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Leray J. Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index
MIT Press, 1981, 271 pages

The book develops the mathematical apparatus of Maslov's lagrangian ayalysis conceptions. This work might have been entitled The Introduction of Planck's Constant into Mathematics, in that it introduces quantum conditions in a purely mathematical way in order to remove the singularities that arise in obtaining approximations to solutions of complex differential equations.

The book's first chapter develops the necessary mathematical apparatus: Fourier transforms, metaplectic and symplectic groups, the Maslov index, and lagrangian varieties.
The second chapter orders Maslov's conceptions in a manner that avoids contraditions and creates step by step an essentially new structure-the lagrangian ayalysis. Unexpectedly and strangely the last step requires the datum of a constant, which in applications to quantum mechanics is identified with Planck's constant.
The final two chapters apply lagrangian analysis directly to the Schr?dinger, the Klein-Gordon, and the Dirac equations. Magnetic field effects and even the Paschen-Back effect are taken into account.


The Fourier Transform and Symplectic Group
Differential Operators, The Metaplectic and Symplectic Groups
Maslov Indices; Indices of Inertia; Lagrangian Manifolds and Their Orientations
Symplectic Spaces

Lagrangian Functions; Lagrangian Differential Operators
Formal Analysis
Lagrangian Analysis
Homogeneous Lagrangian Systems in One Unknown
Homogeneous Lagrangian Systems in Several Unknowns

Schrodinger and Klein-Gordon Equations for One-Electron Atoms in a Magnetic Field
A Hamiltonian H to Which Theorem 7.1 (Chapter II, §3) Applies Easily; the Energy Levels of One-Electron Atoms with the Zeeman Effect
The Lagrangian Equestion aU = 0 mod(1/v2) (a Associated to H, U Having Lagrangian Amplitude , 0 Defined on a Compact V)
When a Is the Schrodinger-Klein-Gordon Operator
The Schrodinger-Klein-Gordon Equation

Dirac Equation with the Zeeman Effect
A Lagrangian Problem in Two Unknowns
The Dirac Equation
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