mathematical scattering theorymathematical surveys and monographs volume 158 american mathematical...
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Mathematical Surveys
and Monographs
Volume 158
American Mathematical Society
Mathematical Scattering Theory
Analytic Theory
D. R. Yafaev
surv-158-yafaev-cov.indd 1 2/4/10 3:47 PM
Mathematical Scattering TheoryAnalytic Theory
http://dx.doi.org/10.1090/surv/158
Mathematical Surveys
and Monographs
Volume 158
American Mathematical SocietyProvidence, Rhode Island
Mathematical Scattering TheoryAnalytic Theory
D. R. Yafaev
EDITORIAL COMMITTEE
Jerry L. BonaRalph L. Cohen, Chair
Michael G. EastwoodJ. T. Stafford
Benjamin Sudakov
2000 Mathematics Subject Classification. Primary 34L25, 35-02, 35P10,35P25, 47A40, 81U05.
For additional information and updates on this book, visitwww.ams.org/bookpages/surv-158
Library of Congress Cataloging-in-Publication Data
IAfaev, D. R. (Dmitriı Rauel′evich), 1948–Mathematical scattering theory : analytic theory / D.R. Yafaev.
p. cm. – (Mathematical surveys and monographs ; v. 158)Includes bibliographical references and index.ISBN 978-0-8218-0331-8 (alk. paper)1. Scattering (Mathematics) I. Title.
QA329.I24 2009515′.724–dc22 2009027382
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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10
To the memory of my parents
Contents
Preface xi
Basic Notation 1
Introduction 5
Chapter 0. Basic Concepts 171. Classification of the spectrum 172. Classes of compact operators 203. The resolvent equation. Conditions for self-adjointness 234. Wave operators (WO) 265. The smooth method 296. The stationary scheme 337. The scattering operator and the scattering matrix (SM) 388. The trace class method 429. The spectral shift function (SSF) and
the perturbation determinant (PD) 4510. Differential operators 5211. Function spaces and embedding theorems 5612. Pseudodifferential operators 5813. Miscellaneous analytic facts 67
Chapter 1. Smooth Theory. The Schrodinger Operator 711. Trace theorems 712. The free Hamiltonian 753. The Schrodinger operator 794. Existence of wave operators 825. Wave operators for long-range potentials 866. Completeness of wave operators 937. The limiting absorption principle (LAP) 958. The scattering matrix 969. Absence of the singular continuous spectrum 9810. General differential operators of second order 10111. The perturbed polyharmonic operator 10312. The Pauli and Dirac operators 104
Chapter 2. Smooth Theory. General Differential Operators 1091. Spectral analysis of differential operators with constant coefficients 1092. Scalar differential operators 1163. Nonelliptic differential operators 1184. Matrix differential operators 122
vii
viii CONTENTS
5. Scattering problems for perturbations of a medium 1246. Strongly propagative systems. Maxwell’s equations 128
Chapter 3. Scattering for Perturbations of Trace Class Type 1331. Conditions on an integral operator to be trace class 1332. Perturbations of differential operators with constant coefficients 1363. The Schrodinger operator 1394. The perturbed polyharmonic operator 1455. General differential operators of second order 1476. Scattering problems for perturbations of a medium 1547. Wave equation 1578. The scattering matrix and the spectral shift function 159
Chapter 4. Scattering on the Half-line 1611. Jost solutions. Volterra equations 1612. Generalized Fourier transform and WO 1703. Low-energy asymptotics 1784. High-energy asymptotics 1885. The SSF for the radial Schrodinger operator 1916. Trace identities 1987. Perturbation by a boundary condition. Point interaction 203
Chapter 5. One-Dimensional Scattering 2091. A direct approach 2092. Low- and high-energy asymptotics 2163. The SSF and trace identities 2214. Potentials with different limits at “ + ” and “− ” infinities 223
Chapter 6. The Limiting Absorption Principle (LAP), the RadiationConditions and the Expansion Theorem 231
1. Absence of positive eigenvalues and radiation conditions 2312. Boundary values of the resolvent 2333. A sharp form of the limiting absorption principle 2354. Nonhomogeneous Schrodinger equation 2395. Homogeneous Schrodinger equation 2416. Expansion theorem 2457. The wave function. The scattering amplitude 2518. A generalized Fourier integral 2559. The Mourre method 259
Chapter 7. High- and Low-Energy Asymptotics 2671. High-energy and uniform resolvent estimates 2672. Asymptotic expansion of the Green function for large values of the
spectral parameter 2753. Small time asymptotics of the heat kernel 2804. Low-energy behavior of the resolvent 2855. Low-energy behavior of the resolvent. Slowly decreasing potentials 291
Chapter 8. The Scattering Matrix (SM) and the Scattering Cross Section 2971. Basic properties of the SM 297
CONTENTS ix
2. The spectrum of the SM. The modified SM 3023. The scattering cross section 3064. High-energy asymptotics of the SM. The ray expansion 3115. The eikonal approximation 3196. The averaged scattering cross section. Singular potentials 3287. The semiclassical limit 337
Chapter 9. The Spectral Shift Function and Trace Formulas 3411. The regularized PD and SSF for the multidimensional Schrodinger
operator 3412. High-energy asymptotics of the SSF 3533. Trace identities for the multidimensional Schrodinger operator 365
Chapter 10. The Schrodinger Operator with a Long-Range Potential 3691. Propagation estimates 3692. Long-range scattering 3743. The eikonal and transport equations 3804. Scattering matrix for long-range potentials 384
Chapter 11. The LAP and Radiation Estimates Revisited 3991. The efficient form of the LAP 3992. Absence of positive eigenvalues and uniqueness theorem 4033. Nonhomogeneous Schrodinger equation with a long-range potential 408
Review of the Literature 415
Bibliography 429
Index 441
Preface
This book can be considered as the second volume of the author’s monograph“Mathematical Scattering Theory (General Theory)” [I]. It is oriented to applica-tions to differential operators, primarily to the Schrodinger operator. A necessarybackground from [I] is collected (but the proofs are of course not repeated) inChapter 0. Therefore it is presumably possible to read this book independently of[I].
Everything said in the preface to [I] pertains also to this book. In particular,we proceed again from the stationary approach. Its main advantage is that, si-multaneously with proofs of various facts, the stationary approach gives formularepresentations for the basic objects of the theory. Along with wave operators, wealso consider properties of the scattering matrix, the spectral shift function, thescattering cross section, etc.
A consistent use of the stationary approach as well as the choice of concretematerial distinguishes this book from others such as the third volume of the course ofM. Reed and B. Simon [43]. The latter course has become a desktop copy for many,in particular, for the author of the present book. However, in view of the broadcompass of material, the course [43] was necessarily written in encyclopedic styleand apparently cannot replace a systematic exposition of the theory. Hopefully,vol. 3 of [43] and this book can be considered as complementary to one another.
There are two different trends in scattering theory for differential operators.The first one relies on the abstract scattering theory. The second one is almostindependent of it. In this approach the abstract theory is replaced by a concreteinvestigation of the corresponding differential equation. In this book we presentboth of these trends.
The first of them illustrates basic theorems of [I]. Thus, Chapters 1 and 2are devoted to applications of the smooth method. Of course the abstract resultsof [I] should be supplemented by some analytic tools, such as the Sobolev tracetheorem. The smooth method works well for perturbations of differential operatorswith constant coefficients. In Chapter 3 applications of the trace class method arediscussed. The main advantage of this method is that it does not require an explicitspectral analysis of an “unperturbed” operator.
Other chapters are much less dependent on [I]. Chapters 4 and 5 are devotedto the one-dimensional problem (on the half-axis and the entire axis, respectively)which is a touchstone for the multidimensional case because specific methods ofordinary differential equations can be used here.
In the following chapters we return to the multidimensional problem and discussdifferent analytic methods appropriate to differential operators. In particular, inChapter 6 scattering theory is formulated in terms of solutions of the Schrodingerequation satisfying some “boundary conditions” (radiation conditions) at infinity.
xi
xii PREFACE
High- and low-energy asymptotics of the Green function (the resolvent kernel) andof related objects are discussed in Chapter 7. Chapter 8 is devoted to a study ofthe scattering matrix and of the scattering cross section. Here some asymptoticmethods, such as the ray expansion and eikonal expansion, are also discussed.
As an example of a useful interaction of abstract and analytic methods, wemention the theory of the spectral shift function. Abstract results are illustrated in§3.8. However, specific properties of this function are studied by concrete methodsin §4.5, §5.3 and in Chapter 9. Here perturbation determinants are also discussedand trace identities are derived.
Note that Chapters 1 and 3 and large parts of Chapters 4 and 5 contain essen-tially a “necessary minimum” on scattering theory, whereas the other chapters areof a slightly more special nature.
The book is mainly devoted to a study of perturbations by differential operatorswith short-range coefficients. Nevertheless, basic results on long-range scattering,in particular, properties of the scattering matrix, can be found in Chapter 10.
We mention that the recent progress in scattering theory is to a large extentrelated to multiparticle systems. This very interesting and difficult problem isdiscussed in [16] and [61].
Similarly to [I], in working on the book the author has tried to resolve twoopposite problems. The first of them is a systematic exposition of the materialstarting from the general background of [I]. The second problem is the expositionof a number of topics to a degree of completeness which might possibly be of interestto experts in spectral theory. We have also tried to fill in numerous gaps presentin monographic literature. This pertains especially to the exposition of works ofRussian and, in particular, Saint Petersburg mathematicians. Compared to [I], theauthor’s tastes are also more thoroughly represented here. As a whole the book isoriented toward a reader (for example, a graduate student in mathematical physics)interested in a deeper study of scattering theory.
In references we use the “three-stage” enumeration of formulas and theoremsand the “two-stage” enumeration of sections. However, the first number is omittedwithin a chapter.
This book is based on the graduate courses taught by the author several timesin Saint-Petersburg and Rennes Universities.
The concept and structure of the entire book, as well as many specific ques-tions, were discussed with the author’s teacher M. Sh. Birman. To a large extent,mathematical tastes of the author were influenced by L. D. Faddeev. The authoris deeply grateful to M. Sh. Birman and L. D. Faddeev. Numerous discussionswith P. Deift, A. B. Pushnitski, G. Raikov and M. Z. Solomyak are also gratefullyacknowledged.
PREFACE xiii
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Bibliography
[I] D. R. Yafaev, Mathematical scattering theory: General theory, Amer. Math. Soc., Provi-dence, Rhode Island, 1992.
Monographs
[1] Z. S. Agranovich and V. A. Marchenko, The inverse problem of scattering theory, Gordonand Breach, New York, 1963.
[2] N. I. Akhieser and I. M. Glasman, The theory of linear operators in Hilbert space, vols. I,II, Ungar, New York, 1961.
[3] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable models in quantum me-chanics, Springer-Verlag, New York, 1988.
[4] W. O. Amrein, Nonrelativistic quantum mechanics, Reidel, Doldrecht, 1981.[5] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, C0-groups, commutator methods and
spectral theory for N-body Hamiltonians, Progress in Math. Physics. Press, 135, Birkhauser,1996.
[6] W. O. Amrein, J. M. Jauch and K. B. Sinha, Scattering theory in quantum mechanics,Benjamin, New York, 1977.
[7] H. Bateman, A. Erdelyi, Higher transcendental functions, vols. 1, 2, McGraw-Hill, NewYork, 1953.
[8] H. Baumgartel and M. Wollenberg, Mathematical scattering theory, Akademie-Verlag,Berlin, 1983.
[9] M. Ben-Artzi and A. Devinatz, The limiting absorption principle for partial differentialoperators, Memoirs AMS. N 364, 1987.
[10] F. A. Berezin and M. A. Shubin, The Schrodinger equation, Kluwer Academic Pub., 1991.[11] M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une variete riemannienne, Lecture
Notes Math. 194, Springer-Verlag, 1971.[12] M. Sh. Birman and M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space,
Reidel, Doldrecht, 1987.[13] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill,
New York, 1955.[14] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrodinger operators, Texts and Monographs
in Physics, Springer-Verlag, Berlin, Heidelberg, New York, 1987.
[15] P. Deift, Classical scattering theory with a trace class condition, PhD thesis, Princeton, NewJersey, 1979.
[16] J. Derezinski and C. Gerard, Scattering theory of classical and quantum N particle systems,Springer-Verlag, 1997.
[17] N. Dunford and J. T. Schwartz, Linear operators, parts 1, 2, Interscience Publishers, NewYork, London, Sydney, 1963.
[18] L. D. Faddeev, Mathematical aspects of the three body problem in quantum scattering theory,Trudy Mat. Inst. Steklov 69, 1963 (Russian); English transl.: Israel Program of Sci. Transl.,1965.
[19] L. D. Faddeev and S. P. Merkur’ev, Quantum scattering theory for several particles systems,MPAM No. 11, Kluwer Academic Press Publishers, 1993.
[20] L. D. Faddeev and O. A. Yakubovski, Lectures on quantum mechanics for mathematicsstudents, Izdat. Leningrad. Univ., Leningrad, 1980 (Russian).
[21] K. Friedrichs, Perturbation of spectra in Hilbert space, Amer. Math. Soc., Providence, RhodeIsland, 1965.
429
430 BIBLIOGRAPHY
[22] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential oper-ators, Moscow, Fizmatgiz, 1963 (Russian); English transl.: Israel Program of Sci. Transl.,1965.
[23] I. C. Gokhberg and M. G. Kreın, Introduction to the theory of linear nonselfadjoint operatorsin Hilbert space, Amer. Math. Soc., Providence, Rhode Island, 1970.
[24] A. Grigis and J. Sjostrand Microlocal analysis for differential operators, Cambridge Univ.Press, 1994.
[25] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Inc., Englewood Cliffs, NewYork, 1962.
[26] L. Hormander, The analysis of linear partial differential operators, vols. I, II, III, IV,Springer-Verlag, Berlin, Heidelberg, New York, 1985.
[27] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, Heidelberg, NewYork, 1984.
[28] S. G. Kreın (editor), Functional analysis, Nauka, 1972 (Russian).[29] S. T. Kuroda, An introduction to scattering theory, Lect. Notes Series No. 51, Aarhus
University, 1978.[30] L. D. Landau and E. M. Lifshitz, Classical mechanics, Pergamon Press, 1960.[31] L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1965.[32] V. A. Marchenko, Sturm-Liouville operators and applications, Birkhauser, Basel, 1986.[33] V. G. Maz’ya, S. L. Sobolev spaces, Leningrad State University, 1965 (Russian).[34] J. Milnor, Topology from the differential viewpoint, The University Press of Virginia, Char-
lottesville, 1965.[35] J. Milnor, Singular points of complex hypersurfaces, Ann. Math. Studies 61, Princeton Univ.
Press, 1968.[36] I. P. Natanson, Theory of functions of a real variable, Ungar, New York, vols. 1,2, 1955,
1961.[37] R. Newton, Scattering theory of waves and particles, Springer-Verlag, 1982.[38] R. Newton, Inverse Schrodinger scattering in three dimensions, Springer-Verlag, 1989.[39] D. Pearson, Quantum scattering and spectral theory, Academic Press, London, 1988.[40] P. A. Perry, Scattering theory by the Enss method, Harwood: Math. Reports, vol. 1, 1983.
[41] I. I. Privalov, Boundary values of analytic functions, GITTL, Moscow, A950; Germantransl., VEB Deutscher Verlag Wiss., Berlin, 1956.
[42] C. R. Putnam, Commutator properties of Hilbert space operators and related topics,Springer-Verlag, Berlin, Heidelberg, New York, 1967.
[43] M. Reed and B. Simon, Methods of modern mathematical physics, vols. 1, 2, 3, 4, AcademicPress, San Diego, CA, 1972, 1975, 1979, 1978.
[44] F. Rellich and J. Berkowitz, Perturbation theory of eigenvalue problems, Gordon and Breach,New York, 1969.
[45] F. Riesz and B. Sz.-Nagy, Functional analysis, Unger, New York, 1955.[46] D. Robert, Autour de l’approximation semi-classique, Progress in Math. 68, Birkhauser,
1987.[47] Y. Saito, Spectral representation for Schrodinger operators with long-range potentials, Lec-
ture Notes in Math. 727, Springer-Verlag, 1979.[48] M. A. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, 1987.[49] B. Simon, Trace ideal methods, London Math. Soc. Lecture Notes, Cambridge Univ. Press,
London and New York, 1979.[50] V. I. Smirnov, A course of higher mathematics, vol. 5, Pergamon Press, Oxford and Addison-
Wesley, Reading, Ma., 1964.[51] A. Sommerfeld, Partial differential equations in physics, Cambridge, New York, 1949.[52] M. Spivak, Calculus on manifolds, W. A. Benjamin, Inc., New York, 1965.[53] M. H. Stone, Linear transformations in Hilbert space, Amer. Math. Soc. Colloq. Publ., vol.
15, 1932.
[54] B. Sz.-Nagy and Foias, Harmonic analysis of operators on Hilbert space, North-Holland,Amsterdam, 1971.
[55] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equa-tions, vol. 1, Oxford, 1946.
[56] H. Triebel, Interpolation theory, functional spaces, differential operators, North-Holland,1978.
[57] B. R. Vaınberg, Asymptotic methods in equations of mathematical physics, Gordon andBreach Sci. Publ, New York, 1988.
BIBLIOGRAPHY 431
[58] S. Vu Ngog, Systemes integrables semi-classique: du local au global, Panoramas et Syntheses22, SMF, 2006.
[59] R. Weder, Spectral and scattering theory for wave propagation in perturbed stratified media,Appl. Math. Sci. 87, Springer-Verlag, 1991.
[60] C. H. Wilcox, Sound propagation in stratified fluids, Springer-Verlag, 1984.[61] D. R. Yafaev, Scattering theory: some old and new problems, Lecture Notes Math. 1735,
Springer-Verlag, 2000.
[62] K. Yosida, Functional analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1966.
Papers
[63] D. R. Adams, Traces of potentials arising from translation invariant operators, Ann. ScuolaNorm. Sup. Pisa (III) 25 no. 1 (1971), 203-217.
[64] S. Agmon, Spectral properties of Schrodinger operators, In: Proc. Intern. Cong. Math. of1970, Nice, vol. 2, 679-684, Gauthier-Villars, Paris, 1971.
[65] S. Agmon, Spectral properties of Schrodinger operators and scattering theory, Ann. ScuolaNorm. Sup. Pisa (IV) 2 no. 4 (1975), 151-218.
[66] S. Agmon, Some new results in spectral and scattering theory of differential operators inRn, Seminaire Goulaouic Schwartz, Ecole Polytechnique, 1978.
[67] S. Agmon and H. Hormander, Asymptotic properties of solutions of differential equations
with simple characteristics, J. Anal. Math. 30 (1976), 1-38.[68] S. Agmon and Y. Kannai, On the asymptotic bahavior of spectral functions and resolvent
kernels of elliptic operators, Israel J. Math. 5 (1967), 1-30.[69] P. Alsholm and G. Schmidt, Spectral and scattering theory for Schrodinger operators, Arch.
Rat. Mech. Anal. 40 (1971), 281-311.[70] W. O. Amrein and D. B. Pearson, A time-dependent approach to the total cross section, J.
Phys. A.: Math. Gen. 12 (1979), 1469-1492.[71] G. S. S. Avila, Spectral resolution of differential operators associated with symmetric hyper-
bolic systems, Applicable Analysis 1 (1972), 283-299.[72] J. Avron, I. Herbst and B. Simon, Schrodinger operators with magnetic fields. I. General
interactions, Duke Math. J. 45 (1978), 847-883.[73] V. M. Babich and Yu. O. Rapoport, The short-time asymptotic of the fundamental solution
of Cauchy problem for parabolic equation of second order, Problemy Mat. Fiz. 7 (1974),21-38, Izdatel’stvo LGU (Russian).
[74] V. Bargmann, On the connection between phase shifts and the scattering potential, Rev.Modern Phys. 21 (1949), 488-493.
[75] A. L. Belopolskii, M. Sh. Birman, Existence of wave operators in scattering theory for apair of spaces, Math. USSR-Izv. 2 (1968), 1117-1130.
[76] M. Ben-Artzi and S. Klainerman, Decay and regilarity for the Schrodinger equation, J. Anal.Math. 58 (1992), 25-37.
[77] F. A. Berezin and L. D. Faddeev, A remark on Schrodinger’s equation with a singularpotential, Soviet Math. Dokl. 2 (1961), 372-375.
[78] M. Sh. Birman, On the spectrum of singular boundary-value problems, Matem. Sb. 55, no.
2 (1961), 125-174 (Russian); English transl.: Eleven Papers on Analysis, Amer. Math. Soc.Transl. (2), vol. 53, Amer. Math. Soc., Providence, Rhode Island, 1966, 23-60.
[79] M. Sh. Birman, Perturbation of the continuous spectrum of a singular elliptic operatorunder the change of the boundary and boundary conditions, Vestnik Leningrad Univ. Math.1 (1962), 22-55 (Russian); English transl.: Adv. Math. Sci. ser. 2, AMS 225 (2008), 19-53.
[80] M. Sh. Birman, Existence conditions for wave operators, Izv. Akad. Nauk SSSR, Ser. Mat.27 no. 4 (1963), 883-906 (Russian).
[81] M. Sh. Birman, A local criterion for the existence of wave operators, Soviet Math. Dokl. 5no. 2 (1965), 1505-1509.
[82] M. Sh. Birman, A local test for the existence of wave operators, Math. USSR-Izv. 2 no. 2(1968), 879-906.
[83] M. Sh. Birman, Scattering problems for differential operators with constant coefficients,Funct. Anal. Appl. 3 no. 3 (1969), 167-180.
[84] M. Sh. Birman, Some applications of a local condition for the existence of wave operators,Soviet Math. Dokl. 10 (1969), 393-397.
[85] M. Sh. Birman, A test of the existence of complete wave operators in scattering theory fora pair of spaces, Topics in Math. Phys. 4 (1971), 17-21, Consultants Bureau, New York-London.
432 BIBLIOGRAPHY
[86] M. Sh. Birman, Scattering problems for differential operators with perturbation of the space,Math. USSR-Izv. 5 (1971), 459-474.
[87] M. Sh. Birman and S. B. Entina, Stationary approach in abstract scattering theory, Math.USSR-Izv. 1 no. 1 (1967), 391-420.
[88] M. Sh. Birman and M. G. Kreın, On the theory of wave operators and scattering operators,Soviet Math. Dokl. 3 (1962), 740-744.
[89] M. Sh. Birman and M. G. Kreın, Some topics of the theory of wave and scattering operators,
Outlines Joint Soviet-Amer. Sympos. Part. Diff. Equations, Novosibirsk, 1963, 39-45.[90] M. Sh. Birman and A. B. Pushnitski, Spectral shift function, amazing and multifaceted, Int.
Eq. Op. Theory 30 (1998), 191-199.[91] M. Sh. Birman and M. Z. Solomyak, Remarks on the spectral shift function, Zap. Nauchn.
Sem. LOMI 27 (1972), 33-46; English. transl.: J. Soviet Math., 3:4 (1975).[92] M. Sh. Birman and M. Z. Solomyak, Estimates for the singular numbers of integral operators,
Russian Math. Surveys 32 (1977), 15-89.[93] M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodiffer-
ential operators with anisotropically homogeneous symbols, I, II, Vestnik Leningrad Univ.Math. 10 (1980), 237-247 and 12 (1982), 155-161.
[94] M. Sh. Birman and D. R. Yafaev, The asymptotic behavior of the spectrum of the scatteringmatrix, J. Soviet Math. 25 (1984), 793-814.
[95] M. Sh. Birman and D. R. Yafaev, A general scheme in the stationary theory of scattering,Problemy Mat. Fiz. 12 (1987), 89-117. English trasl., Amer. Math. Soc. Transl. (Ser.2) 157(1993), 87-112.
[96] M. Sh. Birman and D. R. Yafaev, On the trace-class method in potential scattering theory,J. Soviet Math. 56 no. 2 (1991), 2285-2299.
[97] M. Sh. Birman and D. R. Yafaev, The spectral shift function. The papers of M. G. Kreınand their further development, St. Petersburg Math. J. 4, no. 5 (1993), 833-870.
[98] M. Sh. Birman and D. R. Yafaev, Spectral properties of the scattering matrix, St. PetersburgMath. J. 4, no. 6 (1993), 1055-1079.
[99] M. Sh. Birman and D. R. Yafaev, The scattering matrix for a perturbation of a periodicSchrodinger operator by decreasing potential, St. Petersburg Math. J. 6, no. 3 (1995), 453-
474.[100] D. Bolle, F. Gesztesy and W. Schweiger, Scattering theory for long-range systems at thresh-
old, J. Math. Phys. 26 (1985), 1661-1674.[101] D. Bolle, F. Gesztesy and S. Wilk, A complete treatment of low-energy scattering in one
dimension, J. Operator Theory 13 (1985), 3-31.[102] D. Bolle, F. Gesztesy, C. Danneels and S. Wilk, Threshold behavior and Levinson’s theorem
for two-dimension scattering systems: a surprise, Phys. Rev. Letters 56 N 9 (1986), 900-903.[103] M. Born, Quantenmechanik des Stossvorgange, Z. Phys. 38 (1926), 803-827.[104] J.-M. Bouclet, Trace formulae for relatively Hilbert-Schmidt perturbations, Asympt. Anal.
32 (2002), 1-27.[105] L. de Branges, Perturbations of self-adjoint transformations, American J. Math. 84 (1962),
543-560.[106] Ph. Briet, J. M. Combes, P. Duclos, Spectral stability under tunneling, Comm. Math. Phys.
126 (1989), 133-156.[107] V. S. Buslaev, Trace formulas and some asymptotic estimates of the resolvent kernel of the
Schrodinger operator in dimension three, Topics in Math. Physics 1 (1966), Plenum Press,New York.
[108] V. S. Buslaev, On the asymptotic behavior of the spectral characteristics of exterior problemsfor the Schrodinger operator, Math. USSR-Izv. 9, no. 1, (1975), 139-223.
[109] V. S. Buslaev and L. D. Faddeev, Formulas for traces for a singular Sturm-Liouville differ-ential operator, Soviet Math. Dokl. 1 (1960), 451-454.
[110] V. S. Buslaev and V. B. Matveev, Wave operators for the Schrodinger equation with a
slowly decreasing potential, Theor. Math. Phys. 2 (1970), 266-274.[111] A. P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math.
24 (1964), 113-190.[112] R. W. Carey and J. D. Pincus, Commutators, symbols and determining functions, J. Funct.
Anal. 19 (1975), 50-80.[113] Y. Colin de Verdiere, Une formule de traces pour l’operateur de Schrodinger dans R3, Ann.
Scient. Ec. Norm. Sup. 14 (1981), 27-39.
BIBLIOGRAPHY 433
[114] J. M. Combes, P. D. Hislop and Shu Nakamura, The Lp-theory of the spectral shift func-tion, the Wegner estimate, and the integrated density of states for some random operators,Comm. Math. Phys. 44 (2001), 113-130.
[115] J. M. Combes and R. Weder, New criteria for existence and completeness of wave operatorsand applications to scattering by unbounded obstacles, Comm. PDE 6 (1981), 1179-1223.
[116] P. Constantin, Scattering for Schrodinger operators in a class of domains with noncompactboundaries, J. Funct. Anal. 44 (1981), 87-119.
[117] J. M. Cook, Convergence to the Møller wave matrix, J. Math. Phys. 36 (1957), 82-87.[118] E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics
on the left and right, Comm. Math. Phys. 63 (1978), 277-301.[119] V. G. Deich, Applications of the method of trace-class perturbations in scattering theory in
a couple of spaces, Izvestiya VUZOV, matem., 109 (1971), no. 6, 33-42 (Russian).[120] V. G. Deich, Completeness of wave operators for systems with uniform propagation, Zapiski
Nauchn. Seminarov LOMI, 22 (1971), 36-46 (Russian).[121] V. G. Deich, E. L. Korotyaev and D. R. Yafaev, Potential scattering with spatial anisotropy
taken into account, Soviet Math. Dokl. 235 (1977), 749-752.[122] V. G. Deich, E. L. Korotyaev and D. R. Yafaev, Theory of potential scattering taking into
account spatial anisotropy, J. Soviet Math. 34 (1986), 2040-2050.[123] P. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267-310.[124] P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schro-
dinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), 341-347.[125] P. Deift and B. Simon, On the decoupling of finite singularities from the question of asymp-
totoc completeness in two-body quantum systems, J. Funct. Anal. 23 (1976), 218-238.[126] J. Dollard, Asymptotic convergence and Coulomb interaction, J. Math. Phys. 5 (1964),
723-738.[127] V. N. Efimov, Weakly coupled states of three resonant interacting particles, Nuclear Phys.
12 (1970), 1080-1091 (Russian).[128] D. M. Eidus, On the principle of limiting absorption, Mat. Sb. 57 (1962), 13-44; Engl.
transl.: Amer. Math. Soc. transl. (2) 47 (1965), 157-191.[129] D. M. Eidus, The principle of limit amplitude, Russian Math. Surveys 24, no. 3 (1969),
97-169.[130] D. M. Eidus and A. A. Vinnik, The radiation conditions for domains with infinite bound-
aries, Dokl. Akad. Nauk SSSR 214, no. 2 (1974), 19-21 (Russian).[131] V. Enss, Quantum scattering theory for two- and three-body systems with potentials of short
and long range, In: Schrodinger operators, S. Graffi, ed., Springer Lecture Notes in Math.1159 (1985), 39-176.
[132] V. Enss, Long-range scattering of two- and three-body quantum systems, In: Journees EDP,Saint Jean de Monts (1989), 1-31.
[133] V. Enss and B. Simon, Finite total cross sections in nonrelativistic quantum mechanics,Comm. Math. Phys. 76 (1980), 177-209.
[134] L. D. Faddeev, An expression for the trace of the difference between two singular differentialoperators of Sturm-Liouville type, Dokl. AN SSSR 115 N 5 (1957), 878-881 (Russian).
[135] L. D. Faddeev, On the Friedrichs model in the theory of perturbations of the continuousspectrum, Amer. Math. Soc. Transl. (Ser. 2) 62 (1967), 177-203.
[136] L. D. Faddeev, Properties of the S-matrix of the one-dimensional Schrodinger equation,Amer. Math. Soc. Transl. (Ser. 2) 65 (1967), 139-166.
[137] L. D. Faddeev, Inverse problem of quantum scattering theory. II, J. Soviet. Math. 5, 1976,334-396.
[138] L. D. Faddeev and V. E. Zakharov, Korteweg-de Vries equation is a fully integrable Hamil-tonian system, Funct. Anal. Appl. 5 (1972), 280-287.
[139] K. Friedrichs, Uber die Spektralzerlegung eines Integral-operators, Math. Ann. 115 (1938),249-272.
[140] K. Friedrichs, On the perturbation of the continuous spectra, Comm. Pure Appl. Math. 1(1948), 361-406.
[141] K. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math.7 (1954), 345-393.
[142] R. Froese and I. Herbst, A new proof of the Mourre estimate, Duke Math. J. 49 (1982),1075-1085.
434 BIBLIOGRAPHY
[143] Y. Gatel and D. R. Yafaev, On solutions of the Schrodinger equation with radiation condi-tions at infinity: the long-range case, Ann. Institut Fourier 49, no. 5 (1999), 1581-1602.
[144] I. M. Gel’fand, On identities for eigenvalues of a second order diffferential operator, Usp.Mat. Nauk 11, N 1 (1956), 191-198 (Russian); English transl.: In: Izrail M. Gelfand,Collected Papers vol. I (S. G. Gindikin, V. W. Guillemin, A. A. Kirillov, B. Kostant, S.Sternberg, eds.) Springer-Verlag, Berlin, 1987, 510-517.
[145] I. M. Gel’fand and B. M. Levitan, On a simple identity for eigenvalues of a second order
diffferential operator, Dokl. Akad. Nauk SSSR 88 (1953), 593-596 (Russian); English transl.:In: Izrail M. Gelfand, Collected Papers Vol I (S. G. Gindikin, V. W. Guillemin, A. A.Kirillov, B. Kostant, S. Sternberg, eds.) Springer-Verlag, Berlin, 1987, 457-461.
[146] M. Gell-Mann and M. L. Goldberger, The formal theory of scattering, Phys. Rev. 91 (1953),398-408.
[147] C. Gerard and A. Martinez, Principe d’absorption limite pour des operateurs de Schrodingera longue portee, C. R. Acad. Sci. Paris 306, ser. I (1988), 121-123.
[148] C. Gerard and F. Nier, Scattering theory for the perturbations of periodic Schrodinger op-erators, J. Math. Kyoto Univ. 38 (1998), 595-634.
[149] F. Gesztesy and K. Makarov, The Ξ operator and its relation to Krein’s spectral shift func-tion, J. Anal. Math. 81 (2000), 139-183.
[150] F. Gesztesy and B. Simon, The xi function, Acta Math. 176 (1996), 49-71.[151] I. Ya. Goldsheid, S. A. Molchanov and L. A. Pastur, A typical one-dimensional Schrodinger
operator has a purely point spectrum, Funct. Anal. Appl. 11, no. 1 (1977), 1-10.[152] A. Grossman and T. T. Wu, Schrodinger scattering amplitude. I, J. Math. Phys. 3 (1961),
710-713.[153] M. N. Hack, On the convergence of the Møller wave operators, Nuovo Cimento 9 (1958),
731-733.[154] W. Heisenberg, Die “beobachtbaren Grossen” in der Theorie der Elementarteilchen, Z. Phys.
120 (1943), 513-538.[155] B. Helffer and D. Robert, Calcul fonctionnel par la transformee de Mellin. J. Funct. Anal.
53 (1990), 246-268.[156] R. Hempel and R. Weder, On the completeness of wave operators under loss of local com-
pactness, J. Funct. Anal. 113 (1993), 391-412.[157] I. Herbst, On the connectedness structure of the Coulomb S-matrix, Comm. Math. Phys.
35 (1974), 181-191.
[158] I. Herbst, Spectral theory of the operator (p2+m2)1/2− Ze2
r, Comm. Math. Phys. 53 (1977),
285-294.[159] P. Hislop and S. Nakamura, Semiclassical resolvent estimates, Ann. Inst. H. Poincare, Phys.
Theor. 51 (1989), 283-299.[160] M. Hitrik and I. Polterovich, Resolvent expansions and trace regularizations for Schrodinger
operators, Contemp. Math. 327 (2003), 161-173.[161] M. Hitrik and I. Polterovich, Regularized traces and Taylor expansions for the heat semi-
group, J. London Math. Soc. 68 (2003), 402-418.[162] L. Hormander, Lower bounds at infinity for solutions of differential equations with constant
coefficients, Israel J. Math. 16 no.1 (1973), 103-116.[163] L. Hormander, The existence of wave operators in scattering theory, Math. Z. 146 (1976),
69-91.[164] J. S. Howland, Spectral theory of self-adjoint Hankel type matrices, Michigan Math. J. 33
(1986), 145-153.[165] J. S. Howland, Spectral theory of operators of Hankel type. I, II, Indiana Univ. Math. J. 41
(1992), no. 2, 409-426 and 427-434.[166] J. S. Howland and T. Kato, On a theorem of Ismagilov, J. Funct. Anal. 41 (1981), 37-39.[167] D. Hundertmark and B. Simon, An optimal Lp-bound on the Krein spectral shift function,
J. Anal. Math., 87 (2002), 199-208.[168] W. Hunziker, Potential scattering at high energies, Helv. Phys. Acta 36 (1963), 838-856.[169] W. Hunziker, The S-matrix in classical mechanics, Comm. Math. Phys. 8 (1968), 282-289.[170] T. Ikebe, Eigenfunction expansions associated with the Schrodinger operators and their ap-
plication to scattering theory, Arch. Rational Mech. Anal. 5 (1960), 1-34; Erratum, Remarkson the orthogonality of eigenfunctions for the Schrodinger operator on Rn, J. Fac. Sci. Univ.Tokyo, sect. I, 17 (1970), 355-361.
BIBLIOGRAPHY 435
[171] T. Ikebe, On the eigenfunction expansion connected with the exterior problem for theSchrodinger equation, Japan. J. Math. 36 (1967), 33-55.
[172] T. Ikebe, Spectral representation for Schrodinger operators with long-range potentials, J.Funct. Anal. 20 (1975), 158-177.
[173] T. Ikebe and H. Isozaki, Completeness of modified wave operators for long range potentials,Publ. RIMS Kyoto 15 (1979), 679-718.
[174] T. Ikebe and H. Isozaki, A stationary approach to the existence and completeness of long-
range wave operators, Int. Eq. Op. Theory 5 (1982), 18-49.[175] T. Ikebe and Y. Saito, Limiting absorption method and absolute continuity for the
Schrodinger operator, J. Math. Kyoto Univ. 12 (1972), 513-542.[176] E. M. Il’in, Scattering by unbounded obstacles for elliptic operators of second order, Proc.
of the Steklov Inst. of Math. 179 (1989), 85-107.[177] R. S. Ismagilov, On the spectrum of Toeplitz matrices, Sov. Math. Dokl. 4 (1963), 462-465.[178] H. Isozaki, On the long range potentials wave operators, Publ. RIMS Kyoto 13 (1977),
589-626.[179] H. Isozaki and H. Kitada, Micro-local resolvent estimates for 2-body Schrodinger operators,
J. Funct. Anal. 57 (1984), 270-300.[180] H. Isozaki and H. Kitada, Modified wave operators with time-independent modifies, J. Fac.
Sci, Univ. Tokyo, 32 (1985), 77-104.[181] H. Isozaki and H. Kitada, Scattering matrices for two-body Schrodinger operators, Sci.
Papers College Arts and Sci., Univ. Tokyo, 35 (1985), 81-107.[182] H. Isozaki and H. Kitada, A remark on the micro-local resolvent estimates for two-body
Schrodinger operators, Publ. RIMS, Kyoto Univ., 21 (1986), 889-910.[183] W. Jager, Ein gewohnliher Differentialoperator zweiter Ordnung fur Funktionen mit Werten
in einem Hilbertraum, Math. Z. 113 (1970), 68-98.[184] J. M. Jauch, Theory of the scattering operator, I, II, Helv. Phys. Acta 31 (1958), 127-158,
661-684.[185] A. Jensen, Spectral properties of Schrodinger operators and time decay of the wave functions.
Results in L2(Rn) n ≥ 5, Duke Math. J. 47 (1980), 57-80.[186] A. Jensen, Time delay in potential scattering: Some “geometric” results, Comm. Math.
Phys. 82 (1981), 436-456.[187] A. Jensen, Spectral properties of Schrodinger operators and time decay of the wave functions.
Results in L2(R4), J. Math. Anal. Appl. 101 (1984), 491-513.[188] A. Jensen, Propagation estimates for Schrodinger-type operators, Trans. Amer. Math. Soc.
291 (1985), 129-144.[189] A. Jensen and T. Kato, Asymptotic behavior of the scattering phase for exterior domains,
Comm. Partial Diff. Eq. 32, no. 12 (1978), 1165-1195.[190] A. Jensen and T. Kato, Spectral properties of Schrodinger operators and time decay of the
wave functions, Duke Math. J. 46 (1979), 583-611.[191] A. Jensen, E. Mourre and P. Perry, Multiple commutator estimates and resolvent smoothness
in quantum scattering theory, Ann. Inst. H. Poincare, Phys. Theor. 41 (1984), 207-225.[192] A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev.
Math. Phys. 13 (2001), 717-754.[193] A. Jensen and P. Perry, Commutator Methods and Besov Space estimates for Schrodinger
operators, J. Operator Theory 14 (1985), 181-188.
[194] R. Jost, Uber die falschen Nullstellen der Eigenwerte des S-matrix, Helv. Phys. Acta 20(1947), 250-266.
[195] R. Jost and W. Kohn, Construction of a potential from a phase shift, Phys. Rev. (2) 87(1952), 977-992.
[196] S. Kantorovitz, Cn-operational calculus, noncommutative Taylor formula and perturbationof semigroups, J. Funct. Anal. 113 (1993), 139-152.
[197] T. Kato, On finite-dimensional perturbations of self-adjoint operators, J. Math. Soc. Japan.9 (1957), 239-249.
[198] T. Kato, Perturbations of the continuous spectra by trace class operators, Proc. Japan.Acad. ser. A. Math. Sci. 33 (1957), 260-264.
[199] T. Kato, Growth properties of solutions of the reduced wave equation with variable coeffi-cients, Comm. Pure Appl. Math. 12 (1959), 403-425.
[200] T. Kato, Wave operators and unitary equivalence, Pacific J. Math. 15 (1965), 171-180.
436 BIBLIOGRAPHY
[201] T. Kato, Wave operators and similarity for some non-self-adjoint operators, Math. Ann.162 (1966), 258-279.
[202] T. Kato, Scattering theory with two Hilbert spaces, J. Funct. Anal. 1 (1967), 342-369.[203] T. Kato, Smooth operators and commutators, Studia Math. 31(1968), 535-546.[204] T. Kato, Some results on potential scattering, In: Proc. Intern. Conf. on Funct. Anal. and
Related Topics, 1969, Univ. of Tokyo Press, Tokyo, 206-215.[205] T. Kato, Monotonicity theorems in scattering theory, Hadronic J. 1(1978), 134-154.
[206] T. Kato and S. T. Kuroda, Theory of simple scattering and eigenfunctions expansions,Funct. Anal. and Related Fields (F. F. Browder, ed.), Springer, Berlin, Heidelberg, NewYork, 1970, 99-131.
[207] T. Kato and K. Masuda, Trotter’s product formula for nonlinear semigroups generated bysubdifferentials of convex functionals, J. Math. Soc. Japan. 30 (1978), 169-178.
[208] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothingeffect, Rev. Math. Phys. 1 (1989), 481-496.
[209] H. Kitada, Scattering theory for Schrodinger operators with long-range potentials, I, II, J.Math. Soc. Japan 4 (1988), 505-512.
[210] M. Klaus, Exact behavior of Jost functions at low energy, J. Math. Phys. 29 (1988), 148-154.[211] M. Klaus, Low-energy behavior of the scattering matrix for the Schrodinger equation on the
line, Inverse Problems 4 (1988), 502-512.[212] K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order
and Heisenberg’s theory of S-matrices, Amer. J. Math. 71 (1949), 921-945.[213] L. S. Koplienko, On the spectrum of the scattering matrix, Izv. Vyssh. Uchebn. Zaved. no.
5 (1971), 54-61 (Russian).[214] L. S. Koplienko, On the theory of the spectral shift function, Problemy Mat. Fiz. 5 (1971)
62-79 (Russian); Topics in Math. Phys. 5 (1972), Plenum Press, New York.[215] L. S. Koplienko, Local criteria for the existence of the spectral shift function, Zap. Nauchn.
Sem. LOMI Steklov 73, (1977) 102-117 (Russian).[216] L. S. Koplienko, The trace formula for non trace-class perturbations, Sib. Math. J. 25 (1984)
735-743.[217] E. Korotyaev and A. Pushnitski, On the high-energy asymptotics of the integrated density
of states, Bull. London Math. Soc. 35 (2003), 770-776.[218] E. L. Korotyaev and D. R. Yafaev, Traces on surfaces for function classes with dominant
mixed derivatives, J. Soviet Math. 8, no. 1 (1978), 73-86.[219] M. G. Kreın, On the trace formula in perturbation theory, Mat. Sb. 33 (1953), 597-626
(Russian).[220] M. G. Kreın, On perturbation determinants and the trace formula for unitary and selfadjoint
operators, Soviet Math. Dokl. 3 (1962), 707-710.[221] M. G. Kreın, Some new studies in the theory of perturbations of selfadjoint operators,
First Math. Summer School (Kanev, 1963) “Naukova Dumka”, Kiev, 1964, 103-187; Englishtransl.: In: M. G. Kreın, Topics in differential and integral equations and operator theory,Birkhaser, Basel, 1983, 107-172.
[222] M. G. Kreın and V. A. Yavryan, On spectral shift function arising in the perturbation of apositive operator, J. Operator Theory 6 (1981), 155-191.
[223] J. Kupsh and W. Sandhas, Møller operators for scattering on singular potentials, Comm.Math. Phys. 2 (1966), 147-154.
[224] S. T. Kuroda, On the existence and the unitarity of the scattering operator, Nuovo Cimento12 (1959), 431-454.
[225] S. T. Kuroda, Perturbation of continuous spectra by unbounded operators. I, J. Math. Soc.Japan 11 (1959), 247-262; II, J. Math. Soc. Japan 12 (1960), 243-257.
[226] S. T. Kuroda, An abstract stationary approach to the perturbation of continuous spectraand scattering theory, J. Anal. Math. 20 (1967), 57-117.
[227] S. T. Kuroda, On the Holder continuity of an integral involving Bessel functions, Quart. J.
Math. Oxford 21, no. 2 (1970), 71-81.[228] S. T. Kuroda, Some remarks on scattering theory for Schrodinger operators, J. Fac. Sci.,
Univ. Tokyo, ser. I, 17 (1970), 315-329.[229] S. T. Kuroda, Scattering theory for differential operators, J. Math. Soc. Japan 25, no. 1,2
(1973), 75-104, 222-234.[230] A. A. Kvitsinskii, Scattering by long-range potentials at low energies, Theor. Math. Phys.
59 (1984), 629-633.
BIBLIOGRAPHY 437
[231] O. A. Ladyzhenskaya, On the limiting amplitude principle, Uspekhi Mat. Nauk 12, no. 3(1957), 161-164 (Russian).
[232] O. A. Ladyzhenskaya and L. D. Faddeev, On the theory of perturbations of the continuousspectrum, Dokl. Akad. Nauk SSSR 120, no. 6 (1958), 1187-1190 (Russian).
[233] R. Lavine, Absolute continuity of Hamiltonian operators with repulsive potentials, Proc.Amer. Math. Soc. 22 (1969), 55-60.
[234] R. Lavine, Commutators and scattering theory I: Repulsive interactions, Comm. Math.
Phys. 20 (1971), 301-323; II: A class of one-body problems, Indiana Univ. Math. J. 21(1972), 643-656.
[235] R. Lavine, Completeness of the wave operators in the repulsive N-body problem, J. Math.Phys. 14 (1973), 376-379.
[236] N. Lerner and D. Yafaev, Trace theorems for pseudo-differential operators, J. Anal. Math.74 (1998), 113-164.
[237] N. Levinson, On the uniqueness of the potential in a Schrodinger equation for a givenasymptotic phase, Danske Videnskab. Selskab, Mat.-Fyz. Medd. 25, no. 9 (1949), 29 pp.
[238] N. Levinson, A simplified proof of the expansion theorem for singular second order differ-ential equations, Duke Math. J., 18 (1951), 57-71, 719-722.
[239] I. M. Lifshits, On a problem in perturbation theory, Uspehi Mat. Nauk 7, no. 1 (1952),171-180 (Russian).
[240] B. A. Lippmann and J. Schwinger, Variational principles for scattering processes, Phys.Rev. 79 (1950), 469-480.
[241] A. Melin, Trace distributions associted to the Schrodinger operator. J. Analyse Math. 59(1992), 133-160.
[242] C. Møller, General properties of the characteristic matrix in the theory of elementary par-ticles. Danske Videnskab. Selskab, Mat.-Fyz. Medd. I: 23, no. 1 (1945), 1-48; II: 22, no. 19(1946), 26 pp.
[243] E. Mourre, Absence of singular spectrum for certain self-adjoint operators, Comm. Math.Phys. 78 (1981), 391-400.
[244] E. Mourre, Operateurs conjugues et proprietes de propagation, Comm. Math. Phys. 91(1983), 279-300.
[245] M. Murata, High energy resolvent estimates I, first order operators, and II, higher orderoperators, J. Math. Soc. Japan 35 (1983), 711-733 and 36 (1984), 1-10.
[246] S. Nakamura, Low energy asymptotics for Schrodinger operators with slowly decreasingpotentials, Comm. Math. Phys. 161 (1994), 63-76.
[247] R. Newton, Noncentral potentials: The generalized Levinson theorem and the structure ofthe spectrum, J. Math. Phys. 18 (1977), 1348-1357.
[248] D. Pearson, An example in potential scattering illustrating the breakdown of asymptoticcompleteness, Comm. Math. Phys. 40 (1975), 125-146.
[249] D. Pearson, A generalization of the Birman trace theorem, J. Funct. Anal. 28 (1978), 182-186.
[250] P. Perry, I. M. Sigal and B. Simon, Spectral analysis of N-body Schrodinger operators, Ann.Math. 144 (1981), 519-567.
[251] I. Polterovich, Heat invariants of Riemannian manifolds, Israel J. Math. 119 (2000), 239-252.
[252] A. Ya. Povzner, On the expansion of arbitrary functions in the eigenfunctions of the operator−∆u+qu, Mat. Sb. 32, no. 1 (1953), 109-156 (Russian); AMS Translations, ser. 2, 60 (1967),1-49.
[253] A. Ya. Povzner, On expansion in eigenfunctions of the Schrodinger equation, Dokl. Akad.Nauk SSSR 104, no. 3 (1955), 360-363 (Russian).
[254] Yu. N. Protas, Quasiclassical asymptotics of the scattering amplitude for the scattering ofa plane wave by inhomogeneities of the medium, Math. USSR Sbornik, 45 (1983), 487-506.
[255] A. Pushnitski, Representation for the spectral shift function for perturbations of definite
sign, St. Petersburg Math. J. 9 (1998), 1181-1194.[256] A. Pushnitski, The scattering matrix and the differences of spectral projections, Bulletin
London Math. Soc., 40 (2008), 227-238.[257] A. Pushnitski and D. Yafaev, Spectral theory of discontinuous functions of self-adjoint op-
erators and scattering theory, ArXiv: 0907.1518, 21 pp.[258] M. Reed and B. Simon, The scattering of classical waves from inhomogeneous media, Math.
Z. 155 (1977), 163-180.[259] P. Rejto, On partly gentle perturbation. III, J. Math. Anal. Appl. 27 (1969), 21-67.
438 BIBLIOGRAPHY
[260] P. Rejto, Some potential perturbations of the Laplacian, Helv. Phys. Acta 44 (1971), 708-736.
[261] F. Rellich, Uber das asymptotische Verhalten der Losungen von ∆u+λu = 0 in unendlichenGebieten, Jahresbericht der Deutsch. Math. Ver. 5 (1943), 57-63.
[262] D. Robert, Asymptotique a grande energie de la phase de diffusion pour un potentiel,Asympt. Anal. 3 (1991), 301-320.
[263] D. Robert, Semi-classical approximation in quantum mechanics. A survey of old and recent
mathematical results., Helv. Phys. Acta 71, no. 1 (1998), 44-116.[264] D. Robert, Semiclassical asymptotics for the spectral shift function, Adv. Math. Sci. ser. 2,
AMS 189 (1999), 187-203.[265] D. Robert and H. Tamura, Semiclassical estimates for resolvents of Schrodinger operators
and asymptotics for scattering phases, Comm. PDE 9/10 (1984), 1017-1058.[266] D. Robert and H. Tamura, Semi-classical bounds for resolvents and asymptotics for total
scattering cross-sections, Ann. Inst. H. Poincare, Phys. Theor. 46 (1987), 415-442.[267] D. Robert and H. Tamura, Asymptotic behavior of scattering amplitudes in semi-classical
and low energy limits, Ann. Inst. Fourier 39 (1989), 155-192.[268] H. Rollnik, Streumaxima und gebundene Zustande, Z. Phys. 145 (1956), 639-653.[269] M. Rosenblum, Perturbation of the continuous spectrum and unitary equivalence, Pacific J.
Math. 7 (1957), 997-1010.[270] S. Yu. Rotfel’d, On spectral proprties of the scattering matrix, I. Izv. Vyssh. Uchebn. Zaved.
no. 3 (1972), 52-60; II. Problemy Mat. Fiz. 7 (1974), 135-149, Izdatel’stvo LGU (Russian).[271] Ph. Roux and D. Yafaev, The scattering matrix for the Schrodinger operator with a long-
range electromagnetic potential, J. Math. Phys. 44 (2003), 2762-2786.[272] Y. Saito, Eigenfunction expansions of the Schrodinger operator with long-range potentials,
Q(y) = O(|y|−ε, ε > 0, Osaka J. Math. 14 (1977), 37-53.[273] L. A. Sakhnovich, On the Ritz formula and quantum defects of the spectrum of the radial
Schrodinger equation, Amer. Math. Soc. Transl. (2) 68 (1968).[274] J. R. Schulenberger, A local compactness theorem for wave propagation problems of classical
physics, Indiana Univ. Math. J. 22 (1972), 429-432.[275] J. R. Schulenberger and C. H. Wilcox, Completeness of the wave operators for perturbations
of uniformly propagative systems, J. Funct. Anal. 7 (1971), 447-474.[276] J. R. Schulenberger and C. H. Wilcox, Coerciveness inequalities for nonelliptic systems of
partial differential equations, Ann. Mat. Pura Appl. 88 (1971), 229-305.[277] J. R. Schulenberger and C. H. Wilcox, Eigenfunction expansions and scattering theory for
wave propagation problems of classical physics, Arch. Rat. Mech. Anal. 46 (1972), 280-320.[278] R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10 (1967),
288-307.[279] N. Shenk, Eigenfunction expansions and scattering theory for the wave equation in an
exterior domain, Arch. Rat. Mech. Anal. 21 (1966), 120-150.[280] N. Shenk and D. Thoe, Outgoing solutions of (−∆+ q − k2)u = f in an exterior domain,
J. Math. Anal. Appl. 31 (1970), 81-116.[281] Yu. M. Shirokov, Strongly singular potentials in the three-dimensional quantum mechanics,
Theor. Math. Phys. 42, no. 1 (1980), 28-31.[282] Yu. M. Shirokov, Representation of free solutions for Schrodinger equations with strongly
singular concentrated potentials, Theor. Math. Phys. 46, no. 3 (1981), 191-196.[283] Y. Shizuta, Eigenfunction expansion associated with the operator −∆ in the exterior do-
main, Proc. Japan Acad. 39 (1963), 656-660.[284] B. Simon, Wave operators for classical particle scattering, Comm. Math. Phys. 23 (1971),
37-48.[285] B. Simon, The bound state of weakly coupled Schrodinger operator in one and two dimen-
sions, Ann. Phys. 97 (1976), 279-288.[286] B. Simon, Phase space analysis of simple scattering systems: extensions of some work of
Enss, Duke Math. J. 46 (1979), 119-168.[287] M. M. Skriganov, Uniform coordinate and spectral asymptotics for solutions of the scattering
problem for the Schrodinger equation, J. Soviet Math. 8 (1978), 120-141.[288] A. V. Sobolev, Quasiclassical asymptotics of the scattering cross-section for asymptotically
homogeneous potentials, J. Soviet. Math. 40 (1988), 652-672.
BIBLIOGRAPHY 439
[289] A. V. Sobolev, Efficient bounds for the spectral shift function, Ann. Inst. H. Poincare, Phys.Theor., 58 (1993), 55-83.
[290] A. V. Sobolev and D. R. Yafaev, Phase analysis in the problem of scattering by a radialpotential, J. Soviet. Math.37 (1987), 888-902.
[291] A. V. Sobolev and D. R. Yafaev, On the quasiclassical limit of the total scattering crosssection in nonrelativistic quantum mechanics, Ann. Inst. H. Poincare, Phys. Theor. 44(1986), 195-210.
[292] A. V. Sobolev and D. R. Yafaev, Spectral properties of the abstract scattering matrix, Proc.Steklov Inst. Math. 188, no. 3 (1991), 159-189.
[293] A. Sommerfeld, Die Greensche Funktion der Schwingungsgleichung, Jhrber. Deutsch. Math.-Verein 21 (1912), 309-353.
[294] W. Stinespring, A sufficient condition for an integral operator to have a trace, J. ReineAngew. Math. 200 (1958), 200-207.
[295] A. V. Suslov, About one Ismagilov’s theorem, Problemy Mat. Fiz. 9 (1979), 142-144, Izda-tel’stvo LGU (Russian); Selecta Math. Sov. 5, no. 2 (1986),137-139.
[296] D. Thoe, Eigenfunction expansions associated with Schrodinger operators in Rn, n ≥ 4,Arch. Rat. Mech. Anal. 26 (1967), 335-356.
[297] A. N. Tikhonov, A. A. Samarskii, On the radiation principle, Zh. Eksper. Teoret. Fiz. 18,no. 2 (1948), 243-248 (Russian).
[298] B. R. Vaınberg, On the short wave asymptotic behavior of solutions of stationary problemsand the asymptotic behavior as t → ∞ of solutions of non-stationary problems, RussianMath. Surveys 30 (1975), 1-58.
[299] B. R. Vaınberg, Quasi-classical approximation in stationary scattering problems, Funct.Anal. Appl. 11 (1977), 247-257.
[300] A. F. Vakulenko, A variant of commutator estimates in spectral theory, J. Soviet. Math. 49(1988), 1136-1139.
[301] V. Vogelsang, Das Ausstrahlungsproblem fur elliptische Differentialgleichungen in Gebietenmit unbeschrankten Rand, Math. Z. 144 (1975), 101-125.
[302] D. Voiculescu, On a trace formula of M. G. Kreın, Operator theory: Adv. and Appl. 24(1987), 329-332.
[303] X. P. Wang, Time decay of scattering solutions and classical trajectories, Ann. Inst. H.Poincare, Phys. Theor. 47 (1987), 25-37.
[304] R. Weder, Spectral analysis of strongly propagative systems, J. Reine und Angewandte Math-ematik 354 (1984), 95-122.
[305] R. Weder, Analyticity of the scattering matrix for wave propagation in crystals, J. Math.Pures and Appl. 64 (1985), 121-148.
[306] P. Werner, Randwertprobleme de mathematischen Akustik, Arch. Rat. Mech. Anal. 10(1962), 29-66.
[307] H. Weyl, Uber gewohnliche lineare Differentialgleichungen mit singularen Stellen und ihreEigenfunktionen, Nachr. Ges. Wiss. Gottingen (1909), 37-64.
[308] H. Weyl, Uber gewohnliche Differentialgleichungen mit Singularitatten und die zugehorigenEntwicklungen willkurlicher Funktionen, Math. Ann. 68 (1910), 220-269.
[309] J. A. Wheeler, On the mathematical description of light nuclei by the method of resonatinggroup structure, Phys. Rev. 52 (1937), 1107-1122.
[310] C. H. Wilcox, Wave operators and asymptotic solutions of wave propagation problems inclassical physics, Arch. Rat. Mech. Anal. 22 (1966), 37-78.
[311] D. R. Yafaev, Trace formulas for charged particles in nonrelativistic quantum mechanics,Teor. Mat. Fiz. 11 (1972), 78-92 (Russian).
[312] D. R. Yafaev, Remark on scattering theory for a perturbed polyharmonic operator, Math.Notes 15 (1974), 260-265.
[313] D. R. Yafaev, On the theorie of the discrete spectrum of the three-particle Schrodingeroperator, Math. USSR Sbornik 23 (1974), 535-559.
[314] D. R. Yafaev, The virtual level of the Schrodinger equation, Zap. Nauchn. Sem. LOMI 51(1975), 203-216; English transl.: J. Sov. Math. 11 (1979), 501-510.
[315] D. R. Yafaev, On the break-down of completeness of wave operators in potential scattering,
Comm. Math. Phys. 65 (1979), 167-179.[316] D. R. Yafaev, Wave operators for the Schrodinger equation, Theor. Math. Phys. 45, no. 2
(1981), 992-998.[317] D. R. Yafaev, Spectral properties of the Schrodinger operator with a potential having a slow
falloff, Funct. Anal. Appl. 16 (1982), 280-286.
440 BIBLIOGRAPHY
[318] D. R. Yafaev, The low energy scattering for slowly decreasing potentials, Comm. Math.Phys. 85 (1982), 177-196.
[319] D. R. Yafaev, Scattering theory for time-dependent zero-range potentials, Ann. Inst. H.Poincare, Phys. Theor. 40 (1984), 343-359.
[320] D. R. Yafaev, Remarks on the spectral theory for the multiparticle type Schrodinger operator,J. Soviet Math. 31 (1985), 3445-3459.
[321] D. R. Yafaev, On a resonant scattering on a negative potential, J. Soviet Math. 32 (1986),
549-556.[322] D. R. Yafaev, The eikonal approximation and the asymptotics of the total scattering cross
section for the Schrodinger equation, Ann. Inst. H. Poincare, Phys. Theor. 44 (1986), 397-425.
[323] D. R. Yafaev, The eikonal approximation for the Schrodinger equation, Proc. Steklov Inst.Math. 179, no. 2 (1988), 251-266.
[324] D. R. Yafaev, The quasiclassical asymptotics of the scattering cross section for theSchrodinger equation, Math. USSR-Izv. 32 (1989).
[325] D. R. Yafaev, Eikonal approximation for fast-decreasing potentials, J. Soviet Math. 49(1990), 1225-1236.
[326] D. R. Yafaev, On the asymptotics of the scattering phases for the Schrodinger equation,Ann. Inst. H. Poincare, Phys. Theor. 53 (1990), 283-299.
[327] D. R. Yafaev, On solutions of the Schrodinger equation with radiation conditions at infinity,Adv. Sov. Math., AMS 7 (1991), 179-204.
[328] D. R. Yafaev, On the scattering matrix for perturbations of constant sign, Ann. Inst. H.Poincare, Phys. Theor. 57 (1992), 361-384.
[329] D. R. Yafaev, On a zero-range interaction of a quantum particle with the vacuum, J. Phys.A: Math. Gen. 25 (1992), 963-978.
[330] D. R. Yafaev, Radiation conditions and scattering theory for N-particle Hamiltonians,Comm. Math. Phys. 154 (1993), 523-554.
[331] D. R. Yafaev, New channels in the two-body long-range scattering, St. Petersburg Math. J.8 (1997), 165-182.
[332] D. R. Yafaev, On the classical and quantum Coulomb scattering, J. Phys. A: Math. Gen.
30 (1997), 6981-6992.[333] D. R. Yafaev, The scattering amplitude for the Schrodinger equation with a long-range
potential, Comm. Math. Phys. 191 (1998), 183-218.[334] D. R. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal. 168 (1999),
121-144.[335] D. R. Yafaev, A class of pseudo-differential operators with oscillating symbols, St. Peters-
burg Math. J. 11 (2000), 375-403.[336] D. R. Yafaev, High energy asymptotics of the scattering amplitude for the Schrodinger
equation, Proc. Indian Acad. Sci., Math. Sci., 112 (2002), 245-255.[337] D. R. Yafaev, High energy and smoothness asymptotic expansion of the scattering amplitude,
J. Funct. Anal., 202 (2003), 526-570.[338] D. R. Yafaev, Trace-class approach in scattering problems for perturbations of media, Proc.
of the conference on spectral theory in Sinaya, Roumanie, Adv. in operator algebras andmath. physics, 275-285, Theta, 2005.
[339] D. R. Yafaev, A trace formula for the Dirac operator, Bulletin London Math. Soc., 37(2005), 908-918.
[340] D. R. Yafaev, Perturbation determinants, the spectral shift function, trace identities, andall that, Funct. Anal. and Appl., 41 (2007), 217-236.
[341] D. R. Yafaev, Spectral and scattering theory of fourth order differential operators, Advancesin the Math. Sciences, ser. 2, 225 (2008) AMS, 265-299.
[342] K. Yajima, The quasi-classical limit of scattering amplitude - finite range potentials, LectureNotes in Math. 1159, Springer-Verlag (1984), 242-263.
[343] K. Yajima, The quasi-classical limit of scattering amplitude. L2-approach for short rangepotentials, Japan J. Math. 13 (1987), 77-126.
[344] V. A. Yavryan, On some perturbations of self-adjoint operators, Dokl. Akad. Nauk Arm.SSR 38, no. 1 (1964), 3-7 (Russian).
[345] C. Zemach and A. Klein, The Born approximation in nonrelativistic quantum theory. I,Nuovo Cimento 10 (1958), 1078-1087.
Index
Absence of positive eigenvalues, 231, 399
Absolutely continuous subspace, 19Admissible functions, 28Agmon-Hormander spaces, 235Analytic Fredholm alternative, 33
Asymptotic completeness, see also Wave op-erators
Averaged scattering cross section, 331high-energy asymptotics, 337
semiclassical asymptotics, 336universal upper bounds, 331
Birman-Kato-Kreın theorem, 42Birman-Kreın formula, 47Boundary values of the resolvent of a self-
adjoint operator, 44Boundedness of integral operators, 68
Break-down of completeness, 91additional channels of scattering, 91
Cauchy integral, 358Compact operators, 20
singular numbers, 20
Completeness of wave operators, see alsoAsymptotic completeness
Complex interpolation, 22Hadamard three-line theorem, 22three-line theorem for operator-valued
functions in Sp, 22
Conditions of self-adjointness, 24Cook’s criterion, 29
Determinant, 21regularized determinant, 21
Diagonalization of a self-adjoint operator, 20
Differential operators, 53, 136with constant coefficients, 53, 136
perturbations, 54, 136spectral analysis, 54, 109
Dirac operator, 106Direct integral, 19
Eikonal approximation, 319Exceptional set N , 33Expansion theorem, 248
a generalized Fourier integral, 255
relation to the wave operators, 248standing waves, 250the diagonalization of the Hamiltonian,
245
Free Hamiltonian, 75resolvent, 78
spectral representation, 75unitary group, 76
Friedrichs’ extension, 25
Generalized Fourier transform, 170in the one-dimensional problem, 214
relation with wave operators, 215
on half-line, 170relation with wave operators, 175
Hamiltonian of a relativistic spinless parti-cle, 104
Hardy-Rellich inequalities, 68High-energy asymptotic expansion of the re-
solvent, 277
away from the spectrum, 277in the whole complex plane, 361
High-energy asymptotics of scattering data,188
in the one-dimensional problem, 219on half-line, 188
High-energy asymptotics of the spectralshift function, see also spectral shiftfunction
High-energy estimates of the resolvent, 268the free case, 268
Hilbert identity, 17
Homogeneous Schrodinger equation, 241an exhaustive description of all solutions,
241
Integral operators, 133from Schatten-von Neumann classes, 133from trace class, 133
Intertwining property, 173Invariance principle, 29
for perturbations of trace class type, 43
441
442 INDEX
Jost function, 165, 210
in the one-dimensional problem, 210
regularity properties, 165
Jost solution, 165
regularity properties, 163
Kato-Rosenblum theorem, 42
Laplace transform, 18
Large time local decay of solutions of theSchrodinger equation, 290
Levinson’s formulas, 198, 222, 348
Limit amplitude, 166
Limit phase, 166
Limiting absorption principle, 236, 400
efficient form, 400
commutator method, 400
in Schatten-von Neumann classes, 271
the sharp form, 238
the free case, 236
Lippmann-Schwinger equation, 252Local asymptotic expansion of the parabolic
Green function, 280Laplace transform, 283
Local theory of trace class perturbations, 43
Long-range potentials, 380
diagonal singularity of the scattering am-plitude, 395
eikonal and transport equations, 380
scattering matrix, 394
spectrum, 394
Long-range scattering, 86
existence of modified wave operators, 86
Low-energy asymptotics, 178
in the one-dimensional problem, 216
on half-line
for slowly decaying potentials, 186
on the half-line, 178
Low-energy behavior of the resolvent, 285
Maxwell’s equations, 129
Mourre estimate, 261
Mourre method, 259
limiting absorption principle, 259
propagation or microlocal estimates, 372
Nonhomogeneous Schrodinger equation, 408
existence and uniqueness of solutions, 239
with a long-range potential, 408existence and unicity of solutions, 408
Pauli operator, 104Pearson theorem, 43
Perturbation determinant, 45
for the radial Schrodinger operator, 192
in the one-dimensional problem, 221
modified, 51
regularized, 52
Perturbed polyharmonic operator, 103, 145
wave operators, 145
completeness, 145existence, 145
Phase shift, 166Point interaction, 203
with the vacuum, 207
Propagation of classical waves, 129Propagation or microlocal estimates, 372
Propagative systems, 109of first order, 129
strongly, 109uniformly, 109
Pseudodifferential operators, 58action on the exponential function, 61
boundedness and compactness, 58elementary calculus, 59of negative order, 65
asymptotics of eigenvalues, 66on manifolds, 63
oscillating symbols, 62essential spectrum, 62
principal symbol, 61strongly Carleman, 54
symbols and amplitudes, 60with constant coefficients, 53
perturbations, 54
spectral analysis, 54, 109with homogeneous symbols, 115
Quadratic forms, 25
Radiation conditions, 233, 399Radiation estimates, 373
Ray expansion, 312Reflection coefficients, 166Regular points, 17
Regular solution, 165Regularized perturbation determinant, 341
Resolvent, 17Cauchy-Stieltjes integral, 17
on half-line, 170Resolvent identity, 23
modified, 275
Scattering amplitude, 253for long-range potentials, 385
high-energy and smoothness asymp-totics, 385
for potentials of compact support, 314
high-energy asymptotics, 314for potentials with power-like decay, 318
high-energy and smoothness asymp-totics, 318
Scattering cross section, 307universal upper bounds, 310
Scattering matrix, 38, 159, 297asymptotics of scattering phases, 302
Born series, 301eigenvalues, 39
INDEX 443
for differential operators, 159
for long-range potentials, 385
for perturbations of a definite sign, 301
in the one-dimensional problem, 213
modified, 304
on the half-line, 176
spectral properties, 297
stationary representation, 38, 297
with identifications, 40
Scattering operator, 38
for long-range potentials, 385
Scattering solutions, wave functions oreigenfunctions of the continuous spec-trum, 251
Schatten-von Neumann classes, 20
Hilbert-Schmidt class, 21
trace class, 21
Schrodinger operator, 79, 139
absence of the singular continuous spec-trum, 98
complex conjugation, 81
essential spectrum, 81
limiting absorption principle, 95
magnetic, 101
perturbations of second order, 102
scattering matrix, 96
modified, 97
spectrum, 97
stationary representations, 97
self-adjointness, 80
wave operators, 82, 139
completeness, 93, 139
completeness for anisotropic potentials,94
existence, 82, 139
existence for anisotropic potentials, 84
zero-energy resonance, 287
the resolvent singularity, 288
Slowly decreasing positive potentials, 291
a virtual shift of the continuous spectrum,291
quasiregularity of the spectral point zero,293
superpower local decay of solutions of thetime-dependent Schrodinger equation,295
Smoothness in the Kato sense, 29
local, 30
sufficient conditions of a commutatortype, 32
Sobolev spaces, 56
embedding theorems, 57
invariance with respect to diffeomor-phisms, 57
trace theorem, 73
Holder continuity of traces, 74
Spectral measure, 17
Spectral shift function, 45, 160
continuity, 351
for a trace class perturbation, 46
for differential operators, 160for general perturbations of trace class
type, 47
for perturbations of definite sign, 51
for semibounded operators, 50
for the radial Schrodinger operator, 194high-energy expansion, 356
asymptotic coefficients, 362, 364
in the one-dimensional problem, 222
Spectrum, 17absolutely continuous, 19
essential, 24
singular continuous, 19conditions for its absence, 36
Spherical waves, 242
outgoing and incoming, 242
Strong smoothness, 31Subordination of operators, 43
Time reversal invariance, 97
Time-delay, 346
Total scattering cross section, 255
Trace, 21Trace formula, 45
Trace identities, 199
for the radial Schrodinger operator, 199in the multidimensional problem, 365
in the one-dimensional problem, 222
Transmission coefficients, 213
Uniform estimates of the spectral family, 270
Uniqueness theorem under radiation condi-tions, 233
Volterra equations, 162
Wave equation in inhomogeneous media, 157scattering theory, 157
Wave function, 166
asymptotic behavior, 253in the one-dimensional problem, 213
pointwise asymptotics, 257
Wave operators, 27
completeness, 27for long-range potentials, 379
for magnetic potentials, 152
for matrix differential operators, 122
for nonelliptic differential operators, 118for perturbations of a medium, 124, 154
for scalar differential operators, 116
for second order differential operators, 147
for singular potentials, 154for strongly propagative systems, 128
for the Maxwell operators, 129
in different spaces, 27intertwining property, 27
local, 28
444 INDEX
multiplication theorem, 28perturbations of boundary conditions, 154stationary representations, 36
with identifications, 41weak, 27
Abelian, 27Weyl’s theorem on preservation of power
asymptotics of eigenvalues, 22Weyl’s theorem on preservation of the essen-
tial spectrum, 25
Zero-energy resonance, 287in multidimensional problem, 287in the one-dimensional problem, 217on half-line, 180
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SURV/158
The main subject of this book is applications of methods of scat-tering theory to differential operators, primarily the Schrödinger operator.
There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it. In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book both of these trends are presented. The first half of this book begins with the summary of the main results of the general scattering theory of the previous book by the author, Mathematical Scattering Theory: General Theory, American Mathematical Society, 1992. The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular, their applications to scattering theory of perturbations of differential operators with constant coefficients and to the analysis of the trace class method.
In the second half of the book direct methods of scattering theory for differential opera-tors are presented. After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appro-priate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal expansions.
The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in math-ematical physics).
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-158 www.ams.orgAMS on the Web
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