Principles of Advanced Mathematical Physics

1. Front Matter

   Pages i-xv

   PDF

2. Hilbert Spaces

   • Robert D. Richtmyer

   Pages 1-18

3. Distributions; General Properties

   • Robert D. Richtmyer

   Pages 19-42

4. Local Properties of Distributions

   • Robert D. Richtmyer

   Pages 43-51

5. Tempered Distributions and Fourier Transforms

   • Robert D. Richtmyer

   Pages 52-67

6. L² Spaces

   • Robert D. Richtmyer

   Pages 68-98

7. Some Problems Connected with the Laplacian

   • Robert D. Richtmyer

   Pages 99-124

8. Linear Operators in a Hilbert Space

   • Robert D. Richtmyer

   Pages 125-142

9. Spectrum and Resolvent

   • Robert D. Richtmyer

   Pages 143-157

10. Spectral Decomposition of Self-Adjoint and Unitary Operators

    • Robert D. Richtmyer

    Pages 158-189

11. Ordinary Differential Operators

    • Robert D. Richtmyer

    Pages 190-221

12. Some Partial Differential Operators of Quantum Mechanics

    • Robert D. Richtmyer

    Pages 222-240

13. Compact, Hilbert-Schmidt, and Trace-Class Operators

    • Robert D. Richtmyer

    Pages 241-252

14. Probability; Measures

    • Robert D. Richtmyer

    Pages 253-298

15. Probability and Operators in Quantum Mechanics

    • Robert D. Richtmyer

    Pages 299-319

16. Problems of Evolution; Banach Spaces

    • Robert D. Richtmyer

    Pages 320-334

17. Well-Posed Initial-Value Problems; Semigroups

    • Robert D. Richtmyer

    Pages 335-363

18. Nonlinear Problems: Fluid Dynamics

    • Robert D. Richtmyer

    Pages 364-408

19. Back Matter

    Pages 409-424

    PDF

A first consequence of this difference in texture concerns the attitude we must take toward some (or perhaps most) investigations in "applied mathe­ matics," at least when the mathematics is applied to physics. Namely, those investigations have to be regarded as pure mathematics and evaluated as such. For example, some of my mathematical colleagues have worked in recent years on the Hartree-Fock approximate method for determining the structures of many-electron atoms and ions. When the method was intro­ duced, nearly fifty years ago, physicists did the best they could to justify it, using variational principles, intuition, and other techniques within the texture of physical reasoning. By now the method has long since become part of the established structure of physics. The mathematical theorems that can be proved now (mostly for two- and three-electron systems, hence of limited interest for physics), have to be regarded as mathematics. If they are good mathematics (and I believe they are), that is justification enough. If they are not, there is no basis for saying that the work is being done to help the physicists. In that sense, applied mathematics plays no role in today's physics. In today's division of labor, the task of the mathematician is to create mathematics, in whatever area, without being much concerned about how the mathematics is used; that should be decided in the future and by physics.

• Fourier transform
• Functionals
• Hamiltonian
• Maxwell’s equations
• Monte Carlo method
• Potential
• applied mathematics
• conservation law
• mathematical physics
• mechanics
• momentum
• polarization
• quantum mechanics
• solution
• vector field

• Department of Physics and Astrophysics, University of Colorado, Boulder, USA

  Robert D. Richtmyer

Policies and ethics