Mathematical Methods of Physics

Igor V. Kolokolov is a Russian physicist known for his work on magnetism, soft matter physics and statistical hydrodynamics. He is professor at the Physical Department at Higher School of Economics, Moscow, and Director of Landau Institute of Theoretical Physics, Chernogolovka, Russia.Evgeny A. Kuznetsov is a Russian physicist known for his work on nonlinear physics, soliton stability theory, and Hamiltonian formalism for nonlinear waves. He is a member of the Russian Academy of Sciences (RAS), professor at the Moscow Institute of Physics and Technology, and a principal researcher at the Tamm Theoretical Physics Department of the Lebedev Physics Institute of the RAS.Alexander I. Milstein is a Russian physicist, specialist in theoretical elementary particle physics, nuclear and atomic physics, head of the Theoretical Department at Budker Institute of Nuclear Physics, and professor at Novosibirsk State University (NSU).Evgeny V. Podivilov is a Russian physicist known for his work on nonlinear optics and nonlinear interactions of waves in fibers. He is a professor at NSU and a principal researcher at the Institute of Automation and Electrometry of the RAS.Alexander I. Chernykh holds a PhD in physics and mathematics and is engaged in numerical modeling. He has taught various subjects, including methods of mathematical physics, analytical mechanics, statistical physics, and general theory of relativity.David A. Shapiro is a Russian physicist. He is professor at NSU and heads the Photonics Laboratory at the Institute of Automation and Electrometry of the RAS. His current research interests include fiber optics, nanophotonics, and plasma physics.Elena G. Shapiro holds a PhD in physics and mathematics. In 1985, she became a member of the Institute of Automation and Electrometry of the RAS. She had been teaching undergraduate students at the Physics Department of NSU since 1988.

“An excellent collection of problems and their solutions with elements of theory to accompany the university course on mathematical methods of physics. It covers a broad variety of topics ranging from linear operators, differential and integral equations, special functions, and Green functions to group theory and their applications in physics. The book is recommended to students, PhD students, researchers, and teachers.”Prof. Boris KonopelchenkoUniversity of Salento, Italy

• 1 Linear Operators1.1 Finite Dimensional Space1.2 Functionals and Generalized Functions1.3 Hilbert Space and Completeness1.4 Self-Adjoint Operators1.5 Ket- and Bra- Vectors2 Method of Characteristics2.1 Linear First-Order PDE2.2 Quasilinear Equation2.3 System of Equations3 Second-Order Linear Equations3.1 Canonical Form3.2 Curvilinear Coordinates3.3 Separation of Variables3.4 Fourier Method4 Self-Similarity and Nonlinear Equations4.1 Symmetry of Equations4.2 Nonlinear Equations5 Special Functions5.1 Singular Points5.2 Hypergeometric Functions5.3 Orthogonal Polynomials6 Asymptotic Methods6.1 Asymptotic Power Series6.2 A Laplace Integral6.3 Method of Stationary Phase6.4 Method of Steepest Descents6.5 The Averaging Method7 Green’s Functions Method7.1 Green’s Functions7.2 Continuous Spectrum7.3 Resolvent8 Integral Equations8.1 Fredholm Equations8.2 Degenerate Kernel8.3 Symmetric Kernel8.4 Inverse Problem for Schrödinger Operator9 Groups and Representations9.1 Groups9.2 Representations10 Continuous Groups10.1 Lie Groups and Algebras10.2 Representations of the Rotation Group11 Group Theory in Physics11.1 Molecular Oscillations11.2 Level Splitting11.3 Selection Rules11.4 Invariant Tensors