operator with domain
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The boundary condition , results from the interaction of the quantum system with its environment which we assume associated with the action of the unitary operator W. If we denote by the subset of that vectors in Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>*</subs>)) whose n-th component belongs to the Sobolev space , by we denote the projection of into the n-particles subspace .
Theorem 2.1
The closure of the operator iC in Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>*</subs>))⊗ℋ defined by CΨ = (0,C<subs>1</subs>ψ<subs>1</subs>,...,C<subs>n</subs>ψ<subs>n</subs>,...) on �� = {Ψ ∈ ��:ψ<subs>j</subs> ∈ D<subs>j</subs>, j ∈ ℕ} is the infinitesimal generator of the strongly continuous unitary group {U<subs>t</subs>}<subs>t∈ℝ</subs>.
Therefore is a selfadjoint operator and �� is a core for it. As we announced before, to prove this theorem we work with the n-particles subspace.
Proposition 2.1
The closure of the operator with domain D<subs>n</subs> is the infinitesimal generator of the strongly continuous unitary group {U<subs>t,n</subs>}<subs>t∈ℝ</subs> in .
Proof
It suffices to prove that D<subs>n</subs> is a core for . Due to a well known result about cores of infinitesimal generators, see Ref.[2], Corollary 3.1.7, p. 167, it suffices to prove that
- 1. D<subs>n</subs> is dense in .
- 2. D<subs>n</subs> is invariant under the action of U<subs>t,n</subs>.
- 3. , where is the domain of .
Since , therefore D<subs>n</subs> is a dense subspace of and (1) holds true.
For , the action of U<subs>t,n</subs> for t ≥ 0 is given by
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where l = # ��, with �� = {k ∈ {1,..., n}: x<subs>k</subs> ∈ (− t,0)}. The subspace is invariant under the action of U<subs>t,n</subs>, therefore to prove (2) it suffices to check that Ψ<subs>t,n</subs> = U<subs>t,n</subs>ψ<subs>n</subs> satisfies the boundary conditions. For 1 ≤ j ≤ n we have that
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where is identified with its ampliation and we used that e<sups>−itH</sups> and W commute with because they act on different factors. Similar computations hold for t < 0, therefore Ψ<subs>t,n</subs> satisfies the boundary conditions in D<subs>n</subs>.
Take ψ<subs>n</subs> ∈ D<subs>n</subs>, and for 1 ≤ j ≤ n, let us define , hence, we have that
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where |·| denotes the norm in ℋ.
The second integral goes to zero as t → 0 since the operator with domain is the infinitesimal generator of the unitary group defined by (V<subs>t,n</subs>ψ<subs>n</subs>)(x<subs>1</subs>,...,x<subs>n</subs>) = e<sups>−itH</sups>ψ<subs>n</subs>(x<subs>1</subs> + t,...,x<subs>n</subs> + t), for , continuous.
For every {k<subs>1</subs>,...,k<subs>m</subs>} ⊂ {1,...,n}, 1 ≤ m ≤ n, we define the family of disjoint subsets
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We have that , and . To estimate the first integral we introduce the functions
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where S<subs>{k1,...,km}</subs> = {(x<subs>1</subs>,...,x<subs>n</subs>): x<subs>ks</subs> > 0, 1 ≤ s ≤ m}.
Therefore, we have that
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where we have written simply instead <subs>{k1,···,km}</subs>. The last integral goes to zero as t → 0 since the operator iC<subs>n</subs> with domain is the infinitesimal generator of the unitary group (V<subs>t</subs>)<subs>t∈ℝ</subs>, in defined above. Therefore, we have that and this finishes the proof.
Remark 2.1
The family of subsets R<subs>{k1,...,kn}</subs> resemble the Fock chambers defined by Chebotarev[3].
Proof of Theorem 2.1
To prove Theorem 2.1 it suffices to prove that,
- 1. �� is dense in Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>*</subs>))⊗ℋ.
- 2. �� is invariant under the action of U<subs>t</subs>.
- 3. , where is the domain of .
Let Ψ = (ψ<subs>0</subs>,ψ<subs>1</subs>,...,ψ<subs>n</subs>,...) ∈ Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>*</subs>))⊗ℋ, therefore and given ε > 0, ∃ k<subs>ε</subs> ∈ ℕ such that . Moreover since D<subs>j</subs> is dense in , for each j ∈ ℕ ∃ ϕ<subs>j</subs> ∈ D<subs>j</subs> such that . Let Φ<subs>ε</subs> = (ϕ<subs>0</subs>,...,ϕ<subs>kε</subs>,0,...) ∈ ��, therefore,
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Hence, �� is dense in Γ<sups>s</sups>(L<sups>2</sups>(ℝ<subs>*</subs>))⊗ℋ.
For Ψ ∈ ��, Ψ = (ψ<subs>0</subs>,ψ<subs>1</subs>,...,ψ<subs>n</subs>,...) where ψ<subs>j</subs> ∈ D<subs>j</subs>, ∀ j ≥ 1, ψ<subs>0</subs> ∈ ℂ. We have that U<subs>t</subs>Ψ = (ψ<subs>0</subs>,U<subs>t,1</subs>ψ<subs>1</subs>,...,U<subs>t,n</subs>ψ<subs>n</subs>,...) and the result of Propo-sition 2.1 implies that U<subs>t,j</subs>ψ<subs>j</subs> ∈ D<subs>j</subs> for each j ≥ 1, therefore U<subs>t</subs>Ψ ∈ �� for each t ∈ ℝ and Ψ ∈ ��. Hence {U<subs>t</subs>}<subs>t∈ℝ</subs> leaves �� invariant. It remains to prove that . Let Ψ ∈ ��, therefore ∃ k ∈ ℕ such that Ψ = (ψ<subs>0</subs>,...,ψ<subs>k</subs>, 0,...) with ψ<subs>j</subs> ∈ D<subs>j</subs>, j = 1,2,...,k. Given ε > 0, ∃ t<subs>j</subs>(ε) > 0 such that
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then we have that if , {\TEN
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hence .
3. DEFECT INDICES OF C
The problem of the selfadjointness of C in the case when H is an unbounded symmetric operator is much more subtle. Chebotarev[4] studied sufficient conditions that assure the selfadjointness of C, the isometric property of the associated semigroup and the conservativity of two quantum dynamical semigroups canonically associated with C. In the present section we shall discuss an example that shows that in some cases C is not a selfadjoint operator and iC may not be the generator of a strongly continuous semigroup of isometries. The following example was discussed in a different context in Ref.[3]; see also Ref.[6].
On ℋ = L<subs>2</subs>(0,∞) let us consider operators induced by the differential form
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where f ∈ C<sups>∞</sups>(0,∞), f > 0, f′ is bounded and . Notice that functions of the form f(x) = (1 + x)<sups>α</sups>, 0 ≤ α ≤ 1, satisfy these conditions. We denote by H<subs>1,0</subs> the minimal operator induced by τ<subs>f</subs>; it is defined by
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Notice that
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see for instance Ref.[12], Theorem 6.31.
The maximal operator H<subs>1</subs> induced by τ<subs>f</subs> is defined by
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and
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One can show that H<subs>1,0</subs> is a symmetric operator; hence it is closable and its closure is symmetric, moreover , the proof of this fact is simple but rather long to be included here. Now let us consider the equations
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The solutions of these equations are respectively
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and
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Notice that
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since therefore u<subs>+</subs> ∈ L<subs>2</subs>(0,∞). Similarly one can show that u<subs>−</subs> ∉ L<subs>2</subs>(0,∞). This proves that the defect indices of the symmetric operator are and .
Being a maximal symmetric operator, then is the generator of a strongly continuous semigroup of contractions. Here is a proof.
Theorem 3.1
The operator satisfies the hypotheses of the Lumer–Phillips Theorem, so it generates a strongly continuous semigroup of contractions U<subs>t</subs>.
Proof
By the von Neumann's Theorem we have that
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and
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, v<subs>+</subs> ∈ N<subs>+</subs>, v<subs>−</subs> ∈ N<subs>−</subs>, where
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are the defect subspaces of .
But we have shown that N<subs>−</subs> = {0}, therefore
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and
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, v<subs>+</subs> ∈ N<subs>+</subs>.
Then for u ∈ dom; H<subs>1</subs>, u = ω + v<subs>+</subs>, , v<subs>+</subs> ∈ N<subs>+</subs> we have that
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hence
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This proves that iH<subs>1</subs> is dissipative.
If Θ(iH<subs>1</subs>) = {⟨ iH<subs>1</subs>u,u⟩: u ∈ dom; H<subs>1</subs>, ‖u‖ = 1} is the numerical range of iH<subs>1</subs>, then we have for λ<subs>0</subs> > 0
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since Re⟨ iH<subs>1</subs>u, u⟩ ≤ 0, u ∈ dom; H<subs>1</subs>. Therefore,
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for any u ∈ dom; H<subs>1</subs>, ‖u‖ = 1. Hence the operator (iH<subs>1</subs> − λ<subs>0</subs>I)<sups>−1</sups> there exists, it is bounded and closed on ℛ(iH<subs>1</subs> − λ<subs>0</subs>I), since (iH<subs>1</subs> − λ<subs>0</subs>I) is closed. Then ℛ(iH<subs>1</subs> − λ<subs>0</subs>) is closed and therefore
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since for any λ<subs>0</subs> > 0.
Now, by Corollary 10.6 in Ref.[11], p. 41 we have that the adjoint is the generator of the adjoint semigroup . But, as we will prove below, the operator does not satisfy the necessary and sufficient conditions in the Lumer–Phillips theorem, Ref.[11], and therefore it is not the generator of a strongly continuous semigroup of contractions. □
Corollary 3.1
The range is not total in ℋ. Therefore is not the generator of a strongly continuous semigroup of contractions.
Proof
It suffices to observe that
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From Corollary 10.6 in Ref.[11] it follows that neither is the generator of a strongly continuous semigroup of contractions.
Now we shall use the following theorem proved in Ref.[3], Proposition 12.2.2, p. 254. □
Theorem 3.2
Assume that H = I⊗ H<subs>0</subs>, where H<subs>0</subs> is a closed densely defined symmetric operator which acts in ℋ and commutes with the unitary operator W. Then the defect indices of
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coincides with the defect indices of H<subs>0</subs>.
As a consequence we obtain that the operator with domain D⊗ dom H<subs>0</subs>, is a symmetric operator with the same defect indices as , i.e., n<subs>−</subs> = 0 and n<subs>+</subs> = 1, therefore it has not selfadjoint extensions. Similarly as in the above Corollary 3.1 we can prove that does not satisfy the hypotheses of Lumer–Phillips Theorem, therefore it is not the generator of a strongly continuous semigroup of contractions.
Given a closed symmetric operator it naturally arises the question of whether or not is the restriction of a generator of a strongly continuous semigroup of isometries. The following proposition gives an answer.
Proposition 3.1
Let be a closed symmetric operator in a Hilbert space h with finite defect indices (n<subs>+</subs>, n<subs>−</subs>), then
- i. if n<subs>+</subs> ≤ n<subs>−</subs> the operator is the restriction of a generator of a strongly continuous semigroup of isometries in h.
- ii. if n<subs>+</subs> > n<subs>−</subs> then is not the restriction of a generator of a strongly continuous semigroup of isometries in h.
Proof
(i) If 0 = n<subs>+</subs> ≤ n<subs>−</subs> then is maximal symmetric, or selfadjoint if n<subs>−</subs> = 0, see for instance Ref.[12], Theorem 8.14. Therefore the arguments in Theorem 3.1 prove that generates a strongly continuous semigroup of contractions in h. The adjoint semigroup (U<subs>t</subs> = W*<subs>t</subs>)<subs>t≥0</subs> is generated by ; since n<subs>+</subs> = 0 this semigroup (U<subs>t</subs>)<subs>t≥0</subs> is of isometries. If is selfadjoint, generates a unitary group.
If 0 < n<subs>+</subs> ≤ n<subs>−</subs>, then the defect subspace N<subs>+</subs> of is isometrically isomorphic with a subspace F<subs>−</subs> of N<subs>−</subs>, let us denote by V the isometry V : N<subs>+</subs> → F<subs>−</subs>. By the von Neumann theorem, associated with V there exists a closed symmetric extension of defined as
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and
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for and v ∈ N<subs>+</subs>. Since we have that , hence we are in the case . So we can proceed as above to prove that generates a strongly continuous semigroup of isometries in h, and hence is the restriction of a generator of a strongly continuous semigroup of isometries in h.
(ii)If n<subs>+</subs> > n<subs>−</subs> then there exists an isometry V′ from the defect subspace N<subs>−</subs> of onto a proper subspace of N<subs>+</subs>, and associated with V′ there exists a maximal symmetric extension of . The semigroup of contractions generated by the dissipative operator is not a semigroup of isometries since .
ACKNOWLEDGMENTS
We would like to thank the anonymous referees for a careful reading of the manuscript and their valuable comments. This work was partially supported by CONACYT, Grant 37491-E.
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By OswaldoG. Gaxiola and Roberto Quezada
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