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Vyjayanthi Chari Department of Mathematics University of California Riverside, CA 92521-0135 chari@math.ucr.edu Robert Finn Department of Mathematics Stanford University Stanford, CA 94305-2125 finn@math.stanford.edu Kefeng Liu Department of Mathematics University of California Los Angeles, CA 90095-1555 liu@math.ucla.edu PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982) F. Wolf (1904–1989) EDITORS V. S. Varadarajan (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 pacific@math.ucla.edu Darren Long Department of Mathematics University of California Santa Barbara, CA 93106-3080 long@math.ucsb.edu Jiang-Hua Lu Department of Mathematics The University of Hong Kong Pokfulam Rd., Hong Kong jhlu@maths.hku.hk Sorin Popa Department of Mathematics University of California Los Angeles, CA 90095-1555 popa@math.ucla.edu PRODUCTION pacific@math.berkeley.edu Sorin Popa Department of Mathematics University of California Los Angeles, CA 90095-1555 popa@math.ucla.edu Jie Qing Department of Mathematics University of California Santa Cruz, CA 95064 qing@cats.ucsc.edu Jonathan Rogawski Department of Mathematics University of California Los Angeles, CA 90095-1555 jonr@math.ucla.edu Paulo Ney de Souza, Production Manager Silvio Levy, Senior Production Editor Nicholas Jackson, Production Editor ACADEMIA SINICA, TAIPEI CALIFORNIA INST. OF TECHNOLOGY CHINESE UNIV. OF HONG KONG INST. DE MATEMÁTICA PURA E APLICADA KEIO UNIVERSITY MATH. SCIENCES RESEARCH INSTITUTE NEW MEXICO STATE UNIV. OREGON STATE UNIV. PEKING UNIVERSITY STANFORD UNIVERSITY SUPPORTING INSTITUTIONS UNIVERSIDAD DE LOS ANDES UNIV. OF ARIZONA UNIV. OF BRITISH COLUMBIA UNIV. OF CALIFORNIA, BERKELEY UNIV. OF CALIFORNIA, DAVIS UNIV. OF CALIFORNIA, IRVINE UNIV. OF CALIFORNIA, LOS ANGELES UNIV. OF CALIFORNIA, RIVERSIDE UNIV. OF CALIFORNIA, SAN DIEGO UNIV. OF CALIF., SANTA BARBARA UNIV. OF CALIF., SANTA CRUZ UNIV. OF HAWAII UNIV. OF MONTANA UNIV. OF NEVADA, RENO UNIV. OF OREGON UNIV. OF SOUTHERN CALIFORNIA UNIV. OF UTAH UNIV. OF WASHINGTON WASHINGTON STATE UNIVERSITY These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. See inside back cover or www.pjmath.org for submission instructions. Regular subscription rate for 2006: $425.00 a year (10 issues). Special rate: $212.50 a year to individual members of supporting institutions. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. Prior back issues are obtainable from Periodicals Service Company, 11 Main Street, Germantown, NY 12526-5635. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and the Science Citation Index. The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 969 Evans Hall, Berkeley, CA 94720-3840 is published monthly except July and August. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS at the University of California, Berkeley 94720-3840 A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2006 by Pacific Journal of Mathematics
PACIFIC JOURNAL OF MATHEMATICS Vol. 198, No. 1, 2001 ON THE ASYMPTOTICS OF THE TRACE OF THE HEAT KERNEL FOR THE MAGNETIC SCHRÖDINGER OPERATOR Junichi Aramaki We consider the effect of a magnetic field for the asymptotic behavior of the trace of the heat kernel for the Schrödinger operator. We discuss the case where the operator has compact resolvents in spite of the fact that the electric potential is degenerate on some submanifold. According to the degree of the degenaracy, we obtain the classical and non-classical asymptotics. 1. Introduction. In this paper, we consider the Schrödinger operator on Rd with the magnetic vector potential A(z) and the electric scalar potential V (z): H(A, V ) = 1 2 (i∇ + A(z))2 (1.1) + V (z). We will assume that H(A, V ) and H(0, V ) are essentially self-adjoint in L 2 (R d ) starting from C ∞ 0 (Rd ) and we denote the self-adjoint extensions by H and H0, respectively. It is well known that if (1.2) lim V (z) = +∞, |z|→∞ H0 has compact resolvents (c.f. for example, Reed and Simon [8]). Odencrantz [7] studied the asymptotic behavior of Tr[exp(−tH) − exp(−tH0)] as t ↓ 0 in the case where V (z) ≈ |z| 2p (p > 0) and the curl of A(z) is a uniform magnetic field. Matsumoto [5] extended the result to the case with more general magnetic field. However, (1.2) is not a necessary condition in order that H0 has compact resolvents. In spite of the lack of (1.2), there are some cases where H0 has compact resolvents. The motivation of this paper originates in the works of Simon [11], Robert [9] and Aramaki [3]. In order to explain this, we put Zcl(t) = (2π) −d e −t(|ζ|2 /2+V (z)) dzdζ. It is well known that Tr[exp(−tH0)] ≤ Zcl(t) for t > 0. In [11], he considered the degenerate potential V (z) of the form V (x, y) = |x| 2p |y| 2q (p, q > 0) on 1
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50 ROBERT F. BROWN AND HELGA SCHIRM
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82 GILLES CARRON à l’infini s’
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84 GILLES CARRON isométriques au d
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86 GILLES CARRON Comme D est ellipt
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88 GILLES CARRON Réciproquement si
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90 GILLES CARRON où H est la proje
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92 GILLES CARRON 2.a. Exemples repo
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94 GILLES CARRON alors si σ ∈ C
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96 GILLES CARRON Ainsi s’il exist
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98 GILLES CARRON Proposition 2.7. N
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100 GILLES CARRON l’espace des 1
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102 GILLES CARRON compact. De même
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104 GILLES CARRON 4. Application du
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106 GILLES CARRON Dans le cas où l
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PACIFIC JOURNAL OF MATHEMATICS Vol.
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BRAIDED OPERATOR COMMUTATORS 111 Re
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BRAIDED OPERATOR COMMUTATORS 113 Re
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BRAIDED OPERATOR COMMUTATORS 115 3.
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Then, for 1 < k ≤ m + 1, we have
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BRAIDED OPERATOR COMMUTATORS 119 Mo
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BRAIDED OPERATOR COMMUTATORS 121 It
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E-mail address: prosk@imath.kiev.ua
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124 DANIEL J. MADDEN times. Further
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126 DANIEL J. MADDEN Further, s t
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128 DANIEL J. MADDEN Now the whole
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130 DANIEL J. MADDEN and β = 1 −
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132 DANIEL J. MADDEN We can simplif
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134 DANIEL J. MADDEN surd, but we n
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136 DANIEL J. MADDEN and d = d(u, v
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138 DANIEL J. MADDEN (6l + 7) k +
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140 DANIEL J. MADDEN Then the conti
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142 DANIEL J. MADDEN Thus n + 1 1 =
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144 DANIEL J. MADDEN If we take b =
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146 DANIEL J. MADDEN b, 2b(2bn + 1)
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TANGLES, 2-HANDLE ADDITIONS AND DEH
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m n n n n TANGLES, 2-HANDLE ADDITIO
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SOME CHARACTERIZATION, UNIQUENESS A
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and SOME CHARACTERIZATION, UNIQUENE
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SYMPLECTIC SUBMANIFOLDS FROM SURFAC
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208 PAULO TIRAO Theorem. The affine
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210 PAULO TIRAO It follows from the
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212 PAULO TIRAO K ρ1 (0,−1,1) =
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214 PAULO TIRAO Theorem 2.7. Let ρ
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216 PAULO TIRAO Notation: By A = [
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218 PAULO TIRAO Now is not difficul
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220 PAULO TIRAO Regard H n (Z2 ⊕
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222 PAULO TIRAO is an isomorphism.
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224 PAULO TIRAO Lemma 4.2. Let Λ1
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226 PAULO TIRAO (c)1 B1 B2 B3 B1 0
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228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
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230 PAULO TIRAO being those in whic
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232 PAULO TIRAO Hence, if ρ = m1χ
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(1.6) HÖLDER REGULARITY FOR ∂ 23
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HÖLDER REGULARITY FOR ∂ 239 can
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HÖLDER REGULARITY FOR ∂ 241 wher
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HÖLDER REGULARITY FOR ∂ 243 Theo
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Note (4.7) bD∩U ′ i dzA j J H
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HÖLDER REGULARITY FOR ∂ 247 p. 1
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similarly, we can prove HÖLDER REG
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HÖLDER REGULARITY FOR ∂ 251 For
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Now if n1 ≥ 2, l ∈ S1, then (4.
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HÖLDER REGULARITY FOR ∂ 255 Refe
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