The Schrödinger EquationSpringer Science & Business Media, 6 dic 2012 - 555 pagine This volume deals with those topics of mathematical physics, associated with the study of the Schrödinger equation, which are considered to be the most important. Chapter 1 presents the basic concepts of quantum mechanics. Chapter 2 provides an introduction to the spectral theory of the one-dimensional Schrödinger equation. Chapter 3 opens with a discussion of the spectral theory of the multi-dimensional Schrödinger equation, which is a far more complex case and requires careful consideration of aspects which are trivial in the one-dimensional case. Chapter 4 presents the scattering theory for the multi-dimensional non-relativistic Schrödinger equation, and the final chapter is devoted to quantization and Feynman path integrals. These five main chapters are followed by three supplements, which present material drawn on in the various chapters. The first two supplements deal with general questions concerning the spectral theory of operators in Hilbert space, and necessary information relating to Sobolev spaces and elliptic equations. Supplement 3, which essentially stands alone, introduces the concept of the supermanifold which leads to a more natural treatment of quantization. Although written primarily for mathematicians who wish to gain a better awareness of the physical aspects of quantum mechanics and related topics, it will also be useful for mathematical physicists who wish to become better acquainted with the mathematical formalism of quantum mechanics. Much of the material included here has been based on lectures given by the authors at Moscow State University, and this volume can also be recommended as a supplementary graduate level introduction to the spectral theory of differential operators with both discrete and continuous spectra. This English edition is a revised, expanded version of the original Soviet publication. |
Sommario
1 | |
The OneDimensional Schrödinger Equation | 50 |
The Multidimensional Schrödinger Equation | 150 |
Scattering Theory | 223 |
Symbols of Operators and Feynman Path Integrals | 282 |
150 | 358 |
classical Asymptotics | 374 |
Spectral Theory of Operators in Hilbert Space | 386 |
Variational Principles and Perturbation Theory for a Discrete | 434 |
Sobolev Spaces and Elliptic Equations | 466 |
Regularity of Solutions of Elliptic Equations and a priori Estimates | 475 |
Quantization and Supermanifolds | 483 |
main procedures | 511 |
Supersymmetry of the Ordinary Schrödinger Equation and of | 520 |
181 | 537 |
552 | |
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absolutely continuous algebra arbitrary asymptotics bounded commutes compact condition consider coordinates corresponding defined definition denote dense domain eigenfunctions eigenvalues estimate evolution operator example exists finite finite-dimensional follows formula Fourier transformation function H₁ H₂ Hence Hilbert space Hilbert-Schmidt operator implies inequality isomorphism kernel L²(M L²(R L²(R+ Lebesgue Lebesgue measure Lemma Lie superalgebra limit matrix measure multiplication operator norm Note obtain operator H orthogonal orthonormal particle path integral polynomial potential v(x pq-symbols proof of Theorem Proposition prove qp-symbol quantization quantum mechanics R₂ relations representation respect right-hand side S₁(H satisfies scalar product Schrödinger operator self-adjoint operator solution space H spectral spectrum subspace suffices supermanifold superspace t₁ Theorem 1.1 theory trace class uniquely unitary unitary operator variables vector verify wave operators Weyl symbol Wick symbol