Contents Introduction Preface Chapter I Variational Method 1 1.1. Functional and Its Extremal Problems 1 1.1.1. The conception of functional 1 1.1.2. The extremes of functionals 3 1.2. The Variational of Functionals and the Simplest Euler Equation 7 1.2.1. The variational of functionals 7 1.2.2. The simplest Euler equation 12 1.3. The Cases of Multifunctions and Multivariates 16 1.3.1. Multifunctions 16 1.3.2. Multivariates 19 1.4. Functional Extremes under Certain Conditions 22 1.4.1. Isoperimetric problem 22 1.4.2. Geodesic problem 26 1.5. Natural Boundary Conditioiis 29 1.6. Variational Principle 33 1.6.1. Variational principle of classical mechanics 34 1.6.2. Variational principle of quantum mechanics 40 1.7. The Applications of the Variational Method in Physics 41 1.7.1. The applications in classical physics 44 1.7.2. The applications in quantum mechanics 48 Exercises 50 Chapter 2 Hilbert Space 55 2.1. Linear Space, Inner Product Space and Hilbert Space 55 2.1.1. Linear space 55 2.1.2. Inner product space 63 2.1.3. Hilbert space 71 2.2. Operators in Inner Product Spaces 74 2.2.1. Operators and adjoint operators 74 2.2.2. Selfadjoint operators 84 2.2.3. The alternative theorem for the solutions of linear algebraic equations 94 2.3. Complete Set of Orthonormal Functions 96 2.3.1. Three kinds of convergences 96 2.3.2. The completeness of a set of functions 98 2.3.3. Ndimensional space and Hilbert function space 101 2.3.4. Orthogonal polynomials 103 2.4. Polynomial Approximation 109 2.4.1. Weierstrass theorem 109 2.4.2. Polynomial approximation 112 Exercises 120 Chapter 3 Linear Ordinary Differential Equations of Second Order 127 3.1. General Theory 127 3.1.1. The existence and uniqueness of solutions 127 3.1.2. The struct.ure of solutions of homogeneous equations 130 3.1.3. The solutions of inhomogeneous equations 137 3.2. SturmLiouville Eigenvalue Problem 140 3.2.1. The form of SturmLiouville equations 140 3.2.2. The boundary conditions of SturmLiouville equations 142 3.2.3. SturmLiouville eigenvalue problem 144 3.3. The Polynomial Solutions of SturmLiouville Equations 151 3.3.1. Possible forms of kernel and weight functions 151 3.3.2. The expressions in series and in derivatives of the polynomials 158 3.3.3. Generating functions 165 3.3.4. The completeness theorem of orthogonal polynomials as SturmLiouville solutions 169 3.3.5. Applications in numerical integrations 171 3.4. Equations and Functions that Relate to the Polynomial Solutions 174 3.4.1. Laguerre functions 175 3.4.2. Legendre functions 179 3.4.3. Chebyshev functions 185 3.4.4. Hermite functions 190 3.5. Complex Analysis Theory of the Ordinary Differential Equations of Second Order 196 3.5.1. Solutions of homogeneous equations 196 3.5.2. Ordinary differential equations of second order 216 3.6. NonSelfAdjoint Ordinary Differential Equations of Second Order 224 3.6.1. Adjoint equations of ordinary differential equations 224 3.6.2. SturmLiouville operator 225 3.6.3. Complete set of nonselfadjoint ordinary differential equations of second order 229 3.7. The Conditions under Which Inhomogeneous Equations have Solutions 231 Exercises 236 Appendix 3A Generalization of SturmLiouville Theorem to Dirac Equation 244 Chapter 4 Bessel Functions 247 4.1. Bessel Equation 247 4.1.1. Bessel equat.ion and its solutioiis 247 4.1.2. Bessel functions of the first and second kinds 255 4.2. Fundamental Properties of Bessel Functions 258 4.2.1. Recurrence relations of Bessel functions 258 4.2.2. Asymptotic formulas of Bessel functions 261 4.2.3. Zeros of Bessel functions 262 4.2.4. Wronskian 264 4.3. Bessel Functions of Integer Orders 266 4.3.1. Parity and the values at certain points 266 4.3.2. Generating function of Bessel functions of integer orders 267 4.4. Bessel Functions of HalfInteger Orders 273 4.5. Bessel Functions of the Third Kind and Spherical Bessel Functions 275 4.5.1. Bessel functions of the third kind 275 4.5.2. Spherical Bessel functions 280 4.6. Modified Bessel Functions 288 4.6.1. Modified Bessel functions of the first and second kinds 288 4.6.2. Modified Bessel functions of integer orders 293 4.7. Bessel Functions with Real Arguments 294 4.7.1. Eigenvalue problem of Bessel equation 294 4.7.2. Properties of eigenfunctions 297 4.7.3. Eigenvalue problem of spherical Bessel equation 301 Exercises 302 Chapter 5 The Dirac Delta Function 311 5.1. Definition and Properties of the Delta Function 311 5.1.1. Definition of the delta function 311 5.1.2. The delta function is a generalized function 312 5.1.3. The Fourier and Laplace transformations of the delta function 314 5.1.4. Derivative and integration of generalized functions 315 5.1.5. Complex argument in the delta function 318 5.2. The Delta Function as Weak Convergence Limits of Ordinary Functions 320 5.3. The Delta Function in Multidimensional Spaces 330 5.3.1. Cartesian coordinate system 330 5.3.2. The transform from Cartesian coordinates to curvilinear coordinates 331 5.4. Generalized Fourier Series Expansion of the Delta Function 335 Exercises 339 Chapter 6 Greens Function 345 6.1. Fundamental Theory of Green's Function 345 6.1.1. Definition of Green's fuiictioii 345 6.1.2. Properties of Green's function 347 6.1.3. Methods of obtaining Green's function 352 6.1.4. Physical meaning of Green's function 360 6.2. The Basic Solution of Laplace Operator 362 6.2.1. Threedimensional space 363 6.2.2. Twodimensional space 365 6.2.3. Onedimensional space 369 6.3. Green's Function of a Damped Oscillator 371 6.3.1. Solution of homogeneous equation 371 6.3.2. Obtaining Green's function 372 6.3.3. Generalized solution of the equation 373 6.3.4. The case without damping 374 6.3.5. The infiuence of boundary conditions 375 6.4. Green's Function of Ordiiiary Differential Equations of Second Order 376 6.4.1. The symmetry of Green's function 377 6.4.2. Solutions of boundary value problem of ordinary differential equations of second order 378 6.4.3. Modified Green's function 381 6.4.4. Examples of solving boundary value problem of ordinary differential equations of secondorder 388 6.5. Green's Function in Multidimensional Spaces 394 6.5.1. Ordinary differential equations of second order and Green's function 394 6.5.2. Examples in twodimensional space 400 6.5.3. Examples in threedimensional space 418 6.6. Green's Function of Ordinary Differential Equation of First Order 421 6.6.1. Boundary value problem of inhomogeneous equations 421 6.6.2. Boundary value problem of homogeneous equations 421 6.6.3. Inhomogeneous equations and Green's function 422 6.6.4. General solutions of boundary value problem 425 6.7. Green's Function of NonSelfAdjoint Equations 425 6.7.1. Adjoint Green's function 425 6.7.2. Solutions of inhomogeneous equations 427 Exercises 429 Chapter 7 Norm 435 7.1. Banach Space 435 7.1.1. Banach space 435 7.1.2. Holder inequality 439 7.1.3. Minkowski inequality 442 7.2. Vector Norms 443 7.2.1. Vector norms 443 7.2.2. Equivalence between vector norms 446 7.3. Matrix Norms 447 7.3.1. Matrix norms 447 7.3.2. Spectral norm and spectral radius of matrices 455 7.4. Operator Norms 459 7.4.1. Operator norms 459 7.4.2. Adjoint operators 465 7.4.3. Projection operators 469 Exercises 473 Chapter 8 Integral Equations 477 8.1. Fundamental Theory of Integral Equations 477 8.1.1. Definition and classification of integral equations 477 8.1.2. Relations between integral equat.ions and differential equations 481 8.1.3. Theory of homogeneous integral equations 484 8.2. Iteration Technique for Linear Integral Equations 490 8.2.1. The second kind of FYedhohn integral equations 490 8.2.2. The second kind of Volterra integral equations 502 8.3. Iteration Technique of Inhomogeneous Integral Equations 504 8.3.1. Iteration procedure 504 8.3.2. Lipschitz condition 506 8.3.3. Use of contraction 509 8.3.4. Anharmonic vibration of a spring 510 8.4. Fredholm Linear Equations with Degenerated Kernels 512 8.4.1. Separable kernels 512 8.4.2. Kernels with a finite rank 521 8.4.3. Expansion of kernel in terms of eigenfunctions 532 8.5. Integral Equations of Coiivolution Type 535 8.5.1. Fredholm integral equations of convolution type 535 8.5.2. Volterra integral equations of convolution type 538 8.6. Integral Equations with Polynomials 543 8.6.1. FYedholm integral equations with polynomials 543 8.6.2. Generating function method 544 Exercises 547 Chapter 9 Application of Number Theory in Inverse Problems in Physics 557 9.1. ChenMobius Transformation 557 9.1.1. Introduction 557 9.1.2. Mobius transformation 560 9.1.3. ChenMobius transformation 567 9.2. Inverse Problem in Phonon Density of States in Crystals 571 9.2.1. Inversion formula 571 9.2.2. Lowtemperature approximation 574 9.2.3. Highternperature approximation 577 9.3. Inverse Problem in the Int.eraction Potential between Atoms 580 9.3.1. Onedimensional case 581 9.3.2. Twodimensional case 586 9.3.3. Threedimensional case 592 9.4. Additive Mobius Inversioii and Its Applications 597 9.4.1. Additive Mobius inversion of functions and its applications 598 9.4.2. Additive Mobius inversion of series and its applications 606 9.5. Inverse Problem in Crystal Surface Relaxation and Interfacial Potentials 609 9.5.1. Pair potentials between an isolated atom and atoms in a semiinfinite crystal 609 9.5.2. Relaxation of atoms at a crystal surface 612 9.5.3. Inverse problem of interfacial potentials 614 9.6. Construction of Biorthogonal Complete Function Sets 617 Exercises 620 Appendix 9A. Some Values of Riemann < Function 622 Appendix 9B. Calculation of Reciprocal Coefficients 625 Chapter 10 Fundamental Equations in Spaces with Arbitrary Dimensions 627 10.1. Euclid Spaces with Arbitrary Dimensions 628 10.1.1. Cartesian coordinate system and spherical coordinates 628 10.1.2. Gradient, divergence and Laplace operator 632 10.2. Green's Functions of the Laplace Equation and Helmholtz Equation 636 10.2.1. Green's function of the Laplace equation 636 10.2.2. Green's function of the Helmholtz equation 639 10.3. Radial Equations under Central Potentials 641 10.3.1. Radial equation under a central potential in multidimensional spaces 641 10.3.2. Helmholtz equation 643 10.3.3. Infinitely deep spherical potential 644 10.3.4. Finitely deep spherical potential 644 10.3.5. Coulomb potential 646 10.3.6. Harmonic potential 649 10.3.7. Molecular potential with both negative powers 650 10.3.8. Molecular potential with positive and negative powers 651 10.3.9. Attractive potential with exponential decay 652 10.3.10. Conditions that the radial equation has analytical solutions 652 10.4. Solutions of Angular Equations 654 10.4.1. Fourdimensional space 656 10.4.2. Fivedimensional space 661 10.4.3. Ndimensional space 662 10.5. Pseudo Spherical Coordinates 666 10.5.1. Pseudo coordinates in fourdimensional space 666 10.5.2. Solutions of Laplace equation 668 10.5.3. Five and sixdimensioiial spaces 671 10.6. NonEuclidean Space 674 10.6.1. Metric tensor 674 10.6.2. Fivedimensional Minkowski space and fourdimensional de Sitter space 678 10.6.3. Maxwell equations in de Sitter spacetime 689 Exercises 697 Appendix 10A.Hypergeometric Equation and Hypergeometric Functions 699 References 703 Answers of Selected Exercises 705 Author Index 721 Subject Index 723
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