The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Unformly Elliptic Operators with Non negative Potentials Kazuhiro Kurata Vienna, Preprint ESI June 6, 998 Supported by Federal Ministry of Science and Transport, Austria Available via anonymous ftp or gopher from FTP.ESI.A.AT or via WWW, URL:

2 An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Unformly Elliptic Operators with Non-negative Potentials Kazuhiro Kurata October 2, 200 Keywords: uniformly elliptic operator, magnetic Schrödinger operator, non-negative potential, heat kernel, weighted smoothing estimate, subsolution estimate Introduction and Main Results We consider the uniformly elliptic operator L E = Ax + V x with certain non-negative potential V and the Schrödinger operator L M = i ax 2 + V x with a magnetic field ax = a x,, a n x, n 2. We use the notation L J for J = E or J = M. The purpose of this paper is to give an estimate of the fundamental solution or heat kernel Γ J x, t : y, s to namely Γ J x, t; y, s satisfies t + L J ux, t = 0, x, t R n 0,, t + L J Γ J x, t; y, s = 0, x R n, t > s, 2 lim Γ J x, t; y, s = δx y. 3 t s Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa -, Hachioji-shi, Tokyo, Japan, kurata@math.metro-u.ac.jp,

3 For the elliptic operator L E, we assume the following conditions for Ax = a ij x. Assumption A.: a ij x is a real-valued measurable function and satisfies a ij x = a ji x for every i, j =,, n and x R n. Assumption A.2: There exists a constant λ > 0 such that λ ξ 2 n i,j= a ij xξ i ξ j λ ξ 2, ξ = ξ,, ξ n R n. 4 To state our assumptions on V and a, we prepare some notations. We say U RH if U L locr n and satisfies sup y Bx,r Uy Bx, r Bx,r and say U RH q if U L q loc Rn and satisfies Bx, r Bx,r Uy dy, 5 /q Uy q dy Uy dy, 6 Bx, r Bx,r for some constant and for every x R n and r > 0, respectively. We can define the function mx, U for U RH q with q > n/2 as follows: mx, U = sup{r > 0; r 2 Uy dy }. 7 Bx, r Bx,r We note that if there exist positive constants K and K 2 such that K U x U 2 x K 2 U x, then it is easy to see that there exist positive constants K and K 2 such that K mx, U mx, U 2 K 2mx, U. When n 3, since it is known U RH n/2 actually belongs to RH n/2+ɛ for some ɛ > 0, mx, U can be defined for U RH n/2 [Sh]. For other properties of the class RH q, see, e.g., [KS]. We denote by Bx = B jk x the magnetic field defined by B jk x = j a k x k a j x. We use the notation m J x: m E x = mx, V, m M x = mx, B + V 2

4 for the operator L J, J = E or M, respectively. We assume the following conditions for V x and ax = a x,, a n x. AssumptionV, a, B: For each j =,, n, a j x is a real-valued R n -function, V is non-negative. i For n 3, we assume V x and ax satisfy V + B RH n/2, Bx mx, V + B 3. ii For n = 2, we assume V x and ax satisfy for some q >. V + B RH q, Bx mx, V + B 3 Remark For n = 2, we may assume the condition ii instead of ii by employing Lemma b. ii V L locr 2, Bx 0 and that m J x satisfies m J x + x y m J x m k 0/k 0 + Jy 2 + x y m J x k 0 m J x 8 for some positive constants, 2, k 0 and for every x, y R 2, where m E x = V x and m M x = V x + Bx. We remark that it is known that m J x satisifies 8 under the assumption V, a, B for n 3[Sh] and even for n = 2 in the same way. We also note that if B + V RH, then it is easy to see that Bx + V x mx, B + V 2 holds. For example, the condition B + V RH is satisfied for any a j x = Q j x, V x = P x α, where P x and Q j x, j =,, n, are polymonials and α is a positive constant. In this case, under the assumption V, a, B i or ii, we see that there exists a positive constant m 0 such that m J x m 0, although in general we cannot say B + V is strictly positive for imhomogeneous polynomials. To state our main result, we introduce the notation: Γ 0 x, t; y, s = for some positive constant 0. t s exp x y 2 n/2 0 t s 3

5 Theorem a Suppose Ax and V x satisfy the assumptions A., A.2 and V, 0, 0. Then, there exist positive constants α 0 and j j = 0,, 2 such that 0 Γ E x, t; y, s exp 2 +m E xt s /2 α 0/2 Γ 0 x, t; y, s 9 for x, y R n and t > s > 0. b Suppose V x and ax satisfy the assumption V, a, B. Then, there exist positive constants α 0 and j j = 0,, 2 such that Γ M x, t; y, s exp 2 + m M xt s /2 α 0/2 Γ 0 x, t; y, s 0 for x, y R n and t > s > 0. The number α 0 is actually defined by α 0 = 2/k 0 +, where k 0 is the constant in 8. The exponent α 0 /2 would not be sharp. If we restrict for the case B 0 Bx B 0 > 0, the following sharp estimate is known [Ma], [Er,2] for n 3 and [LT] for n = 2: Γ M x, t; y, s D exp D 2 B 0 tγ D0 x, t; y, s. More detail informations on the constants D j j = 0,, 2 can be seen in those papers. By using the parabolic distance: d P x, t, y, s = max x y, t s /2, we have the following decay estimate. orollary a Under the same assumptions as in Theorem, there exist positive constants j j =, 2 and 0 such that Γ J x, t; y, s exp 2 +m J xd P x, t, y, s 2α 0/α 0 +4 Γ 0 x, t; y, s for J = E and M, for every x, y R n and t > s > 0. b Under the same assumptions as in Theorem, for eack k > 0 there exist positive constants k and 0 such that Γ J x, t; y, s for J = E and M. k + m J xd P x, t, y, s k Γ 0 x, t; y, s 4

6 Remark 2 Actually we can show the estimate in Theorem for the operators L E = Ax, t + V x, t with time-dependent coefficients, if we assume the uniform ellipticity 4 of Ax, t and the existence of constants j, j =, 2, such that Ux V x, t 2 Ux and U satisfies the condition U, 0, 0. For the magnetic Schrödinger operator L M = i ax, t 2 +V x, t, the estimate in Theorem still holds, if there exists positive constants j, j =,, 5, such that Ux V x, t 2 Ux, 3 B x Bx, t 4 B x, and Bx, t 5 mx, B + U 3, where ax, t is and B jk x, t = j ax, t k a j x, t and Ux and B x satisfy the Assumption U, a, B except B x m J x 3 = mx, B + U 3, and if the upper bound : holds for some constants and 0. Γ M x, t; y, s Γ 0 x, t; y, s Remark 3 In particular, orollary b yields Γ J x, t; y, s k + m J x x y k + m J x t s Γ k 0 x, t; y, s k + m J x x y Γ k 0 x, t; y, s for J = E or M. Let n 3. Then this implies Γ J x, y + s Γ J x, t; y, s dt k + m J x x y k x y n 2 where Γ J x, y is the fundamental solution to L J u = 0. This estimate for the elliptic operator was proved by Shen [Sh,2]. Thus, orollary b is a generalization of his estimate. Remark 4 Recently we are informed by Z.Shen that he obtained the following shape estimate[sh3] for the elliptic operators: under the assumption V RH n/2 for n 3 and V RH q with q > for n = 2, exp 2 dx, y x y 2 n Γ E x, y 3 exp 4 dx, y x y 2 n holds for some positive constants j j =, 2, 3, 4, where dx, y is defined by dx, y = inf mγt, V dγt dt. γ 0 dt 5

7 Here the infimum is taken over all curves γ such that γ0 = x and γ = y. Moreover, he gave the following estimate: + mx x y α 0/2 dx, y 2 + mx x y β 0 for some positive constants j j =, 2 and β 0. In particular, it follows Γ E x, y 5 exp 6 + m E x x y α 0/2 x y 2 n for some positive constants 5 and 6. We remark that this decay estimate also can be shown for the fundamental solution Γ M x, y to L M in a similar way. On the other hand, from orollary a it follows a somewhat weaker decay estimate: Γ J x, y exp + m J x x y 2α 0/α 0 +4 x y 2 n for J = E or M. We do not know whether his sharp estimate can be generalized to heat kernel estimates or not. We denote by e tl J the semigroup generated by L J. Here we also denote by L J the self-adjoint operator determined from the form associated with L J see, e.g., [Si], [LS]. We obtain the following weighted smoothing estimate by using orollary b. Theorem 2 Assume the same assumptions as in Theorem. Let J = E or M. Suppose < p q + and /p /q < and put γ = n/p /q [0, n. Then for each l [0, n γ/2] there exists a constant l such that m J x 2l e tl J f L q R n l t l+γ/2 f L p R n, t > 0. 2 orollary 2 Suppose the additional condition B + V RH. Then we have the following estimates: B + V l e tl J f L p R n t l f L p R n, t > 0 3 holds for < p < + and l [0, n/2], and B + V l e tl J f L R n t l+n/2p f L p R n, t > 0 4 holds for p < + and l [0, n/2p ]. Here /p = /p and l is a constant depending on l and p. 6

8 orollary 2 is an easy consequence of Theorem 2 by B +V x m J x 2. Note that 4 for the case l = 0 is a classical result. Theorem yields a weighted smoothing estimate with an exponential decay in time. Theorem 3 Assume the same assumptions as in Theorem and the additional assumption m J x m 0 > 0. a Let p < + and l [0, n/2p ]. Then we have m J x 2l e tl J f L R n exp +m 0 t /2 α 0 2 t l+n/2p f L p R n, t > 0. b Let p 2 and l [0, n/2p ]. Then we have m J x 2l e tl J f L R n exp + m 2 0t t f l+n/2p L p R n, t > 0 for some positive constant and for every p < + and l 0, n/2p ]. Especially, for the case B 0 Bx B 0 > 0, Theorem 3 b yileds an exponential decay estimate in time: e tl M f L R n t n/2 exp 2B 0 t f L R n, t > 0 5 for some positive constant and 2, which is known see, e.g., [Ma], [Er,2], [Ue], [LT]. Indeed, in this case m M x B 0 holds. Note that Theorem 3 a gives weaker decay rate e B 0 t, since k 0 = 0 and α 0 = 2. We also emphasize that Theorem 3 can be applied to any polynomial like magnetic field Bx which may be zero somewhere. Definition We say ux, t is a complex-valued weak solution to t + L M u = 0 in Q r x 0, t 0, if u L t 0 r 2, t 0 ; L 2 Bx 0, r; L 2 t 0 r 2, t 0 ; H Bx 0, r; and satisfies + + t ux, tφx, t dx ux, s s φx, s dxds Bx 0,r t 0 r 2 Bx 0,r t n D t 0 r 2 j a ux, sdj a φx, s dxds Bx 0,r j= t V xux, sφx, s dxds = 0 6 t 0 r 2 Bx 0,r 7

9 for everu φ {φ L 2 t 0 r 2, t 0 ; H Bx 0, r; ; s φ L 2 t 0 r 2, t 0 ; L 2 Bx 0, r;, φx, t 0 r 2 = 0}, where φ is the complex conjugate of φ. Here, we used the notation D a j = i xj a j x and Q r x 0, t 0 = {x, t R n 0, + ; x x 0 < r, t 0 r 2 < t < t 0 }. A real-valued weak solution u to t +L E u = 0 in Q r x 0, t 0 can be defined in a similar way. Our proof of Theorem is based on the following subsolution estimate. Theorem 4 Let ux, t be a weak solution to t u + L J u = 0 in Q 2r x 0, t 0. Then there exits positive constants j, j =, 2, such that sup ux, t exp 2 +rm J x 0 α 0/2 /2. u dxdt 2 x,t Q r/2 x 0,t 0 r n+2 7 Throughout this paper, we use the following notation: D a j = i xj a j x, D = i a, Bx 0, r = {y R n ; y x 0 < r}, A u, u = n j,k= a jk xj u xk u, Q r x 0, t 0 = {x, t R n 0, + ; x x 0 < r, t 0 r 2 < t < t 0 }. 2 Proof of Theorem 4 We use the following inequalities. Lemma a [Sh2] Suppose n 2 and V x and ax satisfy the condition V, a, B. Then there exists a constant 0 such that mx, B + V 2 u 2 dx 0 for u 0 R n ;. i axu 2 + V x u 2 dx 8

10 b [AHS] Suppose n = 2, V 0, V L locr 2, a R 2, and Bx 0. Then the inequality Bx + V x u 2 dx i axu 2 + V x u 2 dx holds for u 0 R n ;. We also prepare the following accioppoli-type inequality. Lemma 2 Let 0 < σ <. Let u be a weak solution to s + L J u = 0 in Q 2r x 0, t 0 for J = E or J = M. Then there exists a constant such that ux, t 2 dx + i au 2 + V u 2 dxds sup t 0 σr 2 tt 0 Bx 0,σr Q σrx 0,t 0 σ 2 r 2 u 2 dxdt. Proof: Although the proof is standard, we give it here for the sake of completeness. We show the estimate for a weak solution u to t +L E u = 0 in Q 2r x 0, t 0. Since we can show the esimate for a weak solution to t +L M u = 0 in the similar way, we just mention some modifications we need at the end of this proof. Take functions χx 0 Bx 0, r and ηt R satisfying 0 χx, χx on Bx 0, σr and χx / σr, and 0 ηt, ηt on t t 0 σr 2, ηt 0 on t t 0 r 2, t ηt /r 2 σ 2. For the sake of simplicity, we also assume t u L 2 Q 2r x 0, t 0. Actually, we can remove this additional assumption by using the argument as in [AS]. Fix t [t 0 σr 2, t 0 ]. Multiplying η 2 tχ 2 xux, t to the equation and integrating over Bx 0, r [t 0 r 2, t], we have + + = 2 Bx 0,r t t 0 r 2 Bx 0,r t t 0 r 2 Bx 0,r t t 0 r 2 Bx 0,r t t 0 r 2 ux, t 2 χx 2 dx Bx 0,r Ax ux, s, ux, s ηs 2 χx 2 dxds V xux, s 2 ηs 2 χx 2 dxds ux, s 2 χx 2 ηs s ηs dxds Ax ux, s, χ 2 x ηs 2 ux, s dxds. 8 9

11 Because of the ellipticity of Ax and the positivity of V, we obtain by 8 + sup t 0 σr 2 tt 0 Bx 0,r { σ r 2 u 2 s η dxds By using 8 again, we have ux, t 2 χx 2 dx 9 u u η 2 χ χ dxds u 2 dxds + } χ 2 η 2 u 2 dxds. λ u 2 χ 2 η 2 dxds + V u 2 χ 2 η 2 dxds A u, u k uχ 2 η 2 dxds + V u 2 χ 2 η 2 dxds u 2 dxds + u χ χη 2 u dxds σr 2 u 2 dxds + λ u 2 χ 2 η 2 dxds. 20 σr 2 2 It follows λ 2 σ 2 r 2 u 2 χ 2 η 2 dxds + V u 2 χ 2 η 2 dxds u 2 dxds. 2 9 and 2 yield the desired result. For L M, we can prove in a similar way by noting the following identities: D a j uχ = D a j uχ + ui χ, Dj a uv dx = udj a v dx. Proof of Theorem 3: Let k N and define p j j =, 2,, k + by p j = 2/3 + j /k 2/3. Let χ j x 0 Bx 0, p j r and η j t R be the functions satisfying 0 χ j, χ j x on Bx 0, p j r, 0

12 χ j x k/r, and 0 η j, η j t on t t 0 p j r 2, η j t 0 on t t 0 p j r 2, η j t k/r 2. By Lemma 2 see also 2, we have i au 2 χ 2 j+ηj+ 2 + V u 2 χ 2 j+ηj+ 2 dxds Q pj+ rx 0,t 0 k2 u 2 dxds. r 2 Q pj+ rx 0,t 0 We write just χ = χ j+ and η = η j+, for simplicity. Since i auηχ 2 2 i au 2 χ 2 η 2 + 2u 2 χ 2 η 2, it follows that i aηχu 2 χ 2 η 2 + V u 2 χ 2 η 2 dxds Q pj+ rx 0,t 0 for j =,, k. By using Lemma, we obtain t0 t 0 p j+ r 2 Bx 0,p j+ r k2 r 2 Q pj+ rx 0,t 0 u 2 dxds m J x 2 ηχu 2 dx dt k2 u 2 dxds. r 2 Q pj+ rx 0,t 0 By using m J x + p j+ rm J x 0 k 0/+k 0 m J x 0 on x x 0 < p j+ r and noting 2/3 p j+ see 8 and the remark after that, we have Q pj rx 0,t 0 u 2 dxdt t0 k 2 r 2 m J x rm Jx 0 2k 0/k 0 + k 2 + rm J x 0 2/k 0+ t 0 p j+ r 2 Bx 0,p j+ r Q pj+ rx 0,t 0 Q pj+ rx 0,t 0 ηχu 2 dxdt u 2 dxdt. u 2 dxdt 22 for each j =, 2,, k. Here we used a trival inequality Q pj rx 0,t 0 dxdt Q pj+ rx 0,t 0 dxdt for the case rm Jx 0. By this proceedure, we can obtain the following: there exists a constant such that for every k N Q 2r/3 x 0,t 0 u 2 dxdt k k 2 k + rm J x 0 kα 0 u 2 dxdt, 23

13 where α 0 = 2/k 0 +. Since V x 0, the well-known subsolution estimate see, e.g., [AS] yields /2 sup u u dxdt 2 24 Q r/2 x 0,t 0 r n+2 Q 2r/3 x 0,t 0 for some constant. For the magnetic Schrödinger operator case, we have used Kato s inequality. ombining 23 and 24, we arrive at sup u Q r/2 x 0,t 0 k/2 k k + rm J x 0 kα 0/2 /2 u dxdt 2 25 r n+2 for every k N. Note that, by Stirling s formula k k e k k!/ 2πk as k, there exists a constant 0 such that k k 0 e k k! for k. Multiplying ɛ k /k! and taking the summation, we obtain ɛ + rm J x 0 α0/2 k sup u Q r/2 x 0,t 0 k= k! 0 ɛe /2. k u dxdt 2 k= r n+2 Take ɛ > 0 so that ɛe <. Then we have /2. sup u exp ɛ + rm J x 0 α0/2 u dxdt 2 Q r/2 x 0,t 0 r n+2 This complete the proof. 3 Proof of Theorem To show Theorem we prove the following proposition. Proposition Under the assumptions as in Theorem, there exist positive constants and 2 such that Γ J x, t; y, s exp 2 + m J x t s /2 α0/2 26 t s n/2 for x, y R n and t > s > 0. 2

14 Proof: Assume t s 2 y x 2. Take r 2 = t s /8. Then uz, u = Γ J z, u; y, s satisfies t + L J uz, u = 0 in Q 2r x, t. Hence, by applying Theorem 4 to uz, u, we obtain Γ J x, t; y, s sup Q r/2 x,t u exp + m J x t s /2 α0/2 r n+2 Q rx,t Γz, u; y, s 2 dzdu /2. By using the maximum principle for L E and the diamagnetic inequality see, e.g., [AS], [LS], [AHS] for L M, we have Γ J z, u; y, s u s exp z y 2 27 n/2 u s for some constant = n, λ. Since t s u s 7r 2 7/8t s on z, u Q r x, t, it is easy to see /2 Γ r n+2 J z, u; y, s 2 dzdu Q rx,t t s. n/2 This yields the desired estimate. Proof of Theorem : The positivity of Γ E x, t; y, s is a consequence of V 0 and the maximum principle. Hence Proposition and 27 imply Γ J x, t; y, s 2 exp + t s /2 m J x α0/2 t s exp n y x 2 t s for some constant. This concludes the desired estimate. Proof of orollary : Let ft = m J xt /2 α 0/2 + x y 2 /t for t > 0. The, an easy computation shows that inf ft m Jx x y 2α 0/α 0 +4 t>0 for some positive constant. Thus, we obtain y 2 Γ J x, t; y, s exp ft s exp x t s n/2 t exp m J xt s /2 α 0/2 Γ 0 x, t; y, s exp m J x x y 2α 0/α 0 +4 exp m J xt /2 α 0/2. This proves the part a since 2α 0 /α α 0 /2. The part b is an easy consequence of the part a. 3

15 4 Proof of Theorem 2, 3 To show Theorem 2, we prove the following inequality. Theorem 5 Let γ [0, n. Then there exists a constant such that m J x 2l e tl J fx t M γ f x 28 l+γ/2 holds for every 0 < l n γ/2. Here M γ f is the fractional maximal function defined by M γ fx = sup f dy, x B B γ/n B where the supremum is taken all balls B containing x. Theorem 2 is a consequence of Theorem 5 and the following lemma see, e.g., [St]. Lemma 3 Let 0 γ < n. There exists a constant such that M γ f q f p for < p q + and /q = /p γ/n. Proof of Theorem 5: Let r = /m J x. By orollary b we have m J x 2l e tl J fx m J x 2l fy + m J x x y k t exp n/2 + fy r 2l t n/2 {2 j r< x y 2 j r} + 2 j exp k j= x y 2 dy t 2j r 2 t dy. 29 By the assumption on l, we take α 0 such that 2α = n γ 2l. Put α = sup s>0 s α e s < + for α 0. Then the right hand side of 29 is dominated by + α t n/2 j= {2 j r< x y 2 j r} α t n/2 α α t n/2 α + j= + j= 2 j n γ + 2 j k 2 j 2α r 2l + 2 j k 2 j r n γ 2 j r 2 α fy dy t { x y 2 j r} fy dy. 2 j n γ + 2 j k 2 j 2α M γ f x. 30 4

16 Now, since n γ 2α = 2l > 0, by taking k > 2l we have and + j= 0 j= 2 j n γ + 2 j k 2 j 2α + j= 2 j n γ + 2 j k 2 j 2α 0 j= < +, 2jk 2l 2 j 2l < +. Thus, we obtain the desired result. Proof of Theorem 3: First, the estimate for the case l = 0 and p = is classical except the exponential factor in time. Under the assumption, by orollary a we have Γ J x, t; y, s Γ 0 x, t; y, s exp + m J x x y 2α 0/α 0 +4 exp + m 0 t /2 α 0/2 3 for some positive constants and 0. Then by using this estimate we can prove the part a of Theorem 3 in a similar way as in the proof of Theorem 2. To show the part b, we use the semigroup property and Theorem 2 and get m J x 2l e tl J f L R n t l+n/4 e 2/3tL J f L 2 R n for some constant. Note that under the assumption m J x m 0, Lemma yields inf σl J m 2 0 for some positive constant. Here σl J is the spectrum of the operator L J. So, we have Using this estimate, we obtain m J x 2l e tl J f L R n e /3tL J g L 2 R n e m2 0 t g L 2 R n. t t l+n/4 e m2 0 e /3tL J f L 2 R n t l+n/4 e m2 0 t t f n/2/p /2 L p R n. In the last inequality, we used p 2 and Theorem 2. REFERENES 5

17 [AHS] Avron J.E., Herbst I., Simon B., Schrödinger operators with magnetic fields, I. General interactions, Duke Math. J., 45978, [AS] Aronson D.G., Serrin J., Local behaviour of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 25967, [Er] Erdös L., Estimates on stochastic oscillatory integrals and on the heat kernel of the magnetic Schrödinger operator, Duke Math. J., 76994, [Er2] Erdös L., Dia- and paramagnetism for nonhomogeneous magnetic fields, J.Math.Phys., , [LS] Leinfelder H., Simader., Schrödinger operators with singular magnetic vector potentials, Math. Z., 7698, 9. [LT] Loss M., Thaller B., Optimal heat kernel estimates for Schrödinger operator with magnetic fields in two dimensions, omm. Math. Phys., 86997, [Ma] Malliavin P., Minoration de l etat fondamental de l équation de Schrödinger du magnëtisme et calcul des variations,.r. Acad. Sci. Ser. I. Math , [KS] Kurata K., Sugano S., Fundamental Solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptic operators with positive potentials, Preprint. [Sh] Shen Z., L p estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, 45995, [Sh2] Shen Z., Estimates in L p for magnetic Schrödinger operators, Indiana Univ. Math. J., 45996, [Sh3] Shen Z., by personal communication. [Si] Simon B., Maximal and minimal Schrödinger forms, J.Opt. Theo., 979, [St] Stein, E.M., Harmonic Analysis: Real variable methods, Orthogonality, and Oscillatry integrals, Prinston Univ. Press, 993. [Ue] Ueki N., Lower bounds on the spectra of Schrödinger operators with magnetic fields, J.Fun.Ana., 20994, AKNOWLEDGEMENT This paper was writen while the author was visiting the Erwin Scgrödinger International Institute for Mathematical Physics in Vienna. The author wishes to thank Professor T. Hoffmann-Ostenhof for his invitation and the 6

18 members of the Schrödinger institute for their hospitality. This work is partially supported by Tokyo Metropolitan University maintenance costs and by the Erwin Schrödinger institute for Mathematical Physics. In the early version of this paper, the author only obtained a polynomial decay estimate for heat kernels. The author wishes to thank Professor Zhongwei Shen for informing his sharp estimate for elliptic operators and his kindness to allow the author to use his argument here in order to improve and obtain exponetial decay for the heat kernel. He also pointed out that even in 2-dimensional case we can obtain the same results under the condition V, a, Bii. The author also wishes to thank Professor László Erdös for his valuable comments. Especially, the author learned from him the observation which improves the exponent in the estimate of Theorem and also simplifies the proof. The author also thank Professor B. Helffer for his valuable advice which leads to the part b of Theorem 3. 7