Eigenvalue estimates for the one-particle density matrix
aa r X i v : . [ m a t h . SP ] A ug EIGENVALUE ESTIMATES FOR THE ONE-PARTICLE DENSITYMATRIX
ALEXANDER V. SOBOLEV
Abstract.
It is shown that the eigenvalues λ k , k = 1 , , . . . , of the one-particle densitymatrix satisfy the bound λ k ≤ Ck − / with a positive constant C . Introduction
Consider on L ( R N ) the Schr¨odinger operator H = N X k =1 (cid:18) − ∆ k − Z | x k | (cid:19) + X ≤ j
0. The notation ∆ k is used for theLaplacian w.r.t. the variable x k . The operator H acts on the Hilbert space L ( R N )and it is self-adjoint on the domain D ( H ) = H ( R N ), since the potential in (1.1) is aninfinitesimal perturbation relative to the unperturbed operator − ∆ = − P k ∆ k , see e.g.[14, Theorem X.16]. Note that we do not need to assume that the particles are fermions,i.e. that the underlying Hilbert space consists of anti-symmetric L -functions. Our resultsare not sensitive to such assumptions. Let ψ = ψ ( x ), x = (ˆ x , x N ), ˆ x = ( x , x , . . . , x N − ),be an eigenfunction of the operator H with an eigenvalue E ∈ R , i.e. ψ ∈ D ( H ) and( H − E ) ψ = 0 . We define the one-particle density matrix as the function γ ( x, y ) = Z R N − ψ (ˆ x , x ) ψ (ˆ x , y ) d ˆ x , ( x, y ) ∈ R × R . (1.2)We do not discuss the importance of this object for multi-particle quantum mechanicsand refer to the monograph [4] for details. Our focus is on spectral properties of the self-adjoint non-negative operator Γ with the kernel γ ( x, y ), which we call the one-electrondensity operator . Note that the operator Γ is represented as a product Γ = Ψ ∗ Ψ whereΨ : L ( R ) → L ( R N − ) is the operator with the kernel ψ (ˆ x , x ). Since ψ ∈ L ( R N ), theoperator Ψ is Hilbert-Schmidt, and hence Γ is trace class. Our objective is to investigate Mathematics Subject Classification.
Primary 35J10; Secondary 47G10, 81Q10.
Key words and phrases.
Multi-particle Schr¨odinger operator, one-particle density matrix, eigenvalues,integral operators. the decay of the eigenvalues λ k (Γ) > k = 1 , , . . . , of the non-negative operator Γ,labelled in descending order counting multiplicity. The significance of such informationfor quantum mechanical computations is discussed in the paper [10]. In particular, itis shown in [10] that Γ has infinite rank. Our main result is contained in Theorem 1.1below. It establishes upper bounds on the decay of the eigenvalues λ k (Γ), k = 1 , , . . . under the condition that ψ decays exponentially as | x | → ∞ : | ψ ( x ) | . e − κ | x | , x ∈ R N . (1.3)Here κ > . ” means that the left-hand side is boundedfrom above by the right-hand side times some positive constant whose precise value is ofno importance for us. This notation is used throughout the paper. Notice that instead ofthe standard Euclidean norm | x | in (1.3) we have the ℓ -norm which we denote by | x | .For the discrete eigenvalues, i.e. the ones below the bottom of the essential spectrum of H , the bound (1.3) follows from [6]. The exponential decay for eigenvalues away fromthe thresholds, including embedded ones, was studied in [5], [11]. For more referencesand detailed discussion we quote [15]. Theorem 1.1.
Suppose that the eigenfunction ψ satisfies the bound (1.3) . Let the func-tion γ ( x, y ) , ( x, y ) ∈ R × R , be defined by (1.2) . Then the eigenvalues λ k (Γ) , k =1 , , . . . , of the operator Γ satisfy the estimate < λ k (Γ) . k − , k = 1 , , . . . , (1.4) with an implicit positive constant independent of k . Remark. (1) The bound (1.4) is sharp. This is confirmed by the asymptotic for-mula for the eigenvalues λ k (Γ) which will be proved in a subsequent publication[16]. In fact, Theorem 1.1 or, more precisely, Theorem 3.1 can be regarded as apreparation for the sharp asymptotic formula in [16].(2) Theorem 1.1 extends to the case of a molecule with several nuclei whose positionsare fixed. The modifications are straightforward.(3) The choice of the norm | x | instead of the Eucledian norm | x | in the estimate(1.3) is made for computational convenience later in the proof.The strategy of the proof is quite straightforward: by virtue of the factorization Γ =Ψ ∗ Ψ, mentioned a few lines earlier, we have λ k (Γ) = s k (Ψ) , k = 1 , , . . . , where s k (Ψ)are the singular values ( s -values) of the operator Ψ. It is well-known that the rate ofdecay of singular values for integral operators depends on the smoothness of their kernels,and the appropriate estimates via suitable Sobolev norms can be found in the monograph[2] by M.S. Birman and M.Z. Solomyak. The regularity of ψ has been well-studied in theliterature. To begin with, according to the classical elliptic theory, due to the analyticityof the Coulomb potential | x | − for x = 0, the function ψ is real analytic away from theparticle coalescence points. A more challenging problem is to understand the behaviourof ψ at the coalescence points. The first result in this direction belongs to T. Kato [13],who showed that the function ψ is Lipschitz. More detailed information on ψ at the ENSITY MATRIX 3 coalescence points was obtained, e.g. in [8], [9], [12], and in the recent paper [7] by S.Fournais and T.Ø. Sørensen. The results of [2] and [7] are of crucial importance for theproof of Theorem 1.1. A combination of the efficient bounds for the derivatives of thefunction ψ obtained in [7], and the estimates for the singular values in [2], leads to thebound s k (Ψ) . k − / , and hence to (1.4).The plan of the paper is as follows. In Sect. 2 we list the facts that serve as ingredientsof the proof. Although our aim is to prove the bound s k (Ψ) . k − / , in Sect. 3 inTheorem 3.1 we state a bound for the operator Ψ with weights which will be useful inthe study of the spectral asymptotics for Ψ. The rest of Sect. 3 provides some preliminaryestimates for auxiliary integral operators. These estimates are put together in Sect. 4to complete the proof of Theorems 3.1 and 1.1.We conclude the introduction with some general notational conventions. Coordinates.
As mentioned earlier, we use the following standard notation for thecoordinates: x = ( x , x , . . . , x N ), where x j ∈ R , j = 1 , , . . . , N . The vector x isusually represented in the form x = (ˆ x , x N ) with ˆ x = ( x , x , . . . , x N − ) ∈ R N − . Inorder to write formulas in a more compact and unified way, we sometimes use the notation x = 0.In the space R d , d ≥ , the notation | x | stands for the Euclidean norm, whereas | x | denotes the ℓ -norm. Indicators.
For any set Λ ⊂ R d we denote by Λ its indicator function (or indicator). Derivatives.
Let N = N ∪ { } . If x = ( x ′ , x ′′ , x ′′′ ) ∈ R and m = ( m ′ , m ′′ , m ′′′ ) ∈ N ,then the derivative ∂ mx is defined in the standard way: ∂ mx = ∂ m ′ x ′ ∂ m ′′ x ′′ ∂ m ′′′ x ′′′ . Bounds.
As explained earlier, for two non-negative numbers (or functions) X and Y depending on some parameters, we write X . Y (or Y & X ) if X ≤ CY withsome positive constant C independent of those parameters. To avoid confusion we maycomment on the nature of (implicit) constants in the bounds.2. Ingredients of the proof
In this section we list three ingredients of the proof of the main Theorem 1.1.2.1.
Regularity of the eigenfunction.
We need some efficient bounds for the deriva-tives of the eigenfunction away from the coalescence points, obtained by S. Fournais andT.Ø. Sørensen in [7]. Letd(ˆ x , x ) = min {| x | , | x − x j | , j = 1 , , . . . , N − } . The following proposition is a consequence of [7, Corollary 1.3]:
Proposition 2.1.
Assume that ψ satisfies (1.3) . Then for all multi-indices m ∈ N , | m | ≥ , we have | ∂ mx ψ (ˆ x , x ) | . d(ˆ x , x ) − l e − κ l | x | , l = | m | , (2.1) ALEXANDER V. SOBOLEV with some κ l > . The precise values of the constants κ l > κ = κ ≥ κ ≥ · · · > . (2.2)Let us rewrite the bounds (2.1) using the notation x = 0. With this convention, wehave d(ˆ x , x ) = min {| x − x j | , j = 0 , , , . . . , N − } and d(ˆ x , x ) − ≤ X ≤ j ≤ N − | x − x j | − . Therefore (1.3) and (2.1) imply that | ∂ mx ψ (ˆ x , x ) | . e − κ l | x | (cid:18) X ≤ j ≤ N − | x − x j | − l (cid:19) , l = | m | , (2.3)for all m ∈ N .2.2. Compact operators.
Our main reference for compact operators is the book [3].Let H and G be separable Hilbert spaces. Let T : H → G be a compact operator. If H = G and T = T ∗ ≥
0, then λ k ( T ), k = 1 , , . . . , denote the positive eigenvalues of T numbered in descending order counting multiplicity. For arbitrary spaces H , G andcompact T , by s k ( T ) > k = 1 , , . . . , we denote the singular values of T defined by s k ( T ) = λ k ( T ∗ T ) = λ k ( T T ∗ ). Note the useful inequality s k ( T + T ) ≤ s k − ( T + T ) ≤ s k ( T ) + s k ( T ) , (2.4)which holds for any two compact T , T , see [3, Formula (11.1.14)]. We classify compactoperators by the rate of decay of their singular values. If s k ( T ) . k − /p , k = 1 , , . . . ,with some p >
0, then we say that T ∈ S p, ∞ and denote k T k p, ∞ = sup k k p s k ( T ) . (2.5)The class S p, ∞ is a complete linear space with the quasi-norm k T k p, ∞ , see [3, § p ∈ (0 ,
1) the quasi-norm satisfies the following “triangle” inequality for operators T j ∈ S p, ∞ , j = 1 , , . . . : (cid:13)(cid:13) X j T j (cid:13)(cid:13) pp, ∞ ≤ (1 − p ) − X j k T j k pp, ∞ , (2.6)see [1, Lemmata 7.5, 7.6], [2, §
1] and references therein. For the case p > § ENSITY MATRIX 5
For T ∈ S p, ∞ the following number is finite: G p ( T ) = (cid:0) lim sup k →∞ k p s k ( T ) (cid:1) p , (2.7)and it clearly satisfies the inequality G p ( T ) ≤ k T k pp, ∞ . (2.8)More precisely, let S ◦ p, ∞ ⊂ S p, ∞ be the closed subspace of all operators R ∈ S p, ∞ with G p ( R ) = 0. As explained in [3, Theorem 11.6.10], G p ( T ) = inf R ∈ S ◦ p, ∞ k T + R k pp, ∞ . (2.9)The functional G p ( T ), p <
1, also satisfies the inequality of the type (2.6):
Lemma 2.2.
Suppose that T j ∈ S p, ∞ , j = 1 , , . . . , with some p < and that X j k T j k pp, ∞ < ∞ . (2.10) Then G p (cid:0) X j T j (cid:1) ≤ (1 − p ) − X j G p ( T j ) . (2.11) Proof.
By (2.6) the operator T = P j T j belongs to S p, ∞ , so that the left-hand side isfinite. Furthermore, due to (2.8) and to the condition (2.10) the right-hand side of (2.11)is finite as well. Fix an ε > N such that ∞ X j = N +1 k T j k pp, ∞ < ε. Then by (2.9) and (2.6), for any R j ∈ S ◦ p, ∞ , j = 1 , , . . . , N , we have the estimate G p ( T ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) N X j =1 ( T j + R j ) + ∞ X j = N +1 T j (cid:13)(cid:13)(cid:13)(cid:13) pp ≤ (1 − p ) − (cid:18) N X j =1 k T j + R j k pp, ∞ + ε (cid:19) . Minimizing the right-hand side over R j , j = 1 , , . . . , N , by (2.9) we get the estimate G p ( T ) ≤ (1 − p ) − (cid:18) N X j =1 G p ( T j ) + ε (cid:19) ≤ (1 − p ) − (cid:18) ∞ X j =1 G p ( T j ) + ε (cid:19) . Since ε > (cid:3)
ALEXANDER V. SOBOLEV
Singular values of integral operators.
The final ingredient of the proof is theresult due to M.S. Birman and M.Z. Solomyak, investigating the membership of integraloperators in the class S p, ∞ with some p >
0. For estimates of the singular values werely on [2, Proposition 2.1], see also [3, Theorem 11.8.4], which we state here in a formconvenient for our purposes. Let C = (0 , d ⊂ R d , d ≥
1, be the unit cube.
Proposition 2.3.
Let T ba : L ( C ) → L ( R n ) , be the integral operator of the form ( T ba u )( t ) = b ( t ) Z C T ( t, x ) a ( x ) u ( x ) dx, where a ∈ L ( C ) , b ∈ L ( R n ) , and the kernel T ( t, x ) , t ∈ R n , x ∈ C , is such that T ( t, · ) ∈ H l ( C ) with some l = 1 , , . . . , l > d , a.e. t ∈ R n . Then s k ( T ba ) . k − − ld (cid:20)Z R n k T ( t, · ) k H l | b ( t ) | dt (cid:21) k a k L ( C ) ,k = 1 , , . . . , with some implicit constant independent of the kernel T , weights a, b andthe index k . In other words, T ba ∈ S q, ∞ with q = 12 + ld , and k T ba k q, ∞ . (cid:20)Z R n k T ( t, · ) k H l | b ( t ) | dt (cid:21) k a k L ( C ) . It is straightforward to check that if one replaces the cube C with its translate C n = C + n, n ∈ Z d , then the bounds of Proposition 2.3 still hold with implicit constantsindependent of n . 3. Preliminary estimates
The weighted operator Ψ . Represent the operator Γ as the product Γ = Ψ ∗ Ψ,where Ψ : L ( R ) → L ( R N − ) is defined by(Ψ u )(ˆ x ) = Z R ψ (ˆ x , x ) u ( x ) dx, u ∈ L ( R ) . Since ψ ∈ L ( R N ), this operator is Hilbert-Schmidt. As explained in the Introduction,in order to prove (1.4) it suffices to show that s k (Ψ) . k − / , k = 1 , , . . . , i.e. thatΨ ∈ S / , ∞ . For future use, we obtain an estimate for the operator b Ψ a with weights a and b . In order to describe these weights, denote C n = (0 , + n , n ∈ Z . Let κ l > a ∈ L ( R ) is such that S ( l ) q ( a ) = (cid:20) X n ∈ Z e − q κ l | n | k a k q L ( C n ) (cid:21) q < ∞ , q = 34 , (3.1) ENSITY MATRIX 7 and that b ∈ L ∞ ( R N − ), so that M ( l ) ( b ) = (cid:20)Z R N − | b (ˆ x ) | e − κ l | ˆ x | d ˆ x (cid:21) < ∞ , ∀ l = 1 , , . . . . (3.2)Recall that the functional G p is defined in (2.7). Our objective is to prove the followingtheorem. Theorem 3.1.
Let b ∈ L ∞ ( R N − ) and let a ∈ L ( R ) be such that S (4)3 / ( a ) < ∞ . Then b Ψ a ∈ S / , ∞ and k b Ψ a k / , ∞ . k b k L ∞ S (3)3 / ( a ) , (3.3) G / ( b Ψ a ) . (cid:0) M (4) ( b ) S (4)3 / ( a ) (cid:1) . (3.4)For a = 1 and b = 1 this theorem implies that s k (Ψ) . k − / , and hence λ k (Γ) = s k (Ψ) . k − / , thereby proving Theorem 1.1.The plan of the proof is as follows. We study first the operators Ψ n = Ψ C n , n ∈ Z .For each fixed n the operator Ψ n is split in the sum of several operators depending on twoparameters: δ > ε >
0, whose singular values are estimated in different ways. Noneof these estimates is sharp, but in the end, when collecting all the estimates together inSect. 4, we get the sharp bound (3.4) by making a clever choice of the parameters δ and ε . For convenience we introduce the notation Int ( T ) : L ( R ) → L ( R N − ) for the integraloperator with the kernel T (ˆ x , x ). Whenever we consider the operators b Int ( · ) C n a withweights a, b , the constants in all the bounds are independent on the weights or on theparameter n ∈ Z .Recall also that we use the notation x = 0. The symbol P j (resp. Q j ) assumessummation (resp. product) over all j = 0 , , . . . , N − Partition of Ψ n : step 1. The first step is to estimate the contribution of thedomain on which the variables x j , j = 0 , , , . . . , N − , are close to each other. Fix a δ > ( δ ) = \ ≤ l δ } . The indicator of this set is denoted by χ ( δ ) , i.e. χ ( δ ) (ˆ x ) = Ω ( δ ) (ˆ x ) = Y ≤ l δ } (ˆ x ) . (3.5) ALEXANDER V. SOBOLEV
Represent ψ as follows: ψ = ψ ( δ )1 + ψ ( δ )2 , (3.6) ψ ( δ )1 (ˆ x , x ) = ψ (ˆ x , x ) χ ( δ ) (ˆ x ) ,ψ ( δ )2 (ˆ x , x ) = ψ (ˆ x , x ) − ψ ( δ )1 (ˆ x , x ) = ψ (ˆ x , x ) (cid:0) − χ ( δ ) (ˆ x ) (cid:1) . It follows from (2.3) that | ∂ mx ψ ( δ )2 (ˆ x , x ) | . e − κ | m | | x | (cid:18) X j | x − x j | −| m | (cid:19) X ≤ l
0. The operator
Int ( ψ ( δ )2 )is considered with the weight b = 1 and arbitrary a ∈ L ( C n ).In the next lemma and further on we use the straightforward inequalitymax x ∈ C n e − κ l | x | ≤ e κ l e − κ l | ˆ x | e − κ l | n | . (3.8) Lemma 3.2.
The operator
Int ( ψ ( δ )2 ) a C n belongs to S / , ∞ and k Int ( ψ ( δ )2 ) a C n k / , ∞ . e − κ | n | δ k a k L ( C n ) , (3.9) for all n ∈ Z and all δ > .Proof. According to (3.7) and (3.8), ψ ( δ )2 (ˆ x , · ) ∈ H ( C n ) for a.e. ˆ x ∈ R N − and e κ | ˆ x | k ψ ( δ )2 (ˆ x , · ) k H . e − κ | n | Z C n (cid:18) X ≤ j ≤ N − | x − x j | − (cid:19) dx × X ≤ l d ), we get that the operator on theleft-hand side of (3.9) belongs to S q, ∞ with q = 6 / k Int ( ψ ( δ )2 ) a C n k / , ∞ . (cid:20) Z R N − k ψ ( δ )2 (ˆ x , · ) k H d ˆ x (cid:21) k a k L ( C n ) . e − κ | n | (cid:20) Z R N − e − κ | ˆ x | X ≤ l
To study the kernel ψ ( δ )1 , we separate the contribution from the values of x that are“far” from x j ’s, j = 0 , , . . . , N −
1. Let θ ∈ C ∞ ( R ) be a function such that 0 ≤ θ ≤ θ ( t ) = 0 , if | t | > θ ( t ) = 1 , if | t | < . Denote ζ ( t ) = 1 − θ ( t ). Observe that for any ν > (cid:12)(cid:12) ∂ mx θ (cid:0) | x | ν − (cid:1)(cid:12)(cid:12) . {| x | < ν } + ν −| m | { ν< | x | < ν } . ν −| m | {| x | < ν } , (3.10) (cid:12)(cid:12) ∂ mx ζ (cid:0) | x | ν − (cid:1)(cid:12)(cid:12) . {| x | >ν } + ν −| m | { ν< | x | < ν } . ν −| m | {| x | >ν } , for all m ∈ N . Consequently, (cid:12)(cid:12) ∂ mx θ (cid:0) | x | ν − (cid:1)(cid:12)(cid:12) . | x | −| m | {| x | < ν } , (cid:12)(cid:12) ∂ mx ζ (cid:0) | x | ν − (cid:1)(cid:12)(cid:12) . | x | −| m | {| x | >ν } , m ∈ N , (3.11)uniformly in ν > ψ ( δ )1 : ψ ( δ )1 = ψ ( δ )11 + ψ ( δ )12 , (3.12) ψ ( δ )11 (ˆ x , x ) = X j θ (cid:0) | x − x j | δ − (cid:1) ψ ( δ )1 (ˆ x , x ) ,ψ ( δ )12 (ˆ x , x ) = (cid:2) − X j θ (cid:0) | x − x j | δ − (cid:1)(cid:3) ψ ( δ )1 (ˆ x , x ) . In view of the definition of χ ( δ ) , see (3.5), we have (cid:2) − X j θ (cid:0) | x − x j | δ − (cid:1)(cid:3) χ ( δ ) (ˆ x ) = Y j ζ (cid:0) | x − x j | δ − (cid:1) χ ( δ ) (ˆ x ) , so that ψ ( δ )12 (ˆ x , x ) = ψ ( δ )1 (ˆ x , x ) Y j ζ (cid:0) | x − x j | δ − (cid:1) . Estimate the derivatives of this function. First observe that in view of (3.11) we have (cid:12)(cid:12) ∂ mx Y j ζ (cid:0) | x − x j | δ − (cid:1)(cid:12)(cid:12) . (cid:18) X j | x − x j | −| m | (cid:19) Y j {| x − x j | >δ } (ˆ x , x ) , m ∈ N . Together with (2.3) this gives (cid:12)(cid:12) ∂ mx ψ ( δ )12 (ˆ x , x ) (cid:12)(cid:12) . e − κ | m | | x | X j | x − x j | −| m | {| x − x j | >δ } (ˆ x , x ) , m ∈ N , (3.13)uniformly in δ > Lemma 3.3.
For any l ≥ the operator Int ( ψ ( δ )12 ) a C n belongs to S q, ∞ with q = 12 + l , (3.14) and k Int ( ψ ( δ )12 ) a C n k q, ∞ . e − κ l | n | δ − l + k a k L ( C n ) , (3.15) for all δ ∈ (0 , δ ] , with an implicit constant depending on l and δ only.Proof. According to (3.13), ψ ( δ )12 (ˆ x , · ) ∈ H l ( C n ) for a.e. ˆ x ∈ R N − with an arbitrary l ≥ l ≥ e κ l | ˆ x | k ψ ( δ )12 (ˆ x , · ) k H l . e − κ l | n | Z C n (cid:18) X j | x − x j | − l {| x − x j | >δ } (ˆ x , x ) (cid:19) dx . e − κ l | n | (cid:0) δ − l +3 (cid:1) . e − κ l | n | δ − l +3 . Now the bound (3.15) follows from Proposition 2.3 with d = 3 and b (ˆ x ) = 1. (cid:3) Partition of Ψ n : step 2. It is important to note that the right-hand side of (3.13)contains the factor | x − x j | −| m | instead of | x − x j | −| m | that is present in (2.3). This isa consequence of the fact that the bound (2.1) holds for | m | ≥
1, but not for m = 0.As we see later on, in spite of this loss of one power of | x − x j | , the estimate (3.15) issufficient for derivation of the sharp bounds (3.3) and (3.4). However, when consideringthe term ψ ( δ )11 in (3.12) the bound by | x − x j | −| m | is not enough, and we need to havethe factor | x − x j | −| m | , just as in (2.3). To achieve this we have to “correct” the kernel ψ ( δ )11 with the help of the auxiliary kernel η ( δ ) (ˆ x , x ) = X j θ (cid:0) | x − x j | δ − (cid:1) ψ ( δ )1 (ˆ x , x j ) . As the next lemma shows, the kernel η ( δ ) has properties similar to those of ψ ( δ )12 . Lemma 3.4.
For any l ≥ the operator Int ( η ( δ ) ) a C n belongs to S q, ∞ with the parameter q defined in (3.14) , and k Int ( η ( δ ) ) a C n k q, ∞ . e − κ l | n | δ − l + k a k L ( C n ) , (3.16) for all δ ∈ (0 , δ ] .Proof. Using (3.8) and (3.10) we get | ∂ mx η ( δ ) (ˆ x , x ) | . δ −| m | e − κ l | n | − κ l | ˆ x | X j {| x − x j | < δ } (ˆ x , x ) , m ∈ N , | m | ≤ l. Therefore, η ( δ ) (ˆ x , · ) ∈ H l ( C n ) for a.e. ˆ x ∈ R N − with an arbitrary l ≥ l ≥ e κ l | ˆ x | k η ( δ ) (ˆ x , · ) k H l . e − κ l | n | Z C n (cid:0) δ − l X j {| x − x j | < δ } (ˆ x , x ) (cid:1) dx . e − κ l | n | (cid:0) δ − l +3 (cid:1) . e − κ l | n | δ − l +3 . ENSITY MATRIX 11
Now the required bound follows from Proposition 2.3 with d = 3 and b (ˆ x ) = 1. (cid:3) Let us now investigate the “corrected” kernel ψ ( δ )11 , and consider instead of it the kernel φ ( δ ) = ψ ( δ )11 − η ( δ ) = X j φ ( δ ) j , (3.17) φ ( δ ) j (ˆ x , x ) = θ (cid:0) | x − x j | δ − (cid:1)(cid:0) ψ ( δ )1 (ˆ x , x ) − ψ ( δ )1 (ˆ x , x j ) (cid:1) . Before proceeding to the next step of the construction, we estimate the difference ψ ( δ )1 (ˆ x , x ) − ψ ( δ )1 (ˆ x , x j ). It follows from (2.1) with | m | = 1 that (cid:12)(cid:12) ψ ( δ )1 (ˆ x , x ) − ψ ( δ )1 (ˆ x , x j ) (cid:12)(cid:12) ≤ | x − x j | max t ∈ [0 , |∇ x ψ ( δ )1 (ˆ x , tx j + (1 − t ) x ) | . | x − x j | e − κ | x | χ ( δ ) (ˆ x ) . (3.18)In order to estimate the derivatives of this difference, we make the following observation.By the definition of θ , we have | x − x j | < δ on the support of φ ( δ ) j . Furthermore, theballs { x ∈ R : | x − x j | < δ } ⊂ R , j = 0 , , . . . , N −
1, are pairwise disjoint sinceˆ x ∈ Ω ( δ ) . As a consequence,d(ˆ x , x ) = | x − x j | , if | x − x j | < δ, ˆ x ∈ Ω ( δ ) . Consequently, the bound (2.1) together with (3.18) lead to (cid:12)(cid:12) ∂ mx (cid:0) ψ ( δ )1 (ˆ x , x ) − ψ ( δ )1 (ˆ x , x j ) (cid:1)(cid:12)(cid:12) . | x − x j | −| m | e − κ | m | | x | χ ( δ ) (ˆ x ) , if | x − x j | < δ, (3.19)for all m ∈ N . Here we have also used our convention that κ = κ , see (2.2).Now return to the functions φ ( δ ) j , see (3.17). The φ ( δ ) j (ˆ x , x ) is again partitioned inthe sum of two new kernels. At this (last) stage of the partition we introduce a newparameter ε ≤ δ/
2. With this choice of ε we have θ ( tε − ) = θ ( tε − ) θ ( tδ − ), so that φ ( δ ) j = ξ ( δ,ε ) j + β ( δ,ε ) j , j = 0 , , , . . . , N − , with ξ ( δ,ε ) j (ˆ x , x ) = θ (cid:0) | x − x j | ε − (cid:1)(cid:0) ψ ( δ )1 (ˆ x , x ) − ψ ( δ )1 (ˆ x , x j ) (cid:1) ,β ( δ,ε ) j (ˆ x , x ) = θ (cid:0) | x − x j | δ − (cid:1) ζ (cid:0) | x − x j | ε − (cid:1)(cid:0) ψ ( δ )1 (ˆ x , x ) − ψ ( δ )1 (ˆ x , x j ) (cid:1) . Therefore φ ( δ ) = ξ ( δ,ε ) + β ( δ,ε ) , where(3.20) ξ ( δ,ε ) = X j ξ ( δ,ε ) j , β ( δ,ε ) = X j β ( δ,ε ) j . In the next lemma we introduce a weight b ∈ L ∞ ( R N − ). Recall that under this conditionthe integral M ( l ) ( b ) defined in (3.2) is finite for all l ≥ Lemma 3.5.
Let a ∈ L ( C n ) and b ∈ L ∞ ( R N − ) . Then b Int ( ξ ( δ,ε ) ) a C n ∈ S / , ∞ and k b Int ( ξ ( δ,ε ) ) a C n k / , ∞ . e − κ | n | ε M (2) ( b ) k a k L ( C n ) , (3.21) for all ε ∈ (0 , and δ ∈ [2 ε, .Proof. According to (2.6), it suffices to prove (3.21) for each j = 0 , , . . . , N − , individ-ually. It follows from (3.11) that (cid:12)(cid:12) ∂ mx θ ( | x − x j | ε − ) (cid:12)(cid:12) . | x − x j | −| m | {| x − x j | < ε } (ˆ x , x ) , uniformly in ε > , δ > ε , for all m ∈ N . Together with (3.19) this implies that | ∂ mx ξ ( δ,ε ) j (ˆ x , x ) | . | x − x j | −| m | e − κ l | x | {| x − x j | < ε } (ˆ x , x ) , for all m ∈ N , | m | ≤ l . Thus ξ ( δ,ε ) j (ˆ x , · ) ∈ H ( C n ) for a.e. ˆ x ∈ R N − and e κ | ˆ x | k ξ ( δ,ε ) j (ˆ x , · ) k H . e − κ | n | Z C n (cid:0) | x − x j | − (cid:1) {| x − x j | < ε } dx . e − κ | n | ( ε + ε ) . e − κ | n | ε. It follows from Proposition 2.3 with l = 2 , d = 3 that b Int ( ξ ( δ,ε ) j ) a C n ∈ S / , ∞ and k b Int ( ξ ( δ,ε ) j ) a C n k / , ∞ . e − κ | n | (cid:20) Z R N − | b (ˆ x ) | k ξ ( δ,ε ) j (ˆ x , · ) k H e − κ | ˆ x | d ˆ x (cid:21) k a k L ( C n ) . e − κ | n | ε M (2) ( b ) k a k L ( C n ) . This completes the proof of (3.21). (cid:3)
Lemma 3.6.
Let a ∈ L ( C n ) and b ∈ L ∞ ( R N − ) . Then for any l ≥ the operator b Int ( β ( δ,ε ) ) a C n belongs to S q, ∞ with the parameter q defined in (3.14) , and k b Int ( β ( δ,ε ) ) a C n k q, ∞ . e − κ l | n | ε − l + M ( l ) ( b ) k a k L ( C n ) , (3.22) for all ε ∈ (0 , and δ ∈ [2 ε, .Proof. As in the previous lemma, due to (2.6), it suffices to prove (3.22) for each j =0 , , . . . , N − , individually. It follows from (3.11) that (cid:12)(cid:12) ∂ mx (cid:0) θ ( | x − x j | δ − ) ζ ( | x − x j | ε − ) (cid:1)(cid:12)(cid:12) . | x − x j | −| m | { ε< | x − x j | < δ } (ˆ x , x ) , uniformly in ε > , δ > ε , for all m ∈ N . Together with (3.19) this implies that | ∂ mx β ( δ,ε ) j (ˆ x , x ) | . | x − x j | −| m | e − κ l | x | {| x − x j | >ε } (ˆ x , x ) , ENSITY MATRIX 13 for all m ∈ N , | m | ≤ l . Thus β ( δ,ε ) j (ˆ x , · ) ∈ H l ( C n ) for a.e. ˆ x ∈ R N − with an arbitrary l ≥ l ≥ e κ l | ˆ x | k β ( δ,ε ) j (ˆ x , · ) k H l . e − κ l | n | Z C n (cid:0) | x − x j | − l (cid:1) {| x − x j | >ε } dx . e − κ l | n | (1 + ε − l ) . e − κ l | n | ε − l . Using Proposition 2.3 with d = 3 and arbitrary l ≥
3, we get that b Int ( β ( δ,ε ) j ) a C n ∈ S q, ∞ and k b Int ( β ( δ,ε ) j ) a C n k q, ∞ . e − κ l | n | (cid:20) Z R N − | b (ˆ x ) | k β ( δ,ε ) j (ˆ x , · ) k H l e − κ l | ˆ x | d ˆ x (cid:21) k a k L ( C n ) . e − κ l | n | ε − l + M ( l ) ( b ) k a k L ( C n ) . This completes the proof of (3.22). (cid:3) Proof of Theorems 3.1 and 1.1
Her we put together the estimates obtained in the previous section to complete theproof of Theorem 3.1. Recall again that the quantities S ( l ) q ( a ) and M ( l ) ( b ) are defined in(3.1) and (3.2) respectively. Lemma 4.1.
Suppose that b ∈ L ∞ ( R N − ) and a ∈ L ( C n ) . Then b Ψ n a ∈ S / , ∞ and k b Ψ n a k / , ∞ . e − κ | n | k b k L ∞ k a k L ( C ) , (4.1) G / ( b Ψ n a ) . (cid:0) e − κ | n | M (4) ( b ) k a k L ( C n ) (cid:1) , (4.2) for all n ∈ Z .Proof. Now we can put together all the estimates for the singular numbers, obtainedabove. Without loss of generality assume that k b k L ∞ ≤ k a k L ( C ) ≤ ψ = ξ ( δ,ε ) + β ( δ,ε ) + η ( δ ) + ψ ( δ )12 + ψ ( δ )2 . According to (3.9),(3.21), and the inequality (2.6), k b Int ( ξ ( δ,ε ) + ψ ( δ )2 ) a C n k / / , ∞ ≤ (cid:0) k b Int ( ξ ( δ,ε ) ) a C n k / / , ∞ + k Int ( ψ ( δ )2 ) a C n k / / , ∞ (cid:1) . e − κ | n | / (cid:0) ε M (2) ( b ) + δ (cid:1) / , so that, by definition (2.5), s k (cid:0) b Int ( ξ ( δ,ε ) + ψ ( δ )2 ) a C n (cid:1) . e − κ | n | (cid:0) ε M (2) ( b ) + δ (cid:1) k − , k = 1 , , . . . . (4.3) Similarly, using (3.15), (3.16) and (3.22) with one and the same l ≥
3, we obtain that k b Int ( β ( δ,ε ) + η ( δ ) + ψ ( δ )12 ) a C n k q, ∞ . e − κ l | n | (cid:0) ε − l + M ( l ) ( b ) + δ − l + (cid:1) , with 1 /q = 1 / l/ s k (cid:0) b Int ( β ( δ,ε ) + η ( δ ) + ψ ( δ )12 ) a C n (cid:1) . e − κ l | n | (cid:0) ε − l + M ( l ) ( b ) + δ − l + (cid:1) k − − l , k = 1 , , . . . . (4.4)Due to (2.4) and (2.2), combining (4.3) and (4.4), we get the estimate s k ( b Ψ n a ) ≤ s k − ( b Ψ n a ) . e − κ l | n | (cid:20)(cid:0) ε M ( l ) ( b ) + δ (cid:1) k − + (cid:0) ε − l + M ( l ) ( b ) + δ − l + (cid:1) k − − l (cid:21) , (4.5)where we have used that M (2) ( b ) ≤ M ( l ) ( b ). Rewrite the expression in the square brack-ets, gathering the terms containing ε and δ in two different groups: (cid:0) ε M ( l ) ( b ) k − + ε − l + M ( l ) ( b ) k − − l (cid:1) + (cid:0) δ k − + δ − l + k − − l (cid:1) = ε M ( l ) ( b ) k − (cid:0) ε − l +2 k − l (cid:1) + δ k − (cid:0) δ − l k − l (cid:1) . Since ε ∈ (0 ,
1] and δ ∈ [2 ε,
2] are arbitrary, we can pick ε = ε k = k − / and δ = δ k =4 k / (3 l ) − / , so that the condition δ > ε is satisfied for all k = 1 , , . . . , and ε − l +2 k − l = 1 , ε k − = k − ,δ − l k − l = 4 − l , δ k − = 4 k l − . Thus the bound (4.5) rewrites as s k ( b Ψ n a ) ≤ s k − ( b Ψ n a ) ≤ e − κ l | n | (cid:0) M ( l ) ( b ) k − + k l − (cid:1) . (4.6)Using the bound M ( l ) ( b ) . k b k L ∞ ≤
1, and taking l = 3 we conclude that s k ( b Ψ n a ) . e − κ | n | k − . This leads to (4.1).In order to obtain (4.2), we use (4.6) to writelim sup k →∞ k s k (cid:0) b Ψ n a (cid:1) . e − κ l | n | lim sup k →∞ (cid:0) M ( l ) ( b ) + k l − (cid:1) . Taking l = 4 we ensure that the second term in the brackets tends to zero. Therefore,lim sup k →∞ k s k (cid:0) b Ψ n a (cid:1) . e − κ | n | M (4) ( b ) . Applying definition (2.7), we arrive at (4.2). (cid:3)
ENSITY MATRIX 15
Proof of Theorems 3.1 and 1.1.
Since Ψ = P n ∈ Z Ψ n , we have, by (2.6) and (4.1), k b Ψ a k / / , ∞ ≤ X n ∈ Z k b Ψ n a k / / , ∞ . k b k L ∞ X n ∈ Z e − κ | n | k a k L ( C n ) = k b k L ∞ (cid:0) S (3)3 / ( a ) (cid:1) < ∞ . This proves (3.3).To prove (3.4) we use Lemma 2.2. According to (2.11) and (4.2), G / ( b Ψ a ) ≤ X n ∈ Z G / ( b Ψ n a ) . (cid:0) M (4) ( b ) (cid:1) X n ∈ Z e − κ | n | k a k L ( C n ) = (cid:0) M (4) ( b ) (cid:1) (cid:0) S (4)3 / ( a ) (cid:1) < ∞ . This completes the proof of Theorem 3.1.Using (3.3) with a ( x ) = 1 and b (ˆ x ) = 1 we get k Ψ k / , ∞ < ∞ , which implies that s k (Ψ) . k − / , and hence λ k (Γ) = s k (Ψ) . k − / . This proves Theorem 1.1. (cid:3) Acknowledgments.
The author is grateful to S. Fournais, T. Hoffmann-Ostenhof,M. Lewin and T. Ø. Sørensen for stumulating discussions and advice.The author was supported by the EPSRC grant EP/P024793/1.
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