Selected Paper s o n Analysi s and Differentia l Equation s

This page intentionally left blank

American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 21 1

Selected Paper s o n Analysi s and Differentia l Equation s

Katsumi Nomiz u Editor

-rtJEMAr,

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/trans2/211

Translat ion edite d b y KATSUM I NOMIZ U

2000 Mathematics Subject Classification. P r imar y 00B55 ; Secondary 35-XX , 42-XX , 46L35 , 58Jxx , 74-XX .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Selected paper s o n analysi s an d differentia l equation s / Katsum i Nomizu , editor . p. cm . — (America n Mathematica l Societ y translatuins , ISS N 0065-929 0 ; ser. 2 , v . 211 )

Includes bibliographica l references . ISBN 0-8218-3508- 4 (acid-fre e paper ) 1. Differentia l equations , Partial . 2 . Fourie r analysis . 3 . Selfadjoin t operators . 4 . Differen -

tial operators . I . Nomizu , Katsumi , 1924 - II . Series .

QA3.A572 ser . 2 , vol . 21 1 [QA374] 510 s—dc22 200305561 3 [515'.353]

Copying an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu -cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg -ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Acquisition s Department , America n Math -ematical Society , 20 1 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n also b e mad e b y e-mai l t o reprint-permission@ams.org .

Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. )

© 200 3 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t h t t p : //www. ams. org/

10 9 8 7 6 5 4 3 2 1 0 8 0 7 0 6 0 5 0 4 0 3

Contents

Preface

Van Vlec k formul a fo r Wiene r integral s an d Jacob i field s NOBUYUKI IKED A

Spectral geometr y fo r Schrodinge r operator s i n a magneti c fiel d focusin g o n geometrical an d dynamica l structure s o n manifold s RUISHI KUWABAR A

Symbolic dynamic s an d C*-algebra s KENGO MATSUMOT O

Inverse problem s fo r elasticit y GEN NAKAMUR A

Time-global solution s o f nonlinea r evolutio n equation s an d thei r stabilit y YOSHIHIRO SHIBAT A

Wavelets an d eigenvalue s o f Schrodinge r operator s KAZUYA TACHIZAW A

Recent topic s o n nonlinea r partia l differentia l equations : Structur e o f radia l solutions fo r semilinea r ellipti c equation s Eui YANAGID A AN D SHOJ I YOTSUTAN I

This page intentionally left blank

Amer. Math . Soc . Transl . (2) Vol . 211 , 200 3

Preface

This i s a collectio n o f severa l paper s tha t originall y appeare d i n th e journa l Sugaku i n Japanese . Thes e translate d article s woul d normall y appea r i n th e AM S journal Sugaku Expositions. I n orde r t o expedit e publication , th e AMS has chosen , with the consent o f the Mathematical Societ y of Japan, t o publish them as a volume of selected papers in the Translations Series 2 of the AMS. To give the reader a quick idea o f the contents , w e give brief quotation s fro m th e introductio n t o eac h article , with occasiona l emendations .

The firs t pape r i s Van Vleck formula for Wiener integrals and Jacobi fields, by Nobuyuk i Ikeda . Thi s surve y i s base d o n a lectur e presente d a t th e annua l meeting o f the Mathematica l Societ y o f Japan i n 1996 , which i n turn wa s based o n joint wor k wit h Shige o Kusuok a an d Shojir o Manabe . Th e ke y word s an d phrase s include: M . Kac , Feynma n pat h integrals , Levy' s formula , Gaussia n integrals , Va n Vleck formula , Jacobia n o f a transform, an d Selberg' s trac e formula .

Ruishi Kuwabar a wrot e o n Spectral geometry for Schrodinger operators in a magnetic field focusing on geometrical and dynamical structures on manifolds. Th e aim o f this articl e i s to presen t som e topic s an d result s o n th e relationship s amon g the spectru m o f H, geometr y o f (M, g, A), an d mechanic s i n H ci i n the cas e wher e M i s a compact manifol d withou t boundar y an d th e scala r potentia l V = 0 . I n th e framework o f symplectic structures and their quantized object s we will consider ho w the magneti c field © or it s vecto r potentia l A affect s th e quantu m spectru m fro m the geometri c an d dynamica l viewpoints . Th e technique s use d her e ar e base d o n the theor y o f Hamiltonian mechanics , th e geometr y o f connections , an d th e theor y of pseudodifferential an d Fourie r integra l operators . Th e main result s in this surve y were obtaine d aroun d 1990 , an d w e canno t cove r th e man y argument s tha t hav e recently bee n deepl y an d widel y develope d o m thi s subject .

In Keng o Matsymoto' s paper , Symbolic dynamics and C*-algebras, th e autho r will dea l wit h tw o differen t type s o f objects . On e i s discret e dynamica l systems , consisting o f th e shift s o n bi-infinit e sequence s o f finite symbols , whos e theor y i s mainly par t o f topologica l dynamica l systems . Th e othe r on e i s operator algebras , consisting o f infinit e dimensiona l algebra s o f mutuall y noncommutativ e operators , whose theor y i s mainl y par t o f functiona l analysis . Thes e tw o object s loo k ver y different. I n thi s paper , th e autho r wil l introduc e a n interestin g relationshi p be -tween them . Man y concept s an d interestin g application s o f th e tw o subject s wil l indeed com e int o play ; fo r example , w e mentio n ho w Hadamard' s ol d idea s wer e instrumental t o th e origi n o f symbolic dynamics .

In Inverse problems for elasticity, Ge n Nakamura consider s such problems wit h and withou t residua l stress . Th e invers e proble m wit h residua l stres s ma y no t b e

©2003 America n Mathematica l Societ y

vii

viii PREFAC E

very familiar , bu t i t i s very interestin g an d importan t i n applications . I n additio n to coverin g know n results , includin g hi s own , th e autho r point s ou t som e o f th e open problems , an d make s conjecture s o n possibl e futur e results .

Yoshihiro Shibat a contribute s a pape r o n Time-global solutions of nonlinear evolution equations and their stability. I n i t h e consider s a certai n proble m o n stability o f stationar y solution s t o evolutio n equations , concentratin g o n hi s ow n recent results . Th e equatio n t o b e treate d i s

ut + A(u) = / ,

where th e subscrip t t denote s partia l differentiatio n wit h respec t t o time . Kazuya Tachizaw a write s o n Wavelets and eigenvalues of Schrodinger opera-

tors. H e start s b y explainin g th e basic s o f wavele t theory , an d goe s o n t o conside r their applications to eigenvalues of Schrodinger operators . On e of the main topics of Fourier analysi s i s the decompositio n o f a function; a typical exampl e i s the Fourie r series decomposition o f a periodic function . I n such a decomposition i t i s importan t to investigat e th e relatio n betwee n th e nor m o f the functio n an d it s components .

Finally, Eij i Yanagid a an d Shoj i Yotsutan i writ e o n Recent topics in nonlinear partial differential equations: Structure of radial solutions for semilinear elliptic equations. Ove r the pas t tw o decades there has been a considerable developmen t i n the theor y o f nonlinear PDEs , an d man y interestin g fact s hav e bee n discovere d fo r various equations . I n thi s articl e th e author s focu s o n th e structur e o f radia l solu -tions fo r semilinea r ellipti c equation s wit h powe r semilinearity , an d relate d topics , and tr y t o surve y th e recen t progres s a s clearl y a s possible .

This page intentionally left blank

Titles i n Thi s Serie s

211 K . Nomizu , Editor , Selecte d paper s o n analysi s an d differentia l equation s

210 S . G . Gindikin , Editor , Li e group s an d symmetri c spaces . I n memor y o f F . I .

Karpelevich

209 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I X

208 M . V . Karasev , Editor , Asymptoti c method s fo r wav e an d quantu m problem s

207 Yu . M . Suhov , Editor , Analyti c method s i n applie d probabilit y

206 M . A . Shubi n an d M . S . Agranovich , Editors , Partia l differentia l equation s (Mar k

Viskik's Seminar )

205 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume VII I

204 K . Nomizu , Editor , Selecte d paper s o n classica l analysi s

203 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume VI I

202 V . Turae v an d A . Vershik , Editors , Topology , ergodi c theory , rea l algebrai c geometry .

Rokhlin's memoria l

201 Michae l Semenov-Tian-Shansky , Editor , L . D . Faddeev' s semina r o n mathematica l

physics

200 L . Lerman , G . Polotovskn , an d L . Shilnikov , Editors , Method s o f qualitativ e

theory o f differentia l equation s an d relate d topic s

199 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume V I

198 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s way .

From probabilit y theor y t o statistica l physic s

197 Vladimi r Arnold , M a x i m Kontsevich , an d Anto n Zorich , Editors , Pseudoperiodi c

topology

196 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinstein , Editors , Norther n

California symplecti c geometr y semina r

195 O . V . Belegrade k e t al. , Mode l Theor y an d Application s

194 A . Astashkevic h an d S . Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y

Collection)

193 N . N . Uraltseva , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume V

192 Le v Beklemishev , Mat i Pentus , an d Nikola i Vereshchagin , Provability ,

Complexity, Grammar s

191 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n

Mathematical Physic s

190 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

189 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection )

188 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I V

187 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

186 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations 185 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) 184 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s (Dedicated t o Seli m Krein )

TITLES I N THI S SERIE S

183 K . Nomizu , Editor , Selecte d Paper s o n Harmoni c Analysis , Groups , an d Invariant s

182 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e

181 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y

180 A . KhovanskiT , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory

179 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology :

On th e Crossroa d

178 V . Kreinovic h an d G . Mints , Editors , Problem s o f Reducin g th e Exhaustiv e Searc h

177 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

176 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e

175 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Net sve taev , Editors , Mathematic s

in St . Petersbur g

173 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s

172 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

171 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s

170 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s

169 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r

168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s

167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations

166 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume II I

165 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut' , M . Hazewinkel , an d Yu . G . Reshetnyak , Editors , Thir d Siberia n

School 'Algebr a an d Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

161 K . Nomizu , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d

Differential Geometr y

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I I

158 A . K . Kelmans , Editor , Selecte d Topic s i n Discret e Mathematics : Proceeding s o f th e

Moscow Discret e Mathematic s Semina r 1972-199 0

157 M . Sh . Birman , Editor , Wav e Propagation . Scatterin g Theor y

156 V , N . Gerasimov , N . G , Nesterenko , an d A , I . Valitskas , Thre e Paper s o n

Algebras an d Thei r Representation s

155 O . A . Ladyzhenskay a an d A . M . Vershik , Editors , Proceeding s o f th e St . Petersburg Mathematica l Society , Volum e I

For a complet e lis t o f title s i n thi s series , visi t th e AMS Bookstor e a t www.ams.org/bookstore / .