References

Abramowitz, M., and Stegun, I. A. (1964): Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series, Washington, D.C.

Akhiezer, N.I., and Glazman, I. M. (1961,1963): Theory of Linear Operators in Hilberl Space, Vol. I (1961), Vol. II (1963). Frederick Ungar Publishing Co., New York.

Arnol'd, V. I. (1973): Ordinary Differential Equations. M.I.T. Press, Cambridge, Mass.

Ashenhurst, R. L., and Metropolis, N. C. (1959): Unnormalized floating point arith­metic. J. Assoc. Comput. Mach., vol. 6, pp. 415-428.

Baernstein, A. (1971): Representation of holomorphic functions by boundary integrals. Trans. Amer. Math. Soc., vol. 160, pp. 27-37.

Banach, S. (1955): Theorie des Operations Lineaires. Chelsea Publishing Co., New York.

Barut, A. O. (1967): The Theory of the Scattering Matrix. Macmillan, London.

Bers, L. (1958): Mathematical Aspects of Subsonic and Transonic Gas Dynamics. John Wiley and Sons, New York.

Bethe, H. A., and Salpeter, E. E. (1957): Quantum Mechanics of One- and Two-Electron Atoms. Springer-Verlag.

Birkhoff, G. (1962): Helmholtz and Taylor instability. Proceedings of Symposia in Appl. Math., Amer. Math. Soc., vol. 92, p. 13 ff.

Birkhoff, G., and Rota, G. (1962): Ordinary Differential Equations. Ginn and Co., Waltham, Mass.

Bleakney, W., and Taub, A. H. (1949): Interaction of shock waves. Rev. Mod. Physics, vol. 21, pp. 584-605.

Carleson, L. (1966): On the convergence and growth of partial sums of Fourier series. Acta Math. vol. 116, pp. 135-157.

Coddington, E. A., and Levinson, N. (1955): Theory of Ordinary Differential Equations. McGraw-Hill, New York.

Courant, R., and Friedrichs, K. O. (1948): Supersonic Flow and Shock Waves. Inter­science, New York.

Courant, R., and Hilbert, D. (1953, 1962): Methods of Mathematical Physics, Volumes I and II. Interscience, New York.

DiPrima, R. C., and Habetler, G. J. (1969): A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability. Arch. Rat. Mech and Anal. vol. 34, pp. 218-227.

409

410 References

Dirac, P. A. M. (1930, 1935, 1947, 1958): The Principles of Quantum Mechanics, Editions 1, 2, 3, and 4. Clarendon Press, Oxford.

Donoghue, Wm. F. (1969): Distributions and Fourier Tran~forms. Acad. Press, New York.

Duff, R. E. (1962): Slip line instability. Proceedings of Symposia in Appl. Math., Amer. Math. Soc. vol. 13, p. 77ff.

Dunford, N., and Schwartz, J. T. (1958): Linear Operators Part I: General Theory. Interscience, New York.

Dym, H., and McKean, H. P. (1972): Fourier Series and Integrals. Academic Press, New York.

Fefferman, C. (1971): On the divergence of multiple Fourier series. Bull. Amer. Math. Soc. vol. 77, pp. 11j1-195.

Fefferman, C. (1971): On the convergence of multiple Fourier series. Bull. Amer. Math. Soc. vol. 77, pp. 744--745.

Feller, W. (1950, 1966): An Introduction to Probability Theory and its Applications Volumes I and II. John Wiley and Sons, New York.

Friedman, A. (1969): Partial Differential Equations. Holt, Rinehart, and Winston, New York.

Gantmacher, F. R. (1959): The Theory of Matrices, Volumes I and II. Chelsea Pub. Co., New York.

Garabedian, P. R. (1964): Partial Differential Equations. John Wiley and Sons, New York.

Gel'fand, I. M., and Shilov, G. E. (1964,1968,1967): Generalized Functions Volumes J, 2, and 3. Academic Press, New York.

GeI'fand, I. M., and Vilenkin, N. Ya. (1964): Generalized Functions Volume 4. Academic Press, New York.

Gel'fand, I. M., Graev, M. I., and Vilenkin, N. Ya. (1966): Generalized Functions Volume 5. Academic Press, New York.

Gray, H. L., and Harrison, C. (1959): Normalized floating-point arithmetic with an index of significance. Proc. Eastern Joint computer Conference.

Gross, L. (1966): The Cauchy problem for the coupled Maxwell and Dirac equations. Comm. Pure Appl. Math., vol. 19, pp. 1-15.

Gustafson, K., and Johnson, G. (1974): On the absolutely continuous subspace of a self-adjoint operator. Helv. Physica Acta, vol. 47, pp. 163-166.

Gustafson, K., and Rejto, P. A. (1973): Some essentially self-adjoint Dirac operators with spherically symmetric potentials. Israel Jour. Math., vol. 14, pp. 63-75.

Halmos, P. R. (1951): Introduction to Hilbert Space and the Theory of Spectral Multi­plicity. Chelsea Pub. Co., New York.

Hille, E. (1962): Analytic Function lheory Volume II. Ginn and Co., Boston.

Hille, E., and Phillips, R. S. (1957): Functional Analysis and Semi-groups. Amer. Math. Soc.

Hormander, L. (1969): Linear Partial Differential Operators. Springer-Verlag.

John, F. (1971): Partial Differential Equations. Springer-Verlag.

References 411

Johnson, G. (1968): Harmonic functions on the unit disc I and II. Ill. Jour. Math., vol. 12, pp. 366-396.

Jordan, P., and von Neumann, J. (1935): On inner products in linear metric spaces. Ann. of Math., vol. 36, pp. 719-723.

Jorgens, K. (1967): Zur Spektraltheorie der Schrodingeroperatoren. Math. Zeitschr., vol. 96, pp. 355-372.

Jorgens, K. (1970): Lineare Integraloperatoren. B. G. Teubner, Stuttgart. J6rgens, K., and Rellich, F. (1976): Eigenwerttheorie gewohnlicher Differential-

operatoren (bearbeitet von J. Weidmann). Springer-Verlag.

Jorgens, K., and Weidmann, J. (1973): Spectral Properties of Hamiltonian Operators. Springer-Verlag.

Jost, R. (1965): The General Theory of Quantized Fields. American Mathematical Society, Providence, RI.

Kantorovich, L. V., and Akilov, G. P. (1964): Functional Analysis in Normed Spaces. MacMillan, New York.

Kato, T. (1966): Perturbation Theory for Linear Operators. Springer-Verlag.

Kelley, J. L. (1955): General Topology. D. Van Nostrand, Princeton, NJ.

Lanczos, C. (1956): Applied Analysis. Prentice Hall, Englewood Cliffs, NJ.

Landau, L. D., and Lifshitz, E. M. (1959): Fluid Mechanics. Pergamon Press, London.

Lax, P. D. (1954): Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math., vol. 7, pp. 159-193.

Lax, P. D. (1957): Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math., vol. 10, pp. 537-566.

Lax, P. D., and Phillips, R. S. (1967): Scattering Theory. Academic Press, New York.

Lewis, G. E. (1959): Analytic Continuation Using Numerical Methods. Ph.D. Thesis, New York University; also in Methods in Computational Physics, Academic Press, New York, vol. 4, pp. 45-81 (1965).

MacDuffee, C. C. (1946): The Theory of Matrices. Chelsea Pub. Co., New York.

Magnus, W., and Oberhettinger, F. (1943): Formulas and Theorems for the Functions of Mathematical Physics. Chelsea Pub. Co., New York.

Meisters, G. H. (1971): Translation-invariant linear forms and a formula for the Dirac measure. J. Functional Anal., vol. 8, pp. 173-188.

Morse, P. M., and Feshbach, H. (1953): Methods of Theoretical Physics, Volumes I and II. McGraw-Hill, New York.

Natanson, I. P. (1955): Theory of Functions of a Real Variable. Frederick Ungar, New York.

Neumann, J. von (1929): Allgemeine Eigenwerttheorie Hermitescher Funktional­operatoren. Math. Annalen, vol. 102, pp. 49-131.

Neumann, J. von (1931): Die Eindeutigkeit der Schrodingerschen Operatoren. Math. Annalen, vol. 104, pp. 570-578.

Rejto, P. A. (1971): On reducing subspaces for one-electron Dirac operators. Israel J. Math., vol. 9, pp. 111-143.

Richtmyer, R. D. (1957): Detached shock calculations by power series. Report NYO-7973, Courant Institute, New York University; also in Annals of the New York Academy of SCiences, vol. 86, pp. 828-842.

412 References

Richtmyer, R. D. (1960): Flow diagrams and the estimation of significance. Report TID 6199, New York University.

Richtmyer, R. D., and Morton, K. W. (1967): Difference Methods for Initial-Value Problems. Wiley-Interscience, New York.

Riesz, F., and Sz. Nagy, B. (1953): Ler;ons d' Analyse Fonctionelle. Akademiai Kiad6, Budapest.

Roos, B. W., and Sangren, W. C. (1962): Spectral theory of Dirac's radial relativistic wave equation. l. Math. Phys., vol. 3, pp. 882-890.

Sattinger, D. H. (1970): The mathematical problem of hydrodynamic stability. lour. Math. and Mech., vol. 19, pp. 797-817.

Schwartz, L. (1950, 1951): Theorie des Distributions, Tomes I et II. Hermann C ie,

Paris.

Sobolev, S. L. (1963): Applications of Functional Analysis in Mathematical Physics. Amer. Math. Soc. Translations.

Solovay, R. M. (1970): A model of set theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2), vol. 92, pp. 1-56.

Spanier, J., and Gelbard, E. M. (1969): Monte Carlo Principles and Neutron Transport Problems. Addison Wesley, Reading, Mass.

Stone, M. H. (1932): Linear Transformations in Hilbert Space and Their Applications to Analysis. American Mathematical Society, Providence, RI.

Taylor, A. E. (1958): Introduction to Functional Analysis. John Wiley and Sons, New York.

Taylor, J. R. (1972): Scattering Theory. John Wiley and Sons, New York.

Thron, W. (1966): Topological Structures. Holt, Rinehart, and Winston, New York.

Titchmarsh, E. C. (1946): Eigenfunction Expansions Associated with Second Order Differential Equations. Clarendon Press, Oxford.

Wasow, W. (1976): Asymptotic Expansionsfor Ordinary Differential ElJuations. Krieger Pub. Co., Huntington, N.Y.

Weidmann, J. (1971): Oszillationsmethoden fUr Systeme gewohnlicher Differential­gleichungen. Math. z., vol. 119, pp. 349-373.

Weinberger, H. F. (1965): A First Course in Partial Differential Equations. Blaisdell Pub. Co., New York.

Werner, P. (1969): Bemerkungen zur Theorie der Lp Riiume. lournalfor die reine und angewandte Mathematik, vol. 239/240, pp. 401--434.

Whittaker, E. T., and Watson, G. N. (1927): A Course of Modern Analysis, Fourth Edition. Cambridge Univ. Press.

Wiener, N. (1930): Generalized harmonic analysis. Acta Math., vol. 55, pp. 117-258.

Wright, J. D. M. (1973): All operators on a Hilbert space are bounded. Bull. Amer. Math. Soc., vol. 79, pp. 1247-1250.

Zygmund, A. (1952): Trigonometrical Series. Chelsea Pub. Co., New York.

Index

Abramowitz, M. 267, 357 Absolutely continuous function 132,

230, 260, 289 Absolutely continuous spectrum 228 Absolutely continuous subspace 230,

231 Adjoint of an operator 128 Akhiezer, N.J. 153, 166,314 Aleph zero 8 Algebra of bounded operators 309 Algebraic eigenspace 242 Algebraic multiplicity 242 Algebraic singularity 39 Almost everywhere 260 Almost periodic function 64 Analytic continuation by

computer 406 Analytic functions, boundary values

of 166-171 Annihilation and creation

operators 146 Approximation in the mean 73 Approximations to the delta

function 29 Area of the unit sphere in 9t n 103 Arzela theorem 113 Ascoli - Arzela theorem 113 Ashenhurst, R.L. 404 Atomic 260

Autocorrelation 61 Autocovariance function 61 Autoionization 230 Axioms of a linear space 3

B*-algebra 309 Balmer formula 217 Banach algebra 127 Banach, S. 289 Banach space 6, 327 Becker, R. 375 Bernstein theorem 9 Berry, A.C. 272 Berry - Esseen theorem 272 Bessel function 214 Bessel's equation 213 Bessel's inequality 12 Bethe, H.A. 234, 235 Bilinear form 22, 23 Bleakney, W. 393 Bolzano-Weierstrass theorem 44 Borel algebra 288 Borel class 286 Borel field 286 Boundary condition at a singular

endpoint 200 Boundary values of analytic

functions 167 -171

413

414 Index

Boundary values of distributions 194-197

Bounded linear functional 16 Bounded observable 183 Bounded operator 125 Bounded support 22 Bound of an operator 125 Bunyakovskii inequality 5

C*-algebra 309 Canonical commutation relations 309 Canonical regularization 40 Canonical representation of a

self-adjoint operator 174-176, 187-189

Cantor function 38, 137, 257 Cardinal numbers 8 Carleman, T. 252 Carleson, L. 72 Case, K. 236 Cauchy inequality 7 Cauchy - Kovalevski theorem 385 Cauchy problem 322 Cauchy-Riemann equations 123

in distribution theory sense 123 Cauchy sequence

in a Hilbert space 7 in a metric space 74

Cauchy's integral formula for operators 181

Cauchy's theorem for operators 181 Cayley transform of a symmetric

operator 154 Central limit theorem 269-273 Change of independent variable in a

distribution 33 Characteristic curve 382 Characteristic form of a hyperbolic

system 377 - 379 Characteristic function of a probability

distribution 262 multidimensional 266

Characteristic surface 383, 384 Characteristics of partial differential

equations 378, 382 Chebyshev'S inequality 275

Circuit for Fourier analysis 73 Classical commutation relations 311

Weyl form 311 Closable operator 136 Closed-graph theorem 137 Closed linear manifold in a Hilbert

space 14 Closed linear span of a set in a Hilbert

space 15 Closed operator 136 Closed set 43 Closed set in a Hilbert space 14 Closure of an operator 136 Closure of a set 43 Coddington, E.A. 207,211,212 Codimension of a linear manifold 153 Coefficient of correlation 264 Column vector 1 Compactness theorem for Sobolev

space 113 Compact set 44 Comparison of cardinals 9 Complete metric space 74 Completeness of function systems 68 Complete normed space 4 Complete orthonormal sequence 12 Complete set of commuting

observables 317 - 319 Completing the measure 286 Completion of metric space 74 Complex conjugate of a distribution 23 Conditionally compact 115 Conditional probability 276 Conditional probability density 277 Cone condition 112 Conservation laws 367 Contact discontinuity 373 Contact resistance 108 Continuity 24

in 9 24 of functionals 24, 31 of operator..:valued function 178

strong 178 in norm 178 weak 178

of the inner product 5

Continuous power spectrum 65 Continuous spectrum 144

in the sense of Hilbert 230, 231 Convergence

in g 24 in Y' 53 of test functions 24, 53

Convergence in a Banach space 333 in the mean 68 of bounded operators 176 strong 176 unform (or in norm) 176 in a Banach space 176 weak 176 of distributions 29, 39 strong 333 weak 333 uniform 333

Convolution ~ ~ v of distributions 101, 109

associativity 109 commutativity 109 differentiation 101 possible nonassociativity when

support is not compact 110 of functions 100

Coordinate representation in quantum mechanics 93

Countably additive 287 Countably infinite set 8 Courant, R. 70, 104, 112, 113, 374,

386,387 Cumulative probability 254 Curl-free vector field 120 Cylinder set 291, 292 Cylinder-set measure 293

Deficiency indices 153 Definite integrals in L 2 spaces 83, 84 Degenerate operator 246 Delta function 19 Density matrix 306-309 Denumerably infinite set 8 Derivative of a distribution 31 Detached shock problem 395-408

Index 415

Differential operator 131 - 135 Differentiation of distributions 31 Dimension 8

of a Hilbert space 14 DiPrima, R. C. 251 Dirac Hamiltonian 233 Direct product of distributions 109 Dirichlet

integral 111 problem 103

Dirichlet's principle 112 Discontinuous linear functional 40 Discrete probability distribution 260 Distribution

inL' (IR) 90 in L2 75 locally in L' 91 whose support is a point 51, 60

Divergence-free vector field 120 Divergent Fourier series 72 Domain 125 Double layer of charge 27 Dual space 88, 332 Duff, R.G. 393 Dunford, N. 113,252,283,290 Dyadic scalar product 118

Eigenfunction expansion 206, 211 Eigenfunctions of the Laplacian 110 Eigenspace, algebraic 242 Eigenvalue 242

algebraic multiplicity 242 geometric multiplicity 242 of a matrix 2

Eigenvector 2 ofa matrix 2

Electrical resistance 107 Entropy 368 Equation of state 367 Equicontinuous family off unction 113 Equivalent Cauchy sequences 75 Esseen, G. 272 Essentially self-adjoint operator 129,

151 Essential spectrum 228 Euler- Lagrange equation 111

416 Index

Expectation 261, 304 Expected value 261 Extension 125 Extension thorem

for bounded operators 126 for measures 283

External cone condition 104

Fefferman, C. 72 Fejer's method for Fourier series 55 Feller, W. 262,272,278,287,288,

293 Field of scalars 2 Field of values of an operator 142 Flow of information along

characteristics 382, 383 Fock space II Formula of Titchmarsh 208 Fourier

coefficients 12 Stieltjes integral 65 transform

analytic 58 as a continuous mapping in.9' 56 as a continuous mapping in!J?' 57 in L' 89,90 in U spaces 93 inverse 55 method for differential

operators _ 192 - 194, 223 of a convolution 101 of a distribution with bounded

support 58 of a not necessarily tempered

distribution 58 of a periodic distribution 60 of a tempered distribution 57 of test functions 55 operator 193

Fractional powers of a nonnegative operator 183

Friedreichs, K.O. 355, 374, 387 Frobenius method 203 Function of bounded variation 295 Functions of operators 181-183 Fundamental solution 323 Fundamental theorem of calculus 35

Garabedian, P.R. 386 Gaussian distribution 266 Gaussian measure 291 Gelbard, E.M. 280 Gel'fand,I.M. 59,110,291,293,295 Generalized Dirichlet integral 118 Generalized eigenvector 164

of a matrix 242 Generalized Fourier coefficients 12,

69 Generalized Fourier series 70 Generalized solutions 336-339 Generating vector 313 Geometric multiplicity 242 Glazman, I.M. 153, 166, 314 Gram - Schmidt orthonormalization

procedure 12, 69, 165 Graph of an operator 138 Gray, H.L. 404 Green's formula 96 Green's function 100, 103, 197, 239

four-point 107 symmetry 106

Gross, L. 364 Growth at infinity

of functions 54 of distributions 54 slow 54 Gustafson, K. 233,237,238

Habetler, G.J. 251 Halmos, P. 290 Hamel basis 41 Hankel function 214 Harmonic distribution 123 Harmonic function 123 Harrison, C. 404 Heine-Borel theorem 44 Helmholtz instability 390-393 Hermitian conjugate of a matrix Hermitian matrix 3, 158 Hilbert, D. 70, 104, 112, 113,231,

386 Hilbert - Schmidt

integral operators 177 norm 245-247 operator 245

Hilbert space of analytic functions 17 Hille, E. 351, 354 Hille-Y osida theorem 354 Holder inequality 87 Hugoniot curve 374 Hydrodynamical stability 118 Hydrogen-like atom

nonrelativistic 216 relativistic 218

Hyperbolic system 377

Idempotent operator (P2=P) 163 Identification of functions with

distributions 26, 38 Indefinite integral of a distribution 32 Index of an eigenvalue 242 Indicial equation 204 Inequivalence of Banach spaces 330 Infinitesimal generator of a

semigroup 351 Inhomogeneous problems 358 Initial-value problem 324

ill-posed 324 well-posed 324 with time-dependent operator 362

Inner product finite dimensional 1 in a Hilbert space 4

Inner-product space 74 Instability of negative

shocks 373-375 Integral operator 130, 247 Integration by parts for distributions in

U 84,97 Integration in U spaces 82 Interior point 43 Inverse Cayley transform 154 Irrotational vector field 118 Isomorphism of Hilbert spaces 13

Johnson, G. 168,233 Joint cumulative probability 263 Jointly continuous 5 Jordan, P. 6 Jordan block 164 Jordan canonical form of a matrix

164

Index 417

Jorgens, K. 210,211,215,217,226, 227,229

Jost, R. 312 Jump conditions 370-372 Jump function 230

Kato, T. 137,141,142,149,226,227, 228,229,237,244,247

K -dimensional subspace 2 Kernel theorem (= nuclear

theorem) 110 Kolmogorov, A. 292

Lagrange multiplier 111 Landau, L.D. 395 Laplacian operator 99-108,

110-112,116-124,222-226 completeness of the

eigenfunctions 117, 121 eigenfunctions 110 existence of eigenfunctions 116 in a bounded region 238 resolvent 224 spectrum 224 spectral projectors 224-226 variational methods 110

Lax, P.D. 123, 367 Lehner, J. 356 Levinson, N. 207,211,212 Lewis, G.E. 406 Lifschitz, E.M. 395 Limit-circle case 209 Limit-point case 205 Limit point in a Hilbert space 14 Linear functional 16, 22

bounded 16 in a Banach space 332 in a function space 22

Linearly dependent vectors 2 Linear operator 125

in a Banach space 331 Linear space 3

axioms 3 normed 3

U norm 74 Locally compact space 16

418 Index

Long-term statistical properties of a function 61

Luzin's Theorem 89,91

Mach stem 394 Magnus, W. 225 Marginal probability 276 Matrix

rank 241, 242 trace 241, 242

Maxwell's equations 347 Mean 261 Mean convergence 68 Measure 28, 89, 281-291 Measure zero 259, 286 Messiah, A. 127 Metropolis, N.C. 404,405 Minkowski inequality 87 Misra, B. 312 Mixed partial derivatives of a

distribution 37 Mollifier 30 Mollifiers in U spaces 93 Moment problem 262 Moments of a distribution 261 Momentum representation in quantum

mechanics 93 Monte Carlo method 278 - 280 Morton, K.W. 355, 359, 375 Multidimensional Stieltjes

integral 266 Multilinear functional 109 Multiple Fourier series 72 Multiplication in U spaces 81 - 82 Multipole point charge 27

Nagy, B.Sz. 86,231,282 Naimark, M.A. 251 Natanson, J.P. 44,91,261,262,289 Negative shock 373 Negative variation 296 Neumann function 214 Neumann, J. von 6, 141, 142, 153,

155, 229, 308, 311 Neumann problem 105 Neutron transport 355

Nilpotent matrix 164 Nondecreasing function

decomposition of 229 of several variables 265 of two variables 264

Nonnegative operator 141 Nonseparable Hilbert space 11 Norm 4 Normal distribution 266

bivariate 268 Normal matrix 3 Normal operator 148 Normed linear space 4 Nuclear Theorem of Schwartz 109 Nullspace of an operator 138 - 139 NUllspace-range Theorem 139 Numerical range 141-142

Oberhettinger, F. 225 Observable 127, 299 open covering of a set 44, 46 Open set 43 Operator

addition 129 algebra 126 bounded 241 compact 242 - 244 completely continuous 242 (d/dx)2 191 degenerate 246 extension 153 Hilbert-Schmidt 131,245 i d/dx 190 of scalar type 165 trace class 245 - 246 with compact resolvent 248 - 252

Operator-valued distribution 38 Orr-Sommerfeld problem 251 Orthogonal complement 174

finite-dimensional 2 in a Hilbert space 14

Orthogonal polynomials 69 Orthogonal projector 161 Orthogonal vectors

finite-dimensional 2 in a Hilbert space 11

Orthonormal vectors finite-dimensional 2 in a Hilbert space 1 in a nonseparable Hilbert space 14

Parallelogram law 6, 86 Partially isometric operator 184 Particle number operator 147 Partition of unity 48 - 50 Parseval relation 12 Pauli principle 229 Peano curve 9 Perturbation of the spectrum 228 Phillips, R.S. 351,354 Piecewise analytic initial-value

problem 393 Piecing-together principle 51 Point charge 27 Point spectrum 143 Pointwise convergence of Fourier

series 72 Polar decomposition of an

operator 184 Polarization formula 6, 18 Poisson integral formula 107 Poisson - Neumann problem 105 Poisson problem 103 Poisson's equation 100, 102

in the distribution-theory sense 106 Positive definite matrix 3 Positive definite operator 141 Positive measure 287 Positive operator 141 Positive semidefinite operator 141 Positive variation 296 Potential theory

classical problems 103 equivalence of 104

Power of the continuum 9 Power series method 401 Power spectrum 60-67 Pre-Hilbert space 74 Probability distribution

absolutely continuous 260 atomic 260 bivariate 263

Index 419

characteristic function 262 discrete 260 multivariate 254 singular 260 univariate 254

Probability in Hilbert space 291 Probability in quantum

mechanics 84-85 Probability measure 262 - 266

on 1R2 265 multidimensional 265 - 266

Probability space 287 Product of a distribution and a C 00

function 28 Projection theorem 15 Projector 159-160

orthogonal 161 Pure line spectrum 65 Pure point spectrum in the sense of

Hilbert 230-231 Putnam, C.R. 312

Quasi-linear 368

Radical momentum operator 139-141, 149, 150, 157

Radon-Nikodym theorem 289-290 Random number generator 279 Random variable 254

on a probability space 288 Range of an operator 125, 138 Range-nullspace theorem 155 Rankine-Hugoniot jump

condition 373 Rank of a matrix 242 Real dimension 2 Real Hilbert space 4 Reciprocity relations 108 Recurrence relations 204 Reflexive space 89 Regularization of singular

functions 39 Regular singular point 203 Regular Sturm - Liouville

operator 194

420 Index

Rehner, N. 87 Rejto, P. 237 Relativistic problems 333 Rellich, F. 210,211,215,217 Rellich's lemma 115 Residual spectrum 144 Resolution of the identity 159,

171-174 Resolvent equation 151 Resolvent of an operator 144, 151,

190-192 Resolvent set 144 Response of a filter 67 Restriction of a distribution to

C~(n) 79 Restriction of a tempered distribution to

C~ 54 Richtmyer,R.D.355,359,375,404 Rickert, C.E. 309 Riemann invariant 379 Riemann-Lebesgue lemma 89-90 Riemann problem 386-388 Riesz, F. 86,88,231,282 Riesz - Fischer theorem 13 Riesz - Frechet representation

theorem 16 Riesz representation theorem for

measures 282 Roos, B.W. 219 Rotated graph 138 Row vector 1

Salpeter, E.E. 234, 235 Sample average 273 Sample correlation coefficient 276 Sample covariance matrix 275 Sample mean 273 Sample variance 274 Sampling 273-276 Sangren, w.e. 219 Sattinger, D.H. 118, 252 Schiff, L. 233 SchrOder-Bernstein Theorem 9 Schrodinger equation 216, 344 SchrOdinger operator 226 Schur's Theorem 165

Schwartz class of test functions 22,52, 53

Schwartz, J.T. 113,252,283,290 Schwartz, L. 283 Schwartz's nuclear theorem 108 -110 Schwartz inequality 5 Self-adjoint operator 129, 148

second definition 157 with a simple spectrum 312-314

semidefinite functional 92 Semigroup 350 Seminorm 92, 333 Separability 8 Separable Hilbert space 10 Separately continuous 5 Sequentially compact set 44 Sesquilinear form 17 Set function 285-291 Shilov, G.E. 59 Shock 373 Shock front 373 Shock tube 387, 393 Shrinkage principle for open covering

of a set 47 Sigalov, A.G. 229 Significance arithmetic 404-406 Simple spectrum 312-314 Simulation 278 - 280 Single layer of charge 27 Singular continuous function 230 Singular continuous spectrum 66 Slip surface 373 Smoothing 30 Sobolev, S.L. 96 Sobolev space 94 Solenoidal vector fields 118 Solovay, R.M. 42 Solovay model of set theory 42 Sonic line 396 Spaces of type L~ 91 Space variable 326 Spanier, J. 280 Spectral decomposition of a Hermitian

matrix 158 ~pectral family {E t} 184-187,

190-192

Spectral function 208 Spectral matrix 211 Spectral multiplicity 211 Spectral operator 165 Spectral projectors E t 184-187 Spectral representation 314 Spectrum 143

in a Banach algebra 310 or a self-adjoint operator 148 of a unitary operator 148

Spherically symmetric distribution 34 Spontaneous generation of

shocks 388 - 390 Standard deviation 262 Stark effect 227 State of a system 299 State space of an initial-value

problem 326 Stegun, I. 267, 357 Stieltjes integral 261

multidimensional 266 Stone, M.H. 311 Stone-von Neumann

theorem 311-312 Strict solution 322 Strong convergence 16, 176 Sturm - Liouville operator 133 Sturm - Liouville operator,

regular 194 completeness of

eigenfunctions 197 - 198 eigenfunction expansion 211 eigenfunctions 195-196 existence and uniqueness of

solutions 195 general boundary conditions 198 Green's function 197 spectral multiplicity 211

Sturm - Liouville operator, singUlar 199-209

eigenfunction expansion 206 limit-circle case 200-202, 209 limit-point case 200-202, 205

Subspace of a Hilbert space 14 Subspace spanned by vectors 2 Support of a distribution 51

Index 421

Support of a function 22 Symmetric operator 129, 153 System of conversation laws 367 Sz. Nagy, B. 86,231,282

Taub, A.H. 393 Taylor instability 390-393 Tempered distribution 54 Test function 21, 52 Theorem of Carleman 251 - 252 Theorem of F. Riesz 88 Theorem of Naimark 251 Theorem of von Neumann 155, 200 Thron, W. 74 Titchmarsh, E.C. 208, 215 Total variation of a function 296 Translation-invariant distribution 31 Translation-invariant linear

functional 32 Transport-theory operator 226 Transpose of a matrix 1 Triangle inequality 4, 5 Triple point 393

Uncertainty of an observable 305 Uncertainty principle 306 Unitary matrix 3 Unitary operator 129, 148 Unnormalized arithmetic 405

Vacuum state 147 Vanishing at infinity in U spaces 85,

97-98 Variance 262 Variational method 110 Vector space 3 Vilenkin, N. Ya. 110,291,293,295

Wave equation 365 Weak convergence 16, 176 Weakly complete space 17 Weak solution

of a partial differential equation 36 of a system of conservation

laws 370

422 Index

Well-posed (in the sense of Hadamard) 324

Well-posed problem 336-339 Weidmann, J. 226,229, 237 Weierstrass approximation theorem

70 Werner, P. 86,87,88,89

Weyl, H. 190,229,311 Wing, M. 356 Wright, J.D.M. 42

Zerecki, M.A. 289 Zeeman effect 227 Zislin, G.M. 229

Texts and Monographs in Physics

Edited by W. Beiglbock, M. Goldhaber, E. Lieb, and W. Thirring

Texts and Monographs in Physics includes books from any field of physics that might be used as basic texts for advanced training and higher education in physics, especially for lectures and seminars at the graduate level.

Polarized Electrons By J. Kessler 1976. ix, 223p. 104 illus. cloth

The Theory of Photons and Electrons The Relativistic Quantum Field Theory of Charged Particles with Spin One-Half Second Expanded Edition By J. Jauch and F. Rohrlich 1976. xix, 553p. 55 illus. cloth

Essential Relativity Special, General, and Cosmological Second Edition By W. Rindler 1977. xv, 284p. 44 illus. cloth

Inverse Problems in Quantum Scattering Theory

By K. Chadan and P. Sabatier 1977. xxii, 344p. 23 illus. cloth

Quantum Mechanics By A.Bohm 1978.approx.576p.approx.85illus.cloth

The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines

Rigourously Constructed upon the Foundation Laid by S. Carnot and F. Reech

By C. Truesdell and S. Bhantha 1977. xxii, 154p. IS illus. cloth

Principles of Advanced Mathematical Physics

Volume I By R.D. Richtmyer 1978. approx. 448p. approx. 45 illus. cloth

Foundations of Theoretical Mechanics Part I: The Inverse Problem in

Newtonian Mechanics By R.M. Santilli 1978. 288p. cloth Part II: Generalizations of the Inverse

Problem in Newtonian Mechanics In preparation

A Springer-Verlag Journal

Communications in Mathematical Physics Editor-in-Chief J. Glimm, New York, New York

Editorial Board H. Araki, Kyoto, Japan R. Geroch, Chicago. Illinois R. Haag, Hamburg. Federal Republic of Germany W. Hunziker, Zurich, Switzerland A. Jaffe, Cambridge, Massachusetts J.L. Lebowitz, New York, New York E. Lieb, Princeton, New Jersey J. Moser. New York. New York R. Stora, Marseille, France

Communications in Mathematical Physics is devoted to such topics as general relativity. equilibrium and nonequilibrium statistical mechanics. foundations of quantum mechanics. Lagrangian quantum field theory. and constructive quantum field theory. Mathematical papers are featured if they are relevant to physics.

Contact Springer- Verlag for subscription information.

Lecture Notes in Physics Managing Editor W. BeiglbOck

This series reports on new developments in physical research and teaching­quickly. informally. and at a high level. The type of material considered for publication includes preliminary drafts of original papers and monographs. lectures on a new field. or presenting a new angle on a classical field. collections of seminar papers. and reports of meetings.

Vol. IO J .M. Stewart, Non-Equilibrium-Relativistic Kinetic Theory. 1971. iii. Il3p.

Vol. I3 M. Ryan, Hamiltonian Cosmology. 1972. vii. 169p.

Vol. 14 Methods of Local and Global Differential Geometry in General Relativity. Edited by D. Farnsworth, J. Fink, J. Porter. and A. Thompson. 1972. v. 188p.

Vol.44 R.A. Breuer, Gravitational Perturbation Theory and Synchrotron Radiation. 1975. vi. I%p.

Vol. 46 E.J. Flaherty, Hermitian and Kilhierian Geometry in Relativity. 1976. viii, 365p.

Springer-Verlag New York Inc. 175 Fifth Avenue New York, NY IOOW