Selected Title s i n Thi s Serie s

151 S . Yu . Slavyanov , Asymptoti c solution s o f th e one-dimensiona l Schrodinge r equation ,

1996

150 B . Ya . Levin , Lecture s o n entir e functions , 199 6

149 Takash i Sakai , Riemannia n geometry , 199 6

148 Vladimi r I . Piterbarg, Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields ,

1996

147 S . G . Gindiki n an d L . R. Volevich , Mixe d proble m fo r partia l differentia l equation s wit h

quasihomogeneous principa l part , 199 6

146 L . Ya. Adrianova , Introductio n t o linea r system s o f differentia l equations , 199 5

145 A . N . Andriano v an d V. G . Zhuravlev , Modula r form s an d Heck e operators , 199 5

144 O . V. Troshkin , Nontraditiona l method s i n mathematica l hydrodynamics , 199 5

143 V. A . Malyshe v an d R. A . Minlos , Linea r infinite-particl e operators , 199 5

142 N . V . Krylov , Introductio n t o th e theor y o f diffusio n processes , 199 5

141 A. A . Davydov , Qualitativ e theor y o f contro l systems , 199 4

140 Aizi k I . Volpert , Vital y A . Volpert , an d Vladimi r A . Volpert , Travelin g wav e solution s o f

parabolic systems , 199 4

139 I . V . Skrypnik , Method s fo r analysi s o f nonlinea r ellipti c boundar y valu e problems , 199 4

138 Yu . P . Razmyslov , Identitie s o f algebra s an d thei r representations , 199 4

137 F . I . Karpelevich an d A. Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues , 199 4

136 Masayosh i Miyanishi , Algebrai c geometry , 199 4

135 Masar u Takeuchi , Moder n spherica l functions , 199 4

134 V . V . Prasolov , Problem s an d theorem s i n linea r algebra , 199 4

133 P . I . Naumki n and I . A . Shishmarev , Nonlinea r nonloca l equation s i n th e theor y o f waves ,

1994

132 Hajim e Urakawa , Calculu s o f variation s an d harmoni c maps , 199 3

131 V. V . Sharko , Function s o n manifolds : Algebrai c an d topologica l aspects , 199 3

130 V . V . Vershinin , Cobordism s an d spectra l sequences , 199 3

129 Mitsu o Morimoto , A n introductio n t o Sato' s hyperfunctions , 199 3

128 V . P . Orevkov, Complexit y o f proof s an d thei r transformation s i n axiomati c theories , 199 3

127 F . L . Zak , Tangent s an d secant s o f algebrai c varieties , 199 3

126 M . L . Agranovskii, Invarian t functio n space s o n homogeneou s manifold s o f Li e group s an d

applications, 199 3

125 Masayosh i Nagata , Theor y o f commutativ e fields , 199 3

124 Masahis a Adachi , Embedding s an d immersions , 199 3

123 M . A . Akivi s an d B . A . Rosenfeld , Eli e Carta n (1869-1951) , 199 3

122 Zhan g Guan-Hou , Theor y o f entir e an d meromorphi c functions : Deficien t an d asymptoti c

values an d singula r directions , 199 3

121 I . B . Fesenk o an d S . V . Vostokov , Loca l fields an d thei r extensions : A constructiv e

approach, 199 3

120 Takeyuk i Hida and Masuyuki Hitsuda , Gaussia n processes , 199 3

119 M . V . Karase v an d V . P . Maslov , Nonlinea r Poisso n brackets . Geometr y an d quantization ,

1993

118 Kenkich i Iwasawa , Algebrai c functions , 199 3

117 Bori s Zilber , Uncountabl y categorica l theories , 199 3

116 G . M . Fel'dman , Arithmeti c o f probabilit y distributions , an d characterizatio n problem s

on abelia n groups , 199 3

115 Nikola i V . Ivanov , Subgroup s o f Teichmulle r modula r groups , 199 2 (Continued in the back of this publication)

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Asymptotic Solution s of the One-Dimensiona l Schrodinger Equatio n

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 15 1

Asymptotic Solution s of the One-Dimensiona l Schrodinger Equatio n

S. Yu . Slavyano v

American Mathematica l Societ y Providence, Rhod e Islan d

' ^ D E D K *

10.1090/mmono/151

C . K ) . CjiaBHHO B

ACHMnTOTMKA PEKIEHM M O i l H O M E P H O r O y P A B H E H R H I H P E U H H r E P A

H3IIATEJIL>CTBO MOCKOBCKOr O YHMBEPCMTETA , 198 8

Translated b y Vadim Khidekel

EDITORIAL COMMITTE E

AMS Subcommitte e Robert D . MacPherso n

Grigorii A . Marguli s James D . Stashef f (Chair )

ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Preidli n (Chair )

1991 Mathematics Subject Classification. Primar y 34E05 , 81Q05; Secondary 33C10.

ABSTRACT. Th e boo k i s devote d t o asymptoti c analysi s o f solution s o f secon d orde r ordinar y dif -ferential equation s wit h a smal l parameter . Th e mai n emphasi s i s on variou s constructiv e scheme s of obtainin g asymptoti c solutions , thei r advantage s an d drawbacks , an d specifi c computations . The autho r give s a complet e overvie w o f th e stat e o f th e theor y an d als o concentrate s o n som e lesser know n aspect s an d problems , i n particula r th e problem s i n whic h exponentiall y smal l term s should b e take n int o accoun t o r th e analysi s o f equation s wit h clos e transitio n points . Suc h ap -plications a s th e derivatio n o f th e formula s fo r th e quasiclassica l quantization , spectru m splittin g in a symmetrica l potential , etc . ar e considered . Th e boo k ca n b e use d b y researcher s an d grad -uate student s workin g i n ordinar y differentia l equation s an d mathematica l problem s o f quantu m mechanics.

Library o f Congres s Cataloging-in-Publicatio n D a t a Slavianov, S . IU . (Serge i IUr'evich )

[Asimptotika resheni i odnomernog o uravnenii a Shredingera . English ] Asymptotic solution s o f th e one-dimensiona l Schrodinge r equatio n / S . Yu . Slavyanov ; [trans -

lator Vadi m Khidekel] . p. cm.—(Translation s o f mathematica l monographs , ISS N 0065-9282 ; v . 151 )

Cyrillic titl e pag e attache d i n Russian . Includes bibliographica l references . ISBN 0-8218-0563- 3 (alk . paper ) 1. Schrodinge r equation . 2 . Differentia l equations—Asymptoti c theory . I . Title . II . Series .

QC174.26.W28S5313 199 6 515/.352—dc20 96-1412 9

CIP

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint -permissionOams.org.

© 199 6 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 .

H Printe d o n recycle d paper .

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

Contents

Preface t o th e Englis h Editio n x i

Preface xii i

Chapter I . Compariso n Function s 1 1.1. Basi c concept s o f asymptoti c method s 1

1. Symbol s O , o , an d ~ 1 2. Asymptoti c serie s an d thei r properties . Asymptoti c ansat z 3 3. Propertie s o f asymptoti c expansion s 5

1.2. Th e Air y function s an d thei r asymptotic s 7 1. Th e Air y equation . Standar d solutions . Relation s betwee n so -

lutions 7 2. Forma l solution s o f the Air y equatio n a t infinit y 1 0 3. Derivatio n o f asymptoti c expansio n fo r th e Air y functio n fro m

integral representatio n a t | argz| < 27r/ 3 — e 1 1 4. Th e Stoke s phenomeno n fo r th e Air y equatio n 1 4 5. Justifyin g forma l asymptoti c solution s o f th e Air y equatio n b y

using integra l equation s 1 7 1.3. Paraboli c cylinde r function s an d thei r asymptotic s 2 1

1. Th e Webe r equation . Standar d solution s an d relation s betwee n solutions 2 1

2. Asymptotic s o f th e paraboli c cylinde r function s fo r larg e argu -ments 2 5

3. Modifie d paraboli c cylinde r function s an d thei r asymptotic s 3 0 1.4. Th e Besse l function s an d thei r asymptotic s 3 1

1. Th e Besse l equation . Standar d solution s an d relation s betwee n solutions 3 1

2. Asymptotic s o f cylinde r function s fo r larg e argument s 3 2 3. Asymptoti c solution s o f the equatio n

y"(z) + [1/z + ( 1 - m 2)/(4z2)}y(z) = 0 3 5 4. Th e equatio n w"(z) —az rnw(z) = 0 : Solution s an d thei r asymp -

totics 3 7 1.5. Confluen t hypergeometri c functio n an d it s asymptotic s 4 0

1. Confluen t hypergeometri c equation. Th e functions $(a , c, z) an d \I/(a,c, z) an d relation s betwee n the m 4 0

2. Asymptotic s o f the function s <l>(a , c, z) an d ^(a , c , z) 4 3 3. Th e Whittake r function s an d thei r asymptotic s 4 5

Comments 4 7

viii C O N T E N T S

Chapter II . Derivatio n o f Asymptotics 4 9 II. 1. Genera l theor y 4 9

1. Reductio n o f second orde r equation s t o th e canonica l for m 4 9 2. Forma l theor y fo r equation s withou t transitio n point s 5 0 3. Th e Liouville-Gree n transformatio n 5 6

11.2. Asymptoti c solution s o n th e comple x plan e 5 9 1. Turnin g points , Stoke s lines , canonica l domain s 5 9 2. Primar y fundamenta l syste m o f solutions i n a canonical domai n 6 2 3. Relatio n matrice s 6 5

11.3. Metho d o f comparison equation s fo r equation s wit h on e transitio n point 6 9 1. Forma l procedur e o f the metho d o f comparison equation s 6 9 2. Metho d o f compariso n equation s fo r equation s wit h on e simpl e

turning poin t 7 1 3. Asymptotic s fa r fro m a turning poin t 7 3 4. Loca l asymptoti c expansion s nea r a turnin g poin t 7 6 5. Turnin g poin t o f multiplicit y m 7 7 6. Equation s wit h on e simpl e pol e 8 0

11.4. Metho d o f comparison equation s fo r equation s wit h tw o transitio n points 8 1 1. Forma l analysi s o f equations wit h tw o simpl e turnin g point s 8 1 2. Regularizatio n o f phase integral s 8 5 3. Forma l analysi s o f equations wit h on e simple turnin g poin t an d

one simpl e pol e 9 0 11.5. Metho d of comparison equations for equations with close transition

points 9 2 1. Scalin g transformation s 9 2 2. Tw o clos e turning point s 9 4 3. Clos e pol e an d turnin g poin t 9 7

Comments 9 9

Chapter III . Physica l Problem s 10 1 111.1. Th e WK B metho d fo r boun d state s i n quantu m mechanic s 10 1

1. Anharmoni c oscillator . Highl y excite d state s 10 1 2. Anharmoni c oscillator . Smal l perturbation s 10 7 3. Quantizatio n fo r potential s Coulomb-typ e singularit y 11 1

111.2. Norma l mode s i n ocea n waveguid e 11 8 1. Formulatio n o f the proble m 11 8 2. Asymptoti c formula s fo r norma l mode s an d phas e velocitie s 12 0

111.3. Exponentia l spectru m splittin g 12 3 1. Tw o symmetri c potentia l well s 12 3 2. Symmetri c two-cente r proble m 12 7

111.4. Quasistationar y state s 13 1 1. Star k effec t i n hydroge n 13 1 2. Ionizatio n i n electri c field 13 4

111.5. One-dimensiona l scatterin g proble m 13 6 1. Semiclassica l asymptotic s o f th e Jos t function s an d scatterin g

phases fo r potential s wit h Coulom b singularit y 13 6 2. Wav e transitio n throug h a potentia l barrie r 13 9

C O N T E N T S i x

3. Overbarrie r reflectio n 14 3 III.6. Ban d spectru m 14 7

1. Equation s wit h periodi c potentia l 14 7 2. Asymptoti c formula s fo r bandwidth s 14 9 3. Th e Mathie u equatio n 15 2

Comments 15 2

Chapter IV . Supplement s 15 5 IV. 1. Numerica l realizatio n o f asymptoti c method s 15 5

1. Approximatio n o f potentia l an d evaluatio n o f phase integral s 15 5 2. Approximatio n o f the derivative s o f a potentia l a t a poin t 15 8

IV.2. Th e Priife r transformatio n an d iterativ e modificatio n o f the WK B method 16 0 1. Th e Priife r transformatio n 16 0 2. Iterativ e procedur e fo r solvin g equation s fo r amplitud e an d

phase an d it s connectio n wit h asymptoti c expansion s 16 2 IV.3. Solution s o f z 2w" - (z s + a 2z

2 + a\z + a 0)w = 0 16 4 1. Standar d solution s 16 4 2. Representatio n o f solution s i n term s o f th e Mellin-Barne s inte -

grals 16 7 3. Connectio n betwee n th e solution s I a

J (z) an d K a {z) 11A 4. Differenc e equatio n fo r connectio n factor s 18 0

Comments 18 3

References 185

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Preface t o th e Englis h Editio n

This boo k wa s written i n 1985-1988 . I t reflecte s th e experienc e an d th e view s of the autho r a t tha t time , an d wa s intende d fo r reader s intereste d i n application s rather tha n i n rigorous theory. After tha t th e autho r almos t stoppe d doin g research in the are a ( a few exceptions ar e paper s [1*—3*] 1). Meanwhil e th e ne w revolution -ary perio d o f muc h deepe r understandin g o f asymptotic s started . Therefore , whe n the America n Mathematica l Societ y kindly suggeste d publishin g th e Englis h trans -lation of the book, the author was at a loss. Shoul d he make changes in the structur e and th e tex t o f th e book , o r i t shoul d b e lef t unmodified ? Th e autho r ha s chose n the secon d possibilit y takin g int o accoun t that :

(i) fo r th e above-mentione d reader s th e change s wer e no t s o vital ; (ii) the author himsel f coul d no t b e regarded a s a real exper t i n the new trends . Still the autho r feel s i t i s his duty a t leas t t o mention th e majo r recen t achieve -

ments an d t o giv e a shor t lis t o f references . Even i n th e boo k itsel f asymptoti c solution s o f th e Schrodinge r equatio n ar e

exposed beyon d th e scop e o f th e Poincar e definitio n o f asymptotics . Althoug h the autho r ha s nothin g t o chang e i n formulas , th e argumentatio n i s essentiall y heuristic rathe r tha n rigorous . Th e reall y mathematica l presentatio n o f wha t i s called "exponentia l asymptotics, " whe n exponentiall y smal l term s (whic h shoul d be neglecte d i n th e Poincar e sense ) ar e deal t with , i s base d o n th e us e o f Bore l transform. Th e exampl e o f th e approac h t o asymptotic s o n th e basi s o f Bore l summability wa s given by Silverstone e t al . [4*] . A much mor e general formulatio n arranged a s th e "resurgenc e theory " ha s bee n presente d b y Ecall e [5*] . Rathe r abstract idea s of Ecalle were later developed and modified b y Pham [6*] , Voros [7*], and others . Mathematica l treatmen t o f exponentia l asymptotic s mor e adequat e to numerica l mathematic s an d close r t o heuristi c argumentatio n wa s propose d b y Kruskal an d Costi n [8*] .

Another crucia l ide a (whic h i s no t presente d i n th e book)—th e metho d o f smoothing the asymptoti c expansions nea r the Stoke s line—was suggested b y Berr y [9*]. Berry' s smoothin g proces s appeare d t o b e ver y genera l an d vali d beyon d th e scope o f Stokes ' phenomena . Mor e rigorou s treatmen t o f th e proces s wa s give n b y McLeod [10*] .

Behavior o f asymptoti c expansion s i n the vicinit y o f coalescing turnin g points , which i s studie d i n th e book , receive d bette r justificatio n i n paper s b y Dunste r [ i i*] .

It i s als o necessar y t o mentio n th e notio n o f hyperasymptotic s propose d b y Berry an d Howl s [12* ] an d extende d t o solution s o f differentia l equation s b y Old e Daalhuis an d Olve r [13*] . Thi s notio n i s remarkabl e sinc e i t goe s beyon d th e

lrThe number s i n thi s prefac e refe r t o additiona l reference s a t th e en d o f th e referenc e list .

xi

xii PREFAC E T O T H E ENGLIS H EDITIO N

scope o f forma l asymptotic s take n a s th e basi s o f th e ansatz . O n th e othe r hand , hyperasymptotics give s the tool for much bette r numerica l estimates than an y othe r form o f asymptotics .

Finally, i t i s author' s pleasur e an d dut y t o than k th e America n Mathematica l Society fo r choosin g t o publis h thi s book , whic h i s rathe r fa r fro m conventiona l mathematics; senio r edito r o f the AMS, Sergei Gelfand , fo r patience i n negotiation s with th e author ; an d th e translato r o f th e book , Vadi m Khidekel , fo r carefull y searching fo r misprint s i n th e Russia n edition .

Saying i t frankly , th e author i s very enthusiastic abou t th e research proces s bu t cares much les s abou t final checking . Therefore , th e reade r shoul d b e awar e o f stil l existing misprint s i n th e text .

S. Yu . Slavyano v St. Petersburg , Februar y 199 6

Preface

This boo k gre w ou t o f th e graduate-leve l cours e "Asymptoti c Method s in the Theory o f Ordinary Differentia l Equations, " give n by the autho r a t th e Departmen t of Physic s o f St. Petersbur g University . Th e reade r i s supposed t o be familia r with basi c notion s o f the theor y o f ordinary differentia l equation s an d th e comple x analysis.

Asymptotic method s ar e importan t i n modern mathematica l physics . The y allow the qualitative and quantitative tracing of the limit passages from one physical theory t o another , fo r example , fro m quantu m t o classica l mechanics , o r fro m wav e to geometrica l optics . The y ca n als o clarif y man y importan t physica l phenomen a such as the long-range propagatio n o f radio waves in the atmosphere an d acoustica l waves in the ocean , th e bindin g o f atoms i n molecules, radioactiv e decay , and man y others.

This boo k i s devoted to a specific proble m in the theor y of asymptotic meth -ods, to asymptotic expansion s o f the solution s o f second-order linea r homogeneou s ordinary differentia l equation s with a small coefficient o f the higher derivative . Thi s equation ca n b e writte n i n a standard for m as

(1) y"(x)+p 2(\-q(x))y(x) = 0.

Equation (1 ) i s often referre d t o a s the one-dimensiona l Schrodinge r equatio n wit h the potentia l q(x) an d th e energ y A . Quantu m terminolog y i s used throughou t th e book, althoug h equatio n (1 ) can als o arise in other field s o f physics. Th e coefficien t p is assumed t o be "large." It s meanin g fro m th e mathematica l poin t o f view is explained b y th e definition s o f asymptoti c expansions . I n practica l application s o f asymptotic formula s i t is sufficient tha t th e paramete r p take th e value s o f a t leas t tens.

This boo k emphasize s th e presentatio n o f variou s algorithm s fo r constructin g asymptotic expansion s o f th e solution s o f equatio n (1) , discussio n o f their advan -tages and disadvantages , an d the computational details , which are important fo r th e readers—physicists an d engineers . Th e proof s ar e given with les s detail; sometime s they ar e omitte d completely .

All the methods o f deriving asymptoti c solution s o f equation (1 ) ar e based o n a simple ide a tha t "similar " equation s hav e "similar " solutions . I n th e simples t case , an equatio n wit h constan t coefficient s i s chosen a s a comparison equation t o (1). However, suc h a n approac h fail s i n a vicinity o f what i s known a s transition points, that is , the points where X—q(x) ha s either zeros or simple poles. Obtainin g unifor m asymptotics o f the solution s o f equation (1 ) in a vicinity o f transition point s an d matching asymptoti c expansion s ar e th e mai n goal s o f th e theor y presented . Tw o methods ar e commonl y used : introducin g a comparison equatio n i n a more genera l

xi i i

xiv P R E F A C E

form, an d bypassin g transitio n point s usin g th e comple x argument . Th e choic e of metho d i s determine d b y th e requirement s fo r th e resul t an d b y th e propertie s of th e potentia l q(x). Usually , th e "better " th e potential , th e mor e complet e an d precise th e results .

In Chapte r I th e propertie s o f compariso n functions—th e Air y functions , th e parabolic cylinde r functions , th e Besse l functions , etc.—ar e presented . Thes e func -tions ar e used t o derive asymptoti c solution s o f equation (1 ) nea r transitio n points . They als o help us to illustrat e th e importan t genera l definition s o f Stokes line s an d Stokes phenomena . Th e asymptotic s o f compariso n function s fo r larg e value s o f the argumen t ar e derive d i n a universa l way : b y calculatin g th e asymptotic s o f the Laplac e contou r integral s tha t represen t thes e functions . I f th e argumen t o f a complex variabl e i s changed , th e integratio n contou r ha s t o b e deformed . Suc h a deformation ma y b e discontinuou s a t som e value s o f th e argument ; thi s result s i n abrupt change s in the form o f the asymptotics . Thes e change s ar e called the Stoke s phenomenon.

Contrary to the conventional Poincare definition o f an asymptotic expansion, i n this book the exponentially decreasin g terms are carefully kep t i n the background of the dominan t asymptoti c series . First , thi s enables us to improve the accurac y tha t can b e achieve d b y asymptoti c formula s (thi s wa s first pointe d ou t b y F . Olver) . Second, thi s helps us to formalize th e algorithms fo r calculatin g exponentially smal l corrections t o eigenvalue s i n various applications . A s a result , asymptoti c formula s for compariso n function s derive d i n thi s boo k ofte n diffe r fro m wha t mos t hand -books o n specia l function s present . Also , th e Stoke s phenomeno n i s interprete d from a differen t poin t o f view .

Chapter I I i s concerne d wit h th e method s o f constructin g th e asymptoti c ex -pansions fo r solutions of equation (1 ) with an arbitrary potentia l q(x). Th e first tw o sections dea l wit h well-know n subjects : writin g asymptoti c solution s i n interval s of th e comple x plan e i n th e absenc e o f transitio n points . Onl y th e mai n idea s ar e discussed. Fo r more information o n the subjec t th e reade r i s referred t o the boo k of Fedoryuk (1983) . Mor e attentio n i s pai d t o studie s o f asymptotic s nea r transitio n points, especiall y t o nontraditiona l issue s suc h a s variou s simplification s o f genera l uniform formulas , regularizatio n o f integrals , passag e t o th e limi t whe n th e transi -tion point s approac h on e another , th e rol e o f a second-orde r pol e singularity i n th e coefficients o f equation (1) , and s o forth .

In Chapte r II I w e stud y som e applie d problems . The y ar e no t alway s closel y connected wit h rea l physica l problems . Bot h model s an d specifi c problem s arisin g in variou s fields o f physic s ar e considered . I n th e first sectio n w e deriv e th e for -mulas fo r a semiclassica l quantization . I n Sectio n 2 the contributio n o f boundar y conditions a t finite point s t o semiclassica l quantizatio n i s studied o n a model prob -lem o f submarin e acoustics . I n th e nex t tw o section s w e presen t th e algorithm s to distinguis h exponentiall y smal l term s i n powe r expansions . The n w e pas s t o the scatterin g proble m an d th e proble m o f a periodi c potential . Throughou t th e book th e cas e whe n th e spectra l paramete r i s proportiona l t o som e powe r o f th e large paramete r (whic h lead s t o clos e turnin g points ) i s analyze d separatel y an d the physica l meanin g o f this assumptio n i s discussed .

In Chapte r I V severa l technica l question s ar e discussed . The y ar e mostl y re -lated t o th e numerica l realizatio n o f asymptoti c method s suc h a s computin g phas e integrals an d th e correspondenc e betwee n exac t solution s an d asymptotics . I n Sec -tion 3 w e concentrat e o n applyin g th e mai n notion s o f Stoke s line s an d Stoke s

P R E F A C E x v

constants t o th e equation s wit h polynomia l coefficient s tha t ar e mor e complicate d than th e equation s fo r specia l functions . Thi s sectio n wa s writte n b y M . A . Ko -valevskh a t th e author' s request .

Much les s analytica l result s exis t her e an d w e have t o tur n mor e t o numerica l methods. O n th e othe r hand , th e problem s unde r consideratio n becom e mor e an d more complicated , an d thi s wil l necessaril y resul t i n usin g thes e equation s a s th e comparison equations . Th e autho r i s very gratefu l t o M. A. Kovalevskh fo r writin g this section .

Many physicist s an d mathematician s develope d an d use d i n their wor k asymp -totic method s fo r equatio n (1) . Sometime s i t i s ver y difficul t t o establis h th e au -thorship o f an idea , method , o r formula. W e will name the scientist s tha t mad e th e most importan t contribution s t o thi s theory . Amon g foreig n scientist s w e shoul d mention H . Poincare , G . Stokes , G . Green , J . Liouville , A . Zwaan , an d H . Jeffreys , who pu t fort h th e mai n ideas ; G . Wentzel , H . Kramers , an d L . Brilloui n wh o wer e the firs t t o appl y asymptoti c method s t o the quantu m mechanica l problem s (WK B method). W e als o mentio n th e monograph s an d paper s o f R . Langer , G . Birkhoff , E. Kemble , T . Cherry , V . Torson , K . Budden , A . Erdelyi , F . Olver , J . Heading , N. Froman, an d P . O. Froman, which summed u p various aspects of the theory. Th e contribution o f Sovie t scientist s t o th e developmen t o f asymptoti c method s i s als o very important . V . A . Foc k an d hi s studen t M . I . Petrashen ' succeede d i n deriv -ing asymptotic solution s o f classical problems o f quantum mechanics . Th e pape r o f A. A. Dorodnytsyn wa s an important mileston e i n the theory. M . V. Fedoryuk gav e rigorous justification o f Zwaan's methods . V . P . Maslov proposed a new method fo r obtaining asymptoti c expansions . E . E . Dubrovskaya , E . E . Nikitin , N . I . Zhirnov , and R . Ya . Dambur g applie d asymptoti c method s t o solv e man y particula r quan -tum mechanical problems. Th e problems of the diffraction an d propagation of waves were studie d usin g asymptoti c method s b y V . S . Buldyrev , L . M . Brekhovskikh , G. I. Makarov, and V. V. Novikov. Th e lectures of L. I. Ponomarev, V . S. Buldyrev, V. M . Babich , an d I . A . Molotko v provid e a goo d stud y o f asymptoti c methods .

The book of M. V. Fedoryuk, i n which a complete picture o f the lates t progres s in th e asymptoti c theor y o f ordinar y differentia l equation s i s presente d o n a ver y high scientifi c level , wa s publishe d i n 1983 . Th e presen t monograp h complement s that boo k an d aim s t o hel p th e reade r develo p skill s o f workin g wit h asymptoti c expansions (wit h man y details) .

The autho r attempte d t o presen t a sufficientl y ful l pictur e o f the whol e theor y and at the same time pay more attention to nontrivial aspects, which are less studied in principa l monographs . Th e problem s t o whos e solutio n th e autho r contribute d are als o discusse d (th e problem s o f th e subordinat e exponentia l functio n i n th e background o f the dominan t one , problems wit h clos e transition points , algorithm s for calculatin g phas e integrals , etc.) .

We wan t t o poin t ou t som e technica l detail s o f th e book' s structure . Th e formulas ar e numbere d independentl y i n eac h chapter . Withi n a chapter , th e firs t digit o f th e equatio n numbe r indicate s th e numbe r o f th e section . Fo r reference s within a chapter , it s numbe r i s no t given . Reference s t o scientifi c publication s are mostl y concentrate d i n th e "Comments " sections . Th e autho r an d th e yea r of publicatio n ar e indicated . Th e bibliograph y a t th e en d o f th e boo k include s not onl y th e entrie s cite d i n th e tex t bu t als o additiona l paper s an d monograph s that th e autho r believe s t o contai n importan t result s concernin g th e asymptoti c solutions o f the one-dimensiona l Schrodinge r equation . Th e autho r i s very gratefu l

xvi P R E F A C E

to V . M . Babich , A . G . Alenitsyn , I . V . Komarov , D . I . Abramov , E . A . Solovyev , T. Grozdanov, N. Froman, P . O. Froman, an d T. F. Pankratova, an d to his audience, physics students, fo r the contribution tha t the y made by their invisible participatio n in creatin g th e book . Th e autho r als o wishe s t o than k th e reviewe r o f th e boo k L. I. Ponomarev an d th e scientifi c edito r V . S . Buldyrev fo r thei r suppor t an d man y helpful comments .

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Selected Title s i n Thi s Serie s (Continued from the front of this publication)

114 Seiz o ltd , Diffusio n equations , 199 2

113 Michai l Zhitomirskii , Typica l singularitie s o f differentia l 1-form s an d Pfafha n equations , 1992

112 S . A . Lomov , Introductio n t o th e genera l theor y o f singula r perturbations , 199 2

111 Simo n Gindikin , Tub e domain s an d th e Cauch y problem , 199 2

110 B . V . Shabat , Introductio n t o comple x analysi s Par t II . Function s o f severa l variables ,

1992

109 Isa o Miyadera , Nonlinea r semigroups , 199 2

108 Take o Yokonuma, Tenso r space s an d exterio r algebra , 199 2

107 B . M . Makarov , M . G . Goluzina , A . A . Lodkin , an d A . N . Podkorytov , Selecte d problem s i n

real analysis , 199 2

106 G.-C . Wen , Conforma l mapping s an d boundar y valu e problems , 199 2

105 D . R . Yafaev , Mathematica l scatterin g theory : Genera l theory , 199 2

104 R . L . Dobrushin , R . Kotecky , an d S . Shlosman , Wulf f construction : A globa l shap e fro m

local interaction , 199 2

103 A . K . Tsikh , Multidimensiona l residue s an d thei r applications , 199 2

102 A . M . Il'in , Matchin g o f asymptoti c expansion s o f solution s o f boundar y valu e problems ,

1992

101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Don g Zhen-xi , Qualitativ e theor y o f

differential equations , 199 2

100 V . L . Popov, Groups , generators , syzygies , an d orbit s i n invarian t theory , 199 2

99 Nori o Shimakura , Partia l differentia l operator s o f ellipti c type , 199 2

98 V . A . Vassiliev , Complement s o f discriminant s o f smoot h maps : Topolog y an d

applications, 199 2 (revise d edition , 1994 )

97 Itir o Tamura , Topolog y o f foliations : A n introduction , 199 2

96 A . I . Markushevich , Introductio n t o th e classica l theor y o f Abelia n functions , 199 2

95 Guangchan g Dong , Nonlinea r partia l differentia l equation s o f secon d order , 199 1

94 Yu . S . Il'yashenko, Finitenes s theorem s fo r limi t cycles , 199 1

93 A . T . Fomenk o an d A . A . Tuzhilin , Element s o f th e geometr y an d topolog y o f minima l

surfaces i n three-dimensiona l space , 199 1

92 E . M . Nikishin an d V. N . Sorokin , Rationa l approximation s an d orthogonality , 199 1

91 Mamor u Mimura and Hiros i Toda , Topolog y o f Li e groups , I an d II , 199 1

90 S . L . Sobolev , Som e application s o f functiona l analysi s i n mathematica l physics , thir d

edition, 199 1

89 Valeri i V . Kozlo v an d Dmitri ! V. Treshchev , Billiards : A geneti c introductio n t o th e

dynamics o f system s wit h impacts , 199 1

88 A . G . Khovanskii , Fewnomials , 199 1

87 Aleksand r Robertovic h Kemer , Ideal s o f identitie s o f associativ e algebras , 199 1

86 V . M . Kadet s an d M . I . Kadets , Rearrangement s o f serie s i n Banac h spaces , 199 1

85 Miki o Is e and Masaru Takeuchi , Li e group s I , II , 199 1

84 Da o Tron g Th i an d A . T . Fomenko , Minima l surfaces , stratifie d rnultivarifolds , an d th e

Plateau problem , 199 1

83 N . I . Portenko , Generalize d diffusio n processes , 199 0

82 Yasutak a Sibuya , Linea r differentia l equation s i n th e comple x domain : Problem s o f

analytic continuation , 199 0

81 I . M . Gelfan d an d S . G . Gindikin , Editors , Mathematica l problem s o f tomography , 199 0 (See th e AM S catalo g fo r earlie r titles )