EIGENVALUE ESTIMATES FOR SCHRÃDINGER ... - UT Mathematics
EIGENVALUE ESTIMATES FOR SCHRÖDINGER OPERATORS WITH COMPLEX POTENTIALS ARI LAPTEV AND OLEG SAFRONOV
1. Introduction Throughout the paper, f± denotes either the positive or the negative part of f , which is, in its turn, either a function or a selfadjoint operator. The symbols 0, the eigenvalues zj of H lying outside the sector {z : |=z| < t 1. We also know the elementary estimate (see Theorem 8.2) Z √ 2γ |= λ| ≤ C |V |3/2+γ dx, γ > 0, d = 3, R3
however it is not quite the same as (1.5). While we do not prove Conjecture 1.1 directly, we find some interesting information about the location of eigenvalues of the operator −∆ R + iV with a positive V ≥ 0. In particular, in d = 3, we obtain that if V dx is small and λ ∈ / R+ is an eigenvalue of −∆ + iV , then |λ| must be large. It might seem that R eigenvalues do not exist at all for small values of V dx, however their presence in such cases can be easily established with the help of the following statement. Proposition 1.1. Let d ≥ 3. Then there is a sequence of positive functions Vn ≥ 0 such that the “largest modulus” eigenvalue λn ∈ / R+ of the R operator −∆ + iVn satisfies |λn | → ∞ as n → ∞, while limn→∞ Vn (x)dx = 0. Proof. If λ is an eigenvalue of −∆ + iV (x), n2 λ is an eigenvalue R then 2 2 of −∆ + n iV (nx). It remains to note that n V (nx)dx = Cn2−d . The proof of existence of a non-real eigenvalue of −∆ + iV (x) at least for one V ≥ 0 is left to the reader. ¤ Remark The proposition does not contradict Conjecture 1.1. Note that our theorems imply also that the eigenvalues of −∆ + iV can not accumulate to zero in d = 3, if V ≥ 0 is integrable.
2. Preliminaries 1. Consider a non self-adjoint operator T = R + iB where R and B are self-adjoint operators. Suppose that R is semibounded from below and has a discrete negative spectrum. If B(R − i)−1 is compact, then the operator T has only discrete set of eigenvalues in the left half of the plane Clef t = {z : j0 by 1-s. This inequality has been discovered for compact operators by H. Weyl. Weyl’s proof is carried over to the case of bounded operators without any difference. Indeed, let P be the orthogonal projection onto the span of eigenvectors corresponding to λj , 1 ≤ j ≤ n. Then for any α>0 n Y det (I + P (αT ∗ T − I)P ) = αn |λj |2 5
1
Consequently, n Y 1/2 1/2 αn |λj |2 ≤ det (I+P (αT ∗ T −I)+ P ) ≤ det (I+(αT ∗ T −I)+ P (αT ∗ T −I)+ ). 1
Removing the orthogonal projection, we obtain n Y Y n αs2j |λj |2 ≤ det (I + (αT ∗ T − I)+ ) = α 1
αs2j >1
It remains to choose α = s−1 n . Note that if the number of sj > 1 is finite, we can take α = 1 to obtain that n Y Y |λj |2 ≤ s2j 1
s2j >1
for all n. Corollary 2.2. Let γ ≥ 1. Then n n X X (|λj |2 − 1)γ ≤ (s2j − 1)γ 1
1
for all n. Proof. Consider first the case γ > 1. As a consequence of (2.3), we obtain that n n X X (2.4) log |λj | ≤ log sj . 1
1
Moreover, n n X X (log |λj | − η)+ ≤ (log sj − η)+
(2.5)
j=1
j=1
for any −∞ < η < ∞. Note now that the function φ(t) = (e2t − 1)γ is representable in the form Z ∞ φ(λ) = (λ − t)+ φ00 (t) dt and φ00 (t) ≥ 0 for t ≥ 0. 0
Since φ(log λ) = (λ2 − 1)γ , the statement of Corollary 2.2 for γ > 1 follows from (2.5). If γ = 1, then one only needs to prove that n n X X |λj |2 ≤ s2j 1
6
1
In order to do that we consider the function ψ(t) = e2t , which is representable in the form Z ∞ ψ(λ) = 1 + 2λ + (λ − t)+ ψ 00 (t) dt with ψ 00 (t) > 0. 0 2
Since ψ(log λ) = λ , the statement of Corollary 2.2 for γ = 1 follows from (2.4) and (2.5). ¤ 3. Let T be a compact operator in a Hilbert space. Then the square roots of eigenvalues of T ∗ T are called s-numbers. Let us introduce the distribution function n(s, T ) of s-numbers sj of an operator T by the equality n(s, T ) = card{j : sj > s}, s > 0. Note that n satisfies the so called Ki-Fan inequality n(s1 + s2 , T1 + T2 ) ≤ n(s1 , T1 ) + n(s2 , T2 ), for any pair of compact operators T1 and T2 and any pair of positive numbers s1 and s2 . The class of operators T for which the following quantity [T ]pp := sup sp n(s, T ) < ∞ s>0
is finite, is called the weak Neumann-Schatten class Σp . Theorem 2.1 ( M.Cwikel [2]). Let Φ be the operator of Fourier transformation and let α and β be the operators of multiplication by the functions α(ξ) and β(x). Suppose that β is in the space Lq (Rd ) with q > 2 and [α]qq = sup tq meas{ξ ∈ (Rd : |α(ξ)| > t} < ∞. t>0
Then the operator T = αΦβ as (well as the operator βΦ∗ α) is in Σp and Z q q (2.6) [T ]q ≤ C[α]q |β(x)|q dx. The main applications of Theorem 2.1 in mathematical physics are related to the following fact: Proposition 2.1 (Birman-Schwinger principle). Let A and V be two positive self-adjoint operators acting in Hilbert space. Suppose √ the same −1/2 that V is bounded and the operator V (A + I) is compact. Then the number N (E) of eigenvalues of the operator A − V lying to the left of a negative point −E satisfies the relation √ √ N (E) = n(1, V (A + E)−1 V ). 7
In applications, A is a differential operator and V is the operator of multiplication by a function. 4. Let T be a compact operator in a Hilbert space. Then the square roots of eigenvalues of T ∗ T are called s-numbers. The class of compact operators T for which the following quantity X p sj < ∞, p ≥ 1, ||T ||pp = j
is finite, is called the Neumann-Schatten class Sp . It is easy to see that ¡ ¢1/p the functional ||T ||p = tr(T ∗ T )p/2 has all properties of a norm on Sp . The next theorem gives a sufficient condition guaranteeing that an operator of the form b(x)a(i∇) belongs to the class Sp . Theorem 2.2. Let Φ be the operator of Fourier transformation and let a and b be the operators of multiplication by the functions a(ξ) and b(x). Suppose that a and b are in the space Lp (Rd ) where p ≥ 2 Then the operator T = bΦ∗ a is in Sp and Z Z p −d p (2.7) ||T ||p ≤ (2π) |a(ξ)| dξ |b(x)|p dx. This theorem can be found in [6]. See also [5] and [7]. 5. Below, we will need the following result about eigenvalue estimates for a certain operator with constant coefficients perturbed by a potential V . It is one of the consequences of the inequality (2.6). Proposition 2.2. Let a(ξ) = (ξ 2 − µ)2 , ξ ∈ Rd , and V (x) ≥ 0 be two functions on Rd . Suppose that V ∈ C0∞ (Rd ). Let N (E) be the number of eigenvalues of the operator a(i∇) − V (x) lying to the left of the point −E where E > 0. Then, for any p > 1/2, (2.8) Z Z ´ C³ p+d/4 d/2−1 N (E) ≤ p V dx + µ V p+1/2 dx , if d ≥ 2; E Rd Rd Z C V p+1/2 dx, if d = 1. (2.9) N (E) ≤ p 1/2 E µ d R Proof. The reasoning is based on the elementary application of the Cwikel estimate. Indeed, according to Birman-Schwinger principle N (E) = n(1, X), where X is the compact operator defined by the equality √ √ X = V (a(i∇) + E)−1 V . 8
Let χ be the characteristic function of the ball {ξ ∈ Rd : Represent X in the form X = X1 + X2 , where √ √ X1 = V (a(i∇) + E)−1 χ(i∇) V .
ξ 2 ≤ µ}.
According to the Ki-Fan inequality, (2.10)
n(1, X) ≤ n(1, 2X1 ) + n(1, 2X2 ).
Therefore it is sufficient to estimate each term in the right hand side of (2.10) separately. We begin with the first term. Set α = p + d/4. Then, according to (2.6), Z Z dξ α n(1, 2X1 ) ≤ C0 V dx ≤ 2 2 α ξ 2 >µ ((ξ − µ) + E) Z Z ∞ Z Z ∞ d/2−1 sd/2−1 ds s ds α α ≤ C1 V dx ≤ C2 V dx = 2 α 2 ((s − µ) + E) (s + E)α µ 0 Z C = p V p+d/4 dx. E Rd Let us now estimate the second term in (2.10). Set β = p + 1/2. According to (2.6), Z Z dξ β ≤ n(1, 2X2 ) ≤ C3 V dx 2 2 β ξ 2 1, Z ∞ Z ∞ dξ dξ C ≤ = . √ 2 2 β 2 β µE β−1/2 ∞ ((ξ − µ) + E) ∞ ((ξ − µ) µ + E) Set now β = p + 1/2. We obtain according to (2.6) that R R C V β dx C V p+1/2 dx N (E) ≤ √ β−1/2 = , √ p µE µE which means that (2.9) is also proven. 9
¤
3. Proof of Theorem 1.1. Some relates results Proof of Theorem 1.1. The central role in the proof is played by Corollary 2.1. The second trick which we apply in the proof is that we use the relation between some of the eigenvalues of the operator −∆ + V and the eigenvalues of the operator (−∆ + 2i − µ + V )2 , µ > 0, situated to the left of the line | 0. Finally, note that 1/ s ≥ ε0 /s for s ≥ ε0 . Therefore without loss of generality one can assume that √ ∀s > 0. m(b) = [Cb/ s] + 1, 2
Since there is no zj ∈ Πb with the property =zj > C|Ψb (W )| 2r−1 , we obtain X
m(b)
(=zj − s)+ ≤
zj ∈Πb
X X l=1 zj ∈Ωµl
1
(b + |Ψb (W )| 2r−1 ) √ (=zj − s)+ ≤ C F (s, b) s
Observe now that X X Z γ |=zj | = γ(γ − 1) zj ∈Πb
zj ∈Πb
∞
(=zj − s)+ sγ−2 ds,
0
which leads to Z X 1 γ |=zj | ≤ (b + |Ψb (W )| 2r−1 )C
∞
sγ−5/2 F (s, b)ds
0
zj ∈Πb
The integral in the right hand side converges only if γ > 3/2 and the previous relation means that ³Z X 2δ 1 γ |=zj | ≤ C|Ψb (W )| 2r−1 (b + |Ψb (W )| 2r−1 ) |W |d/4−1/2+γ−δ dx+ Rd
zj ∈Πb
+b
Z |W |γ−δ dx
d/2−1
´
Rd
with 0 < δ < 1/2. It remains to set r = γ − δ to complete the proof. ¤ 12
In Theorem P 1.1γ and Theorem 1.2, we estimate the eigenvalue sum of the form |=zj | with γ > 3/2. If we restrict the set and consider not all eigenvalues but only those that belong to a certain domain, then one can estimate even the sum of the first powers. Corollary 3.1. Let zj be the eigenvalues of the operator −∆ + V lying inside the domain {z : (=zj + 1)2 − ( 0} and let d ≥ 2. Then Z ´ ³Z X 1/2+p d/4+p d/2−1 W dx ||W ||1−p W dx + µ |=zj | ≤ C ∞ Rd
Rd
j
for 1/2 < p < 1. Corollary 3.1 does not follow immediately from the Theorem 1.1, however it follows from a relation that is similar to (3.4). In the same way, one can prove Corollary 3.2. Let d = 1 and let zj be the eigenvalues of the operator −d2 /dx2 + V lying inside the domain {z : (=zj + 1)2 − ( 0}. Then Z X 1−p −1/2 |=zj | ≤ C||W ||∞ µ W 1/2+p dx Rd
j
for 1/2 < p < 1. There is an open gap in this theory in the case d = 1, where we are forced to keep away from the point z = 0 and deal with the strip a < 0. Probably, the reason why we have to do that is hidden in a special behaviour of the spectrum near zero.PThis approach is unable to say anything about the sums of the form 0 7/4 and the previous relation means that ³Z ´ X 1/2 γ 1−p |W |γ−5/4+p dx |=zj | ≤ C||W ||∞ (b + ||W ||∞ ) Rd
zj ∈Πb
with 1/2 < p < 1. It remains to set r = γ + p − 1 to complete the proof. ¤ 14
4. Proof of Theorem 1.3 The main tool of the proof is the linear fractional mapping that takes the upper half-plane {z : =z > 0} into the compliment of the unit disk {z : |z| > 1}. This transformation is given by the formula z+i+1 z 7→ . z−i+1 Insert the operator H = −∆ + V instead of z into this formula, i.e. consider the operator U = (H + I + i)(H + I − i)−1 = I + 2i(H + I − i)−1 . The number z ∈ / R is an eigenvalue of the operator H if and only if the point (z + i + 1)/(z − i + 1) is an eigenvalue of U . Let us now find the operator U ∗ U . It is easy to see that U ∗ = I − 2i(H ∗ + I + i)−1 . Therefore U ∗ U = I +2i(H +I −i)−1 −2i(H ∗ +I +i)−1 +4(H ∗ +I +i)−1 (H +I −i)−1 . Using the Hilbert identity, we obtain U ∗ U = I + 2i(H ∗ + I + i)−1 (H ∗ − H)(H + I − i)−1 or, put it differently, U ∗ U = I + 4(H ∗ + I + i)−1 =V (H + I − i)−1 . Thus, we obtain that U ∗ U − I ≤ 4Y ∗ Y, where Y = =V + (H + I − i)−1 . According to Corollary 2.2, the eigenvalues λj of the operator H satisfy the estimate ´p X³¯ λj + 1 + i ¯2 ¯ ¯ − 1 ≤ tr (U ∗ U − I)p+ ≤ 4p tr (Y ∗ Y )p = 4p ||Y ||2p 2p . λ + 1 − i + j j p
It follows from this inequality that ´p X³ =λj ≤ ||Y ||2p (4.1) 2p 2+1 + |λ + 1| j j Indeed, denote a = 2=λj /(|λj + 1|2 + 1) and suppose that =λj > 0. Then ¯ λ + 1 + i ¯2 ³1 + a´ ¯ j ¯ − 1 ≥ 2a. ¯ ¯ −1= λj + 1 − i 1−a We come to the conclusion that one needs to estimate the norm of the operator p Y = =V + (H + I − i)−1 15
in the class S2p . Let us represent this operator in the form p Y = =V + (−∆ + I)−1/2 B, where B = (−∆ + I)1/2 (H + I − i)−1 . We will show that the operator B is bounded and its norm does not exceed 1. In other words, we will show that ||(−∆ + I)1/2 (H + I − i)−1 f ||2 ≤ ||f ||2 ,
(4.2)
for all f ∈ L2 . Denote u = (H + I − i)−1 f . It is obvious that Z Z 2 2 (|∇u| + (1 + d/2.
Using homogeneity one can easily obtain that Z dξ d/2−p J = |λ| , 2 iφ p Rd |ξ − e | where φ = arg λ. Therefore the question about anR estimate of J is reduced to the problem of estimating the integral Rd |ξ 2 − eiφ |−p dξ, which behaves as Z Z dξ dt ∼C ∼ Cφ1−p . 2 2 2 p/2 2 2 )p/2 |(ξ − 1) + φ | (t + φ d R R In other words,
Z Rd
|ξ 2
dξ ≤ C| sin φ|1−p . − eiφ |p
Consequently, J ≤ C|λ|d/2−p | sin φ|1−p = C|=λ|1−p |λ|d/2−1 . It remains to note that
Z −d
|V |p dx.
1 ≤ (2π) J Rd
The proof in the case p = d/2 > 1 is similar to the proof in the case p > d/2 with the only exception that instead of Theorem 2.2 one needs to use Theorem 2.1. Indeed, let 1 and p = d/2 > 1. a(ξ) = 2 |ξ − λ| Then, using homogeneity, one can easily obtain that 1 where a0 = 2 [a]pp = [a0 ]pp , |ξ − eiφ | and φ = arg λ. Therefore the question about an estimate of [a]pp is reduced to the problem of estimating the quantity [a0 ]pp which is not bigger than Z Z dξ dt C1 + C2 ∼ C ∼ Cφ1−p . 2 2 2 p/2 2 2 )p/2 |(ξ − 1) + φ | (t + φ |ξ| 0 (see Theorem 8.1 for that matter). Theorem 7.1. Let d = 3 and z = k 2 ∈ / R+ be an eigenvalue of H. Then Z 0. Note that the real part of the operator X is positive. Consequently, the spectrum of this operator lies in the right half plane. Therefore whenever z is the eigenvalue of H the sum of the real parts of the eigenvalues ζj of X is greater than or equal 1, i.e. X
1. Introduction Throughout the paper, f± denotes either the positive or the negative part of f , which is, in its turn, either a function or a selfadjoint operator. The symbols 0, the eigenvalues zj of H lying outside the sector {z : |=z| < t 1. We also know the elementary estimate (see Theorem 8.2) Z √ 2γ |= λ| ≤ C |V |3/2+γ dx, γ > 0, d = 3, R3
however it is not quite the same as (1.5). While we do not prove Conjecture 1.1 directly, we find some interesting information about the location of eigenvalues of the operator −∆ R + iV with a positive V ≥ 0. In particular, in d = 3, we obtain that if V dx is small and λ ∈ / R+ is an eigenvalue of −∆ + iV , then |λ| must be large. It might seem that R eigenvalues do not exist at all for small values of V dx, however their presence in such cases can be easily established with the help of the following statement. Proposition 1.1. Let d ≥ 3. Then there is a sequence of positive functions Vn ≥ 0 such that the “largest modulus” eigenvalue λn ∈ / R+ of the R operator −∆ + iVn satisfies |λn | → ∞ as n → ∞, while limn→∞ Vn (x)dx = 0. Proof. If λ is an eigenvalue of −∆ + iV (x), n2 λ is an eigenvalue R then 2 2 of −∆ + n iV (nx). It remains to note that n V (nx)dx = Cn2−d . The proof of existence of a non-real eigenvalue of −∆ + iV (x) at least for one V ≥ 0 is left to the reader. ¤ Remark The proposition does not contradict Conjecture 1.1. Note that our theorems imply also that the eigenvalues of −∆ + iV can not accumulate to zero in d = 3, if V ≥ 0 is integrable.
2. Preliminaries 1. Consider a non self-adjoint operator T = R + iB where R and B are self-adjoint operators. Suppose that R is semibounded from below and has a discrete negative spectrum. If B(R − i)−1 is compact, then the operator T has only discrete set of eigenvalues in the left half of the plane Clef t = {z : j0 by 1-s. This inequality has been discovered for compact operators by H. Weyl. Weyl’s proof is carried over to the case of bounded operators without any difference. Indeed, let P be the orthogonal projection onto the span of eigenvectors corresponding to λj , 1 ≤ j ≤ n. Then for any α>0 n Y det (I + P (αT ∗ T − I)P ) = αn |λj |2 5
1
Consequently, n Y 1/2 1/2 αn |λj |2 ≤ det (I+P (αT ∗ T −I)+ P ) ≤ det (I+(αT ∗ T −I)+ P (αT ∗ T −I)+ ). 1
Removing the orthogonal projection, we obtain n Y Y n αs2j |λj |2 ≤ det (I + (αT ∗ T − I)+ ) = α 1
αs2j >1
It remains to choose α = s−1 n . Note that if the number of sj > 1 is finite, we can take α = 1 to obtain that n Y Y |λj |2 ≤ s2j 1
s2j >1
for all n. Corollary 2.2. Let γ ≥ 1. Then n n X X (|λj |2 − 1)γ ≤ (s2j − 1)γ 1
1
for all n. Proof. Consider first the case γ > 1. As a consequence of (2.3), we obtain that n n X X (2.4) log |λj | ≤ log sj . 1
1
Moreover, n n X X (log |λj | − η)+ ≤ (log sj − η)+
(2.5)
j=1
j=1
for any −∞ < η < ∞. Note now that the function φ(t) = (e2t − 1)γ is representable in the form Z ∞ φ(λ) = (λ − t)+ φ00 (t) dt and φ00 (t) ≥ 0 for t ≥ 0. 0
Since φ(log λ) = (λ2 − 1)γ , the statement of Corollary 2.2 for γ > 1 follows from (2.5). If γ = 1, then one only needs to prove that n n X X |λj |2 ≤ s2j 1
6
1
In order to do that we consider the function ψ(t) = e2t , which is representable in the form Z ∞ ψ(λ) = 1 + 2λ + (λ − t)+ ψ 00 (t) dt with ψ 00 (t) > 0. 0 2
Since ψ(log λ) = λ , the statement of Corollary 2.2 for γ = 1 follows from (2.4) and (2.5). ¤ 3. Let T be a compact operator in a Hilbert space. Then the square roots of eigenvalues of T ∗ T are called s-numbers. Let us introduce the distribution function n(s, T ) of s-numbers sj of an operator T by the equality n(s, T ) = card{j : sj > s}, s > 0. Note that n satisfies the so called Ki-Fan inequality n(s1 + s2 , T1 + T2 ) ≤ n(s1 , T1 ) + n(s2 , T2 ), for any pair of compact operators T1 and T2 and any pair of positive numbers s1 and s2 . The class of operators T for which the following quantity [T ]pp := sup sp n(s, T ) < ∞ s>0
is finite, is called the weak Neumann-Schatten class Σp . Theorem 2.1 ( M.Cwikel [2]). Let Φ be the operator of Fourier transformation and let α and β be the operators of multiplication by the functions α(ξ) and β(x). Suppose that β is in the space Lq (Rd ) with q > 2 and [α]qq = sup tq meas{ξ ∈ (Rd : |α(ξ)| > t} < ∞. t>0
Then the operator T = αΦβ as (well as the operator βΦ∗ α) is in Σp and Z q q (2.6) [T ]q ≤ C[α]q |β(x)|q dx. The main applications of Theorem 2.1 in mathematical physics are related to the following fact: Proposition 2.1 (Birman-Schwinger principle). Let A and V be two positive self-adjoint operators acting in Hilbert space. Suppose √ the same −1/2 that V is bounded and the operator V (A + I) is compact. Then the number N (E) of eigenvalues of the operator A − V lying to the left of a negative point −E satisfies the relation √ √ N (E) = n(1, V (A + E)−1 V ). 7
In applications, A is a differential operator and V is the operator of multiplication by a function. 4. Let T be a compact operator in a Hilbert space. Then the square roots of eigenvalues of T ∗ T are called s-numbers. The class of compact operators T for which the following quantity X p sj < ∞, p ≥ 1, ||T ||pp = j
is finite, is called the Neumann-Schatten class Sp . It is easy to see that ¡ ¢1/p the functional ||T ||p = tr(T ∗ T )p/2 has all properties of a norm on Sp . The next theorem gives a sufficient condition guaranteeing that an operator of the form b(x)a(i∇) belongs to the class Sp . Theorem 2.2. Let Φ be the operator of Fourier transformation and let a and b be the operators of multiplication by the functions a(ξ) and b(x). Suppose that a and b are in the space Lp (Rd ) where p ≥ 2 Then the operator T = bΦ∗ a is in Sp and Z Z p −d p (2.7) ||T ||p ≤ (2π) |a(ξ)| dξ |b(x)|p dx. This theorem can be found in [6]. See also [5] and [7]. 5. Below, we will need the following result about eigenvalue estimates for a certain operator with constant coefficients perturbed by a potential V . It is one of the consequences of the inequality (2.6). Proposition 2.2. Let a(ξ) = (ξ 2 − µ)2 , ξ ∈ Rd , and V (x) ≥ 0 be two functions on Rd . Suppose that V ∈ C0∞ (Rd ). Let N (E) be the number of eigenvalues of the operator a(i∇) − V (x) lying to the left of the point −E where E > 0. Then, for any p > 1/2, (2.8) Z Z ´ C³ p+d/4 d/2−1 N (E) ≤ p V dx + µ V p+1/2 dx , if d ≥ 2; E Rd Rd Z C V p+1/2 dx, if d = 1. (2.9) N (E) ≤ p 1/2 E µ d R Proof. The reasoning is based on the elementary application of the Cwikel estimate. Indeed, according to Birman-Schwinger principle N (E) = n(1, X), where X is the compact operator defined by the equality √ √ X = V (a(i∇) + E)−1 V . 8
Let χ be the characteristic function of the ball {ξ ∈ Rd : Represent X in the form X = X1 + X2 , where √ √ X1 = V (a(i∇) + E)−1 χ(i∇) V .
ξ 2 ≤ µ}.
According to the Ki-Fan inequality, (2.10)
n(1, X) ≤ n(1, 2X1 ) + n(1, 2X2 ).
Therefore it is sufficient to estimate each term in the right hand side of (2.10) separately. We begin with the first term. Set α = p + d/4. Then, according to (2.6), Z Z dξ α n(1, 2X1 ) ≤ C0 V dx ≤ 2 2 α ξ 2 >µ ((ξ − µ) + E) Z Z ∞ Z Z ∞ d/2−1 sd/2−1 ds s ds α α ≤ C1 V dx ≤ C2 V dx = 2 α 2 ((s − µ) + E) (s + E)α µ 0 Z C = p V p+d/4 dx. E Rd Let us now estimate the second term in (2.10). Set β = p + 1/2. According to (2.6), Z Z dξ β ≤ n(1, 2X2 ) ≤ C3 V dx 2 2 β ξ 2 1, Z ∞ Z ∞ dξ dξ C ≤ = . √ 2 2 β 2 β µE β−1/2 ∞ ((ξ − µ) + E) ∞ ((ξ − µ) µ + E) Set now β = p + 1/2. We obtain according to (2.6) that R R C V β dx C V p+1/2 dx N (E) ≤ √ β−1/2 = , √ p µE µE which means that (2.9) is also proven. 9
¤
3. Proof of Theorem 1.1. Some relates results Proof of Theorem 1.1. The central role in the proof is played by Corollary 2.1. The second trick which we apply in the proof is that we use the relation between some of the eigenvalues of the operator −∆ + V and the eigenvalues of the operator (−∆ + 2i − µ + V )2 , µ > 0, situated to the left of the line | 0. Finally, note that 1/ s ≥ ε0 /s for s ≥ ε0 . Therefore without loss of generality one can assume that √ ∀s > 0. m(b) = [Cb/ s] + 1, 2
Since there is no zj ∈ Πb with the property =zj > C|Ψb (W )| 2r−1 , we obtain X
m(b)
(=zj − s)+ ≤
zj ∈Πb
X X l=1 zj ∈Ωµl
1
(b + |Ψb (W )| 2r−1 ) √ (=zj − s)+ ≤ C F (s, b) s
Observe now that X X Z γ |=zj | = γ(γ − 1) zj ∈Πb
zj ∈Πb
∞
(=zj − s)+ sγ−2 ds,
0
which leads to Z X 1 γ |=zj | ≤ (b + |Ψb (W )| 2r−1 )C
∞
sγ−5/2 F (s, b)ds
0
zj ∈Πb
The integral in the right hand side converges only if γ > 3/2 and the previous relation means that ³Z X 2δ 1 γ |=zj | ≤ C|Ψb (W )| 2r−1 (b + |Ψb (W )| 2r−1 ) |W |d/4−1/2+γ−δ dx+ Rd
zj ∈Πb
+b
Z |W |γ−δ dx
d/2−1
´
Rd
with 0 < δ < 1/2. It remains to set r = γ − δ to complete the proof. ¤ 12
In Theorem P 1.1γ and Theorem 1.2, we estimate the eigenvalue sum of the form |=zj | with γ > 3/2. If we restrict the set and consider not all eigenvalues but only those that belong to a certain domain, then one can estimate even the sum of the first powers. Corollary 3.1. Let zj be the eigenvalues of the operator −∆ + V lying inside the domain {z : (=zj + 1)2 − ( 0} and let d ≥ 2. Then Z ´ ³Z X 1/2+p d/4+p d/2−1 W dx ||W ||1−p W dx + µ |=zj | ≤ C ∞ Rd
Rd
j
for 1/2 < p < 1. Corollary 3.1 does not follow immediately from the Theorem 1.1, however it follows from a relation that is similar to (3.4). In the same way, one can prove Corollary 3.2. Let d = 1 and let zj be the eigenvalues of the operator −d2 /dx2 + V lying inside the domain {z : (=zj + 1)2 − ( 0}. Then Z X 1−p −1/2 |=zj | ≤ C||W ||∞ µ W 1/2+p dx Rd
j
for 1/2 < p < 1. There is an open gap in this theory in the case d = 1, where we are forced to keep away from the point z = 0 and deal with the strip a < 0. Probably, the reason why we have to do that is hidden in a special behaviour of the spectrum near zero.PThis approach is unable to say anything about the sums of the form 0 7/4 and the previous relation means that ³Z ´ X 1/2 γ 1−p |W |γ−5/4+p dx |=zj | ≤ C||W ||∞ (b + ||W ||∞ ) Rd
zj ∈Πb
with 1/2 < p < 1. It remains to set r = γ + p − 1 to complete the proof. ¤ 14
4. Proof of Theorem 1.3 The main tool of the proof is the linear fractional mapping that takes the upper half-plane {z : =z > 0} into the compliment of the unit disk {z : |z| > 1}. This transformation is given by the formula z+i+1 z 7→ . z−i+1 Insert the operator H = −∆ + V instead of z into this formula, i.e. consider the operator U = (H + I + i)(H + I − i)−1 = I + 2i(H + I − i)−1 . The number z ∈ / R is an eigenvalue of the operator H if and only if the point (z + i + 1)/(z − i + 1) is an eigenvalue of U . Let us now find the operator U ∗ U . It is easy to see that U ∗ = I − 2i(H ∗ + I + i)−1 . Therefore U ∗ U = I +2i(H +I −i)−1 −2i(H ∗ +I +i)−1 +4(H ∗ +I +i)−1 (H +I −i)−1 . Using the Hilbert identity, we obtain U ∗ U = I + 2i(H ∗ + I + i)−1 (H ∗ − H)(H + I − i)−1 or, put it differently, U ∗ U = I + 4(H ∗ + I + i)−1 =V (H + I − i)−1 . Thus, we obtain that U ∗ U − I ≤ 4Y ∗ Y, where Y = =V + (H + I − i)−1 . According to Corollary 2.2, the eigenvalues λj of the operator H satisfy the estimate ´p X³¯ λj + 1 + i ¯2 ¯ ¯ − 1 ≤ tr (U ∗ U − I)p+ ≤ 4p tr (Y ∗ Y )p = 4p ||Y ||2p 2p . λ + 1 − i + j j p
It follows from this inequality that ´p X³ =λj ≤ ||Y ||2p (4.1) 2p 2+1 + |λ + 1| j j Indeed, denote a = 2=λj /(|λj + 1|2 + 1) and suppose that =λj > 0. Then ¯ λ + 1 + i ¯2 ³1 + a´ ¯ j ¯ − 1 ≥ 2a. ¯ ¯ −1= λj + 1 − i 1−a We come to the conclusion that one needs to estimate the norm of the operator p Y = =V + (H + I − i)−1 15
in the class S2p . Let us represent this operator in the form p Y = =V + (−∆ + I)−1/2 B, where B = (−∆ + I)1/2 (H + I − i)−1 . We will show that the operator B is bounded and its norm does not exceed 1. In other words, we will show that ||(−∆ + I)1/2 (H + I − i)−1 f ||2 ≤ ||f ||2 ,
(4.2)
for all f ∈ L2 . Denote u = (H + I − i)−1 f . It is obvious that Z Z 2 2 (|∇u| + (1 + d/2.
Using homogeneity one can easily obtain that Z dξ d/2−p J = |λ| , 2 iφ p Rd |ξ − e | where φ = arg λ. Therefore the question about anR estimate of J is reduced to the problem of estimating the integral Rd |ξ 2 − eiφ |−p dξ, which behaves as Z Z dξ dt ∼C ∼ Cφ1−p . 2 2 2 p/2 2 2 )p/2 |(ξ − 1) + φ | (t + φ d R R In other words,
Z Rd
|ξ 2
dξ ≤ C| sin φ|1−p . − eiφ |p
Consequently, J ≤ C|λ|d/2−p | sin φ|1−p = C|=λ|1−p |λ|d/2−1 . It remains to note that
Z −d
|V |p dx.
1 ≤ (2π) J Rd
The proof in the case p = d/2 > 1 is similar to the proof in the case p > d/2 with the only exception that instead of Theorem 2.2 one needs to use Theorem 2.1. Indeed, let 1 and p = d/2 > 1. a(ξ) = 2 |ξ − λ| Then, using homogeneity, one can easily obtain that 1 where a0 = 2 [a]pp = [a0 ]pp , |ξ − eiφ | and φ = arg λ. Therefore the question about an estimate of [a]pp is reduced to the problem of estimating the quantity [a0 ]pp which is not bigger than Z Z dξ dt C1 + C2 ∼ C ∼ Cφ1−p . 2 2 2 p/2 2 2 )p/2 |(ξ − 1) + φ | (t + φ |ξ| 0 (see Theorem 8.1 for that matter). Theorem 7.1. Let d = 3 and z = k 2 ∈ / R+ be an eigenvalue of H. Then Z 0. Note that the real part of the operator X is positive. Consequently, the spectrum of this operator lies in the right half plane. Therefore whenever z is the eigenvalue of H the sum of the real parts of the eigenvalues ζj of X is greater than or equal 1, i.e. X
