THE NORM ESTIMATE OF THE DIFFERENCE BETWEEN THE KAC OPERATOR AND SCHR ¨ ODINGER SEMIGROUP II: THE GENERAL CASE INCLUDING THE RELATIVISTIC CASE

More thorough results than in our previous paper in Nagoya Math. J. are given on the L p -operator norm estimates for the Kac operator e − tV= 2 e − tH 0 e − tV= 2 compared with the Schr¨odinger semigroup e − t ( H 0 + V ) . The Schr¨odinger operators H 0 + V to be treated in this paper are more general ones associated with the L(cid:19)evy process, including the relativistic Schr¨odinger operator. The method of proof is probabilistic based on the Feynman-Kac formula. It di(cid:11)ers from our previous work in the point of using the Feynman-Kac formula not directly for these operators, but instead through subordination from the Brownian motion, which enables us to deal with all these operators in a uni(cid:12)ed way. As an application of such estimates the Trotter product formula in the L p -operator norm, with error bounds, for these Schr¨odinger semigroups is also derived. operator, Trotter product formula, Lie-Trotter-Kato product formula, Feynman-Kac formula, subordination of Brownian motion, Kato’s inequality


Introduction
By the Kac operator we mean an operator of the kind K(t) = e −tV /2 e −tH 0 e −tV /2 , where H = H 0 + V ≡ −∆/2 + V (x) is the nonrelativistic Schrödinger operator in L 2 (R d ) with mass 1 with scalar potential V (x) bounded from below. This K(t) may correspond to the transfer operator for a lattice model in statistical mechanics studied by M. Kac [Ka]. There it is one of the important problems to know asymptotic spectral properties of K(t) for t ↓ 0. To this end, in [H1, H2] Helffer estimated the L 2 -operator norm of the difference between K(t) and the Schrödinger semigroup e −tH to be of order O(t 2 ) for small t > 0, if V (x) satisfies |∂ α V (x)| ≤ C α (1 + |x| 2 ) (2−|α|) + /2 for every multi-index α with a constant C α . Then such norm estimates may be applied to get spectral properties of K(t) in comparison with those of H.
In  and  we have extended his result to the case of more general scalar potentials V (x) even in the L p -operator norm, 1 ≤ p ≤ ∞, making a probabilistic approach based on the Feynman-Kac formula. In  we have also considered this problem for both the nonrelativistic Schrödinger operator H = H 0 + V and the relativistic Schrödinger operator H r = H r 0 + V ≡ √ −∆ + 1 − 1 + V (x) with light velocity 1. The L p -operator norm of this difference is estimated to be of order O(t a ) of small t > 0 with a ≥ 1, though the relativistic case shows for small t > 0 a slightly different behavior from the nonrelativistic case. As another application of these results the Trotter product formula for the nonrelativistic and relativistic Schrödinger operators in the L p -operator norm with error bounds is obtained. There are also related L 2 results with operator-theoretic methods, for which we refer to [D-I-Tam].
Conditions (A) 0 , (A) 1 and (A) 2 on V (x) are used in [Tak] and are more general than in [I-Tak1,2], while conditions (V) 1 and (V) 2 are used in [D-I-Tam]. But these conditions may not be best possible. A simple example of a function which has property (A) 0 , (A) 1 or (A) 2 is, needless to say, V (x) = |x| r (0 < r < ∞), and a slightly complicated one V (x) = |x| r (2 + sin log |x|), according as 0 < r ≤ 1, 1 < r < 2 or r ≥ 2. Also V (x) = 1 + |x 1 − x 2 | r (x = (x 1 , x 2 , . . . , x d )) satisfies (A) 0 , (A) 1 or (A) 2 with the same r as above, but neither (V) 1 nor (V) 2 . To the contrary V (x) = 1 + |x| |x| 0 (1 + sin(θ 2 ))dθ satisfies (V) 1 , but neither (V) 2 , (A) 0 , (A) 1 nor (A) 2 . The operator H ψ 0 +V is essentially selfadjoint on C ∞ 0 (R d ), and so its unique selfadjoint extension is also denoted by the same H ψ 0 + V . The semigroup e −t(H ψ 0 +V ) on L 2 (R d ) is extended to a strongly continuous semigroup on L p (R d ) (1 ≤ p < ∞) and C ∞ (R d ), to be denoted by the same e −t(H ψ 0 +V ) . Here C ∞ (R d ) is the Banach space of the continuous functions on R d vanishing at infinity. To be complete, these and further facts are proved in Appendix.
In fact, the first estimate in (iii) holds independent of (L).
A consequence of Theorem 1 is the following Trotter product formula in the L p -operator norm with error bounds.
In fact, the asymptotic estimates (iii) and (iv) hold independent of (L).
Notice here that though the estimates with small t, in Theorem 1, for e −tV e −tH ψ 0 and e −tH ψ 0 /2 e −tV e −tH ψ 0 /2 are of worse order than that for e −tV /2 e −tH ψ 0 e −tV /2 , one has, in Theorem 2, the same error bounds with large n for these three products.
Finally we give a comment on what kind of operators are to be covered by our H ψ 0 +V . To this end we briefly illustrate how our result reads on the Trotter product formula in the case H Therefore Theorem 2 says that for 1 ≤ p ≤ ∞ and uniformly on each finite t-interval in [0, ∞), An important remark is the following. In the above example, the case α = 1 is missing. This is equivalent to the nonrelativistic case H 0 + V = −∆/2 + V (x), treated in 2]). However we may think that this case is also implicitly contained in our results, Theorems 1 and 2, for α = 1/2. Indeed, by using H r we can obtain the case α = 1/2 so as to involve the parameter c (light velocity). Since, in the nonrelativistic limit c → ∞, the relativistic Schrödinger semigroup e −t(H r 0 (c)+V ) is strongly convergent to the nonrelativistic Schrödinger semigroup e −t(H 0 +V ) uniformly on each finite t-interval in [0, ∞) (e.g. [I2]), we can reproduce the nonrelativistic result in [Tak] (cf. Remark following Theorem 2.3).
In Section 2, we state our results in more general form: we generalize Theorems 1 and 2 to Theorems 2.1 and 2.2 / 2.3 by introducing the subordinator σ t , namely, a time-homogeneous Lévy process associated with the Lévy measure e −l/2 n(dl). Moreover we state Theorem 2.4 on asymptotics of the moments of the process σ t . Once we know these asymptotics, we can obtain Theorems 1 and 2 from Theorems 2.1 and 2.2 / 2.3. These four theorems are proved in Sections 3 -6.
In Appendix, we give a full study of the semigroups e −t(H ψ 0 +V ) , t ≥ 0, on L p (R d ), 1 ≤ p < ∞ and C ∞ (R d ) defined through the Feynman-Kac formula. We show they constitute a strongly continuous contraction semigroup there. It is also shown that its infinitesimal generator G ψ,V p has C ∞ 0 (R d ) as a core, by establishing Kato's inequality for the operator H ψ 0 . Some of these results seem to be new.
The authors would like to thank the referee for his / her careful reading of the manuscript and for a number of comments.

General results
In this section we shall prove the theorems in a little more general setting based on probability theory. To describe it we introduce some notations and notions. For a continuous function Suppose we are given the independent random objects N (·) and B(·) on some probability space (Ω, F, P): Then (σ t ) t≥0 is a time-homogeneous Lévy process with increasing paths such that (e.g. Note 1.7.1 in [It-MK]). Note that σ t has moments of all order (cf. (6.1)), which is to be seen at the beginning of Section 6. We use a subordination of B(·) by a subordinator σ · , i.e., a process (B(σ t )) t≥0 on R d . This is a Lévy process such that which corresponds to the semigroup {e −tH ψ 0 } t≥0 with generator H ψ 0 in (1.1). We prove the following generalization of Theorems 1 and 2.
(ii) Under (V) 2 for n ≥ 1, Remark. As noted at the end of Section 1, the nonrelativistic case for H 0 + V = −∆/2 + V , being equivalent to the case α = 1 which Theorems 1 and 2 fail to cover, can be thought to be implicitly contained in the relativistic case, of the above three theorems, for the relativistic Schrödinger operator H r Lévy measure e −l/2 n(dl; c) = (2π) −1/2 ce −c 2 l/2 l −3/2 dl. In this case, Theorem 2.1 and Theorems 2.2 / 2.3 hold with the corresponding c-dependent subordinator σ t (c), just as they stand, namely, only with E [σ a s ] replaced by E [σ s (c) a ] for each respective s > 0 and a > 0. Then the nonrelativistic case in question is obtained as the nonrelativistic limit c → ∞ of this c-dependent relativistic case, turning out to be just Theorems 2.1 and 2.2 / 2.3 with E [σ a s ] replaced by s a . This is because one can show that, as c → ∞, the relativistic Schrödinger semigroup e −t(H r 0 (c)+V ) on the LHS converges strongly to the nonrelativistic Schrödinger semigroup e −t(H 0 +V ) uniformly on each finite t-interval in [0, ∞) (cf. [I2]), and E [σ t (c) a ] on the RHS tends to t a . Then taking the most dominant contribution on the RHS for small t or large n reproduces the same nonrelativistic result as in [Tak].
Theorems 1 and 2 follow immediately from Theorems 2.1 and 2.2 / 2.3, if one knows the asymptotics for t ↓ 0 of the moments of σ t to investigate which of the terms on the RHS makes a dominant contribution for small t or large n. These asymptotics are given by the following theorem.
In fact, for a ≥ 1 this always holds independent of (L).
(ii) If α = a and a < 1, then The proofs of Theorems 2.1, 2.2, 2.3 and 2.4 are given in Sections 3, 4, 5 and 6, respectively. To show Theorem 2.1, in fact, we prove estimates of the integral kernels of Q K (t), Q G (t) and Q R (t) by a finite positive linear combination of is the heat kernel (see (A.2)). Such estimates of the integral kernels of the three operators of difference in Theorems 2.2 / 2.3 also can be obtained (cf. [Tak]), but are omitted.

Proof of Theorem 2.1
It is easily seen (see (A.6 and generally where E σ and E B are the expectations with respect to σ · and B · , respectively, In the following we shall prove Theorem 2.1 only in Cases (A) 2 and (A) 0 . The proof of Case (A) 1 is omitted; it is similar to that of (A) 2 .

Case (A) 2
In this subsection, we suppose condition (A) 2 on V (x).
Proof. In view of (3.14) and (3.20 Here (and hereafter) the following inequality has been (will be) used: Collecting all the above into (3.23) yields the estimate in Claim 3.1 and the proof is complete.
( 3.29) This estimate together with (3.26) and (3.27) gives us that By the Schwarz inequality, it follows that Take expectation E B above, and integrate in θ. Then whence follows immediately the estimate in Claim 3.2.
We are now in a position to prove Theorem 2.1(iii). To do so, we need the following lemma.
Lemma 3.1. Let 1 ≤ p ≤ ∞. Then, for a, b ≥ 0 with C(a, d) = R d |y| a p(1, y)dy, Proof. For p = ∞, the described estimate is obvious. So let 1 ≤ p < ∞. First we note the Minkowski inequality for integrals: If h(x, y) is a measurable function on a σ-finite product Note also that for c ≥ 0 By these inequalities, the estimate is obtained as follows: Proof of Theorem 2.1(iii). By Claims 3.1, 3.2 with (3.7) By Claim 3.3 with (3.8), (3.9) Combining these with Lemma 3.1 we have the assertion of Theorem 2.1(iii).

Case (A) 0
In this subsection, we suppose condition (A) 0 on V (x). In this case Here taking expectation E B , we have by (3.28) or (3.36), t ) and hence, by (3.7), (3.8) and (3.9) From this and Lemma 3.1 the assertion of Theorem 2.1(i) follows immediately.

Proof of Theorem 2.2 for K(t)
Since K(t) and e −sH are contractions, we have Combined with the estimates for Q K (t) in Theorem 2.1, the desired bound for K(t/n) n − e −tH in Case (A) 0 , (A) 1 or (A) 2 is obtained immediately.

Proof of Theorem 2.2 for G(t) and R(t) in Case (A) 0
In the same way as above from which together with Theorem 2.1(i), the desired bounds follow immediately.

Proof of Theorem 2.2 for G(t) and R(t) in Case (A) 1 or (A) 2
In this subsection we suppose that V (x) satisfies (A) 1 or (A) 2 .
We first observe that for t ≥ 0 and n ∈ N As for the first term on the RHS of (4.1) and (4.2), we see by Theorem 2.2 which was proved in Section 4.1 in Case (A) 2 .
As for the third term on the RHS of (4.1) and the fourth term of (4.2), we see by Theorem 2.1 Therefore we need to estimate the middle terms of (4.1) and (4.2).

Claim 4.1. Let s ≥ 0 and t > 0.
Then Proof. First we estimate the L p -operator norm of [e −sV , e −tH ]. We have (by (A.13)) that for Hence we have To estimate the integrand in (4.3), note by Taylor's theorem that In Case (A) 1 , it follows that where the last inequality is due to Jensen's inequality. In Case (A) 2 (4.5) By (3.25), (4.4) and (4.5) imply the desired estimate: We take expectation E B in the above. This time we use the following moment estimate: For
Next we estimate the L p -operator norm of [e −sH 0 , e −tH ].
First we suppose that V : R d → [0, ∞) is in C ∞ and all its derivatives have polynomial growth. Then it is easily verified that (cf. Claim A.2 and its Remark) Here G ψ,V p (1 ≤ p < ∞) is the infinitesimal generator of {e −t(H 0 +V ) } on L p (R d ) and G ψ,V ∞ the one on C ∞ (R d ). By these facts the following formula holds in L p (R d ) (1 ≤ p < ∞) and C ∞ (R d ): Hence, taking L p -norm in the above yields that for each f ∈ S(R d ) (4.8) Now let V satisfy (A) 1 or (A) 2 . In this case V is not necessarily smooth. So, take a nonnegative , and satisfies condition (A) 1 or (A) 2 with the same const's as V does. Further, by (A) 1 (i) or (A) 2 (ii) all the derivatives of V ε have polynomial growth. Hence, by (4.7) and Lemma 3.1 it holds that for g ∈ S(R d ) Since (4.8) holds with V = V ε , by combining this with the above we have Finally let ε ↓ 0. Since V ε → V compact uniformly, we see by the Feynman-Kac formula Hence the desired bound for [e −sH 0 , e −t(H 0 +V ) ] follows immediately by the Fatou inequality.
We return to estimate G(t/n) n − e −tH and R(t/n) n − e −tH . By Claim 4.1 Therefore, collecting all the estimates above yields the desired bounds for G(t/n) n − e −tH and R(t/n) n − e −tH .

Proof of Theorem 2.3
As in the previous section, we are setting H 0 = H ψ 0 and H = H 0 + V .

Case (V) 1
In this subsection we suppose condition (V) 1 on V (x).

Let us adopt an idea in [D-I-Tam]. Take again a nonnegative
Then V ε is a smooth function and it satisfies the following: The proof is not difficult, so is omitted (cf. [Tak]).
As a consequence of Lemma 5.1, it is easily seen that V ε satisfies condition (A) 2 , i.e.
In what follows we write c, C, c 1 , c 2 , C 1 and C 2 simply for c , C , c 1 , c 2 , C 1 and C 2 .

Proof of Theorem 2.4
For a > 0, the proof will be given, divided into the three cases a = 1, a > 1 and 0 < a < 1.
First we note that for every a > 0 In fact, it is enough to show when a = ν is a positive integer. To do so, let ϕ t be the characteristic function of σ t , i.e., ϕ t (ξ) = E [e √ −1 ξσt ]. We have ϕ t (ξ) = e −tf (ξ) , where Since smoothness of ϕ t (ξ) near ξ = 0 implies existence of moments of σ t (cf. Exercise 2.6(viii) in [It]), we have only to show that ϕ t or f is in C ∞ near ξ = 0. But this is obvious, because, by a property of the Lévy measure n, the integral (0,∞) l ν e −l/2 n(dl) is convergent, so that by the Lebesgue convergence theorem le −l/2 n(dl)drdθ. Since σ t is increasing in t with σ 0+ = σ 0 = 0 and a − 1 > 0, we have (σ tr + θl) a−1 ↓ θ a−1 l a−1 as t ↓ 0. It follows by the Lebesgue convergence theorem that

The case 0 < a < 1
By the same reason as above (but in this case, a − 1 < 0), we have (σ tr + θl) a−1 ↑ θ a−1 l a−1 as t ↓ 0, and hence, by the monotone convergence theorem This time the integral on the RHS is not always convergent. To find the exact asymptotics we suppose assumption (L).
Proof. To rewrite (6.2), we see first with (2.2) and then we have The λ-integral in the last line is further computed by the change of variable λ = φ −1 (x) as follows: Here L(·, G) denotes the Laplace transform of a right-continuous increasing function G : The last fourth and third equalities are respectively be- Hence (6.2) is rewritten as follows: (6.11) 1 • The case 0 < a < α. Then 0 < α ≤ 1. By (6.10), (φ −1 (·)) −a is regularly varying with exponent −a/α ∈ (−1, 0). By Theorem 1 of §VIII.9 in [Fe], Hence, by combining this with (6.10), By applying the Abelian theorem (cf. Theorem 2 of §XIII.5 in [Fe]), this implies that Now if, for simplicity, we set then, by (6.11) and also, Therefore, applying Theorem 1 of §VIII.9 in [Fe] again, we have which is just the assertion (i).
3 • The case α < a < 1. Then 0 ≤ α < 1. By (6.6), it is enough to show that First this identity is seen from the following computation: l a e −l/2 n(dl).
Next this integral is convergent.
for any R > 0. On the other hand, since φ(λ) ∼ λ α L 1 (λ) as λ ↑ ∞, and L 1 (·) is slowly varying at infinity, there exists an R ε > 0 for 0 < ε < a − α (cf. Lemma 2 of §VIII.8 in [Fe]) such that φ(λ) ≤ 2λ α L 1 (λ) and L 1 (λ) < λ ε for any λ ≥ R ε . Hence Appendix: Semigroups e −t(H ψ 0 +V ) and their generators in L p (R R R d ) and C ∞ (R R R d ) In this appendix we suppose only that V : and define an R d -valued right-continuous process (X t ) t≥0 by where the second term on the RHS is a stochastic integral w.r.t. M . This is a d-dimensional time-homogeneous Lévy process starting at the origin such that which is easily seen by Itô's formula (cf. [Ik-Wa]), so that We now define a system of operators P ψ,V t , t ≥ 0, by the Feynman-Kac formula: From this definition the following is easily seen: (i) If f is a nonnegative Borel measurable function, so is P ψ,V t f , and it satisfies By (i) and (ii), {P ψ,V t } t≥0 is a strongly continuous contraction semigroup on C ∞ (R d ). By the Riesz-Banach theorem there exists a finite measure P ψ,V (t, x, dy) on R d such that Indeed, by noting (A.5), P ψ,V (t, x, dy) is absolutely continuous w.r.t. the Lebesgue measure dy on R d and expressed as where B τ,y 0,x (θ) is defined in (3.13). By (i) and (ii) again P ψ,V t is uniquely extended to a bounded operator on L p (R d ), which is denoted by the same P ψ,V t , and thus {P ψ,V t } t≥0 is a strongly continuous contraction semigroup on L p (R d ). Clearly, for f ∈ L p (R d ) and, when p = 2, P ψ,V t is symmetric.
i.e., the integral in (A.14) is convergent for a.e. x, and H ψ , the integral in (A.14) is convergent for every x and H ψ 0 f ∈ C(R d ).
Since H ψ f ∈ C ∞ (R d ) by Claim A.1, it is enough to check pointwise convergence (cf. Lemma 31.7 in [Sa]). To do so we apply Itô's formula for (A.4) to obtain Note that the third term on the RHS is a martingale, so that the expectation is zero. Taking expectation and changing the variable s = tσ we have (1 − θ) y, ∇ 2 f (x + X tσ + θy)y dθ J(dy), (A.17) where the second equality is due to Taylor's theorem with the aid of symmetry of J(dy). On letting t ↓ 0 in the first equality of (A.17) we have (A.16) pointwise.
Next we prove for 1 ≤ p < ∞ that Since H ψ f ∈ L p (R d ) by Claim A.1, it is enough to check weak convergence (cf. Lemma 32.3 in [Sa]).
Proof of Theorem A.1. First consider the L p -case, 1 ≤ p < ∞. It suffices to show that Im (H ψ 0 + V + 1) = (H ψ 0 + V + 1)(C ∞ 0 (R d )) is dense in L p (R d ). By the Hahn-Banach theorem, this is further reduced to show the following: Let q be the conjugate exponent of p. If v ∈ L q (R d ) satisfies that v, (H ψ 0 + V + 1)ϕ = 0 for any ϕ ∈ C ∞ 0 (R d ), then v = 0 in L q (R d ).
Now note that the resolvent (1 − G ψ,0 p ) −1 is expressed as Then it is not difficult to check that if f ∈ S(R d ), then (1 − G ψ,0 p ) −1 f ∈ S(R d ) and further, if f is nonnegative, so is (1 − G ψ,0 p ) −1 f . Also, by Remark following Claim A.2 (with V (x) ≡ 0) whence it immediately follows that v = 0 and the proof in the L p -case is complete.
In this paper we have denoted the semigroups P ψ,0 t and P ψ,V t by e −tH ψ 0 and e −t(H ψ 0 +V ) , respectively, taking Theorem A.1 into account. With the general theory ( [Trot], [Ch]) we have taken for granted that the Trotter product formula holds in the strong topology of L p (R d ) or C ∞ (R d ):