The Complex WKB Method for Nonlinear Equations I: Linear by Victor P. Maslov (auth.)

By Victor P. Maslov (auth.)

While this booklet was once first released (in Russian), it proved to develop into the fountainhead of a huge move of vital papers in arithmetic, physics or even chemistry; certainly, it shaped the foundation of latest method and opened new instructions for study. the current English version contains new examples of purposes to physics, hitherto unpublished or to be had basically in Russian. Its important mathematical inspiration is to take advantage of topological ways to learn isotropic invariant manifolds which will receive asymptotic sequence of eigenvalues and eigenvectors for the multi-dimensional Schrödinger equation, and in addition take into consideration the so-called tunnel results. Finite-dimensional linear idea is reviewed intimately. Infinite-dimensional linear conception and its purposes to quantum statistics and quantum box idea, in addition to the nonlinear concept (involving instantons), might be thought of in a moment quantity.

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14. A Lagrangian manifold with complex germ (A k , rn) will be called dissipative and be denoted by [A k , r n ], if the following condition holds: (r3) The matrix (1/2i)(C* B - B*C) is nonnegative and is of rank (n - k). This condition will be called the dissipation condition. Further, we shall consider only dissipative Lagrangian manifolds with complex germ, which will be simply called Lagrangian manifolds with complex germ. §7. 5. , it is independent of the basis ai = (aw / aai, a z / aai), i = 1, ...

9. A k-(complex)-dimensional plane (TAk) in C 2n such that (TAk)compl(U) n lR2n = TAk(u) is said to be the C-complexified tangent plane to the manifold Ak at the point u E Akin the 2n-dimensional phase space 2 n. If the plane T Ak (u) at the point u E Ak with coordinates T is given by the equations c k P= L PTj (T)aj, j=l k q= L QTj (T)aj, j=l where aj E lRl, j = 1, ... ,k, are parameters, then the plane (TAk)compl is obviously defined by the same equations, and the parameters aj belong to C 1 .

By equating coefficients at the same powers of a, we obtain the following assertion. 2. Let J = 8q{1) j8a =1= 0 for t E [0, T]. 9) exists in a closed neighborhood of the trajectory x = Q(t) and can be represented in the form a = fJl + fJ2 + fJ3, where Wl(al, t) being a smooth function. 24)) in powers of al near the point al = O. 1, we obtain where w2(al, t) is a smooth function. 1. P(t) 8q{1) 8p(1) + 8 a 8a ' II. 2, we transform the function S(a(x, t), t) 8(a(x, t), t) = 8(0, t) 8q(1) + 8a P(t)({31 + (32) 1 (8 2q +"2 8a2 I 0=0 P(t) + 8q(1) 8P(1)) 8a 8a a~ + a~w2(a1' t) I 8q(1) 1 8 2q 2 =8(0,t)+P(t)-8 a1--P(t)82 a1 a 2 a 0=0 2 18 q I 2 1 8q(1) 8p(1) 2 3 + "2 8a2 0=0 a1 P (t) +"2 8a a;-a1 + a1 W2(a1, t) = 8 0 (0) t 8q(1) + 10 (PQ - H(P, Q, r)) dr + P(t) 8a a1 1 8q(1) 8p(1) 2 3 +"2 8a 8a a1 + a1 W2(a1, t).

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