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

By Victor P. Maslov (auth.)

While this e-book used to be first released (in Russian), it proved to develop into the fountainhead of an important circulation of vital papers in arithmetic, physics or even chemistry; certainly, it shaped the foundation of recent technique and opened new instructions for learn. the current English variation comprises new examples of purposes to physics, hitherto unpublished or to be had basically in Russian. Its valuable mathematical concept is to take advantage of topological how you can study isotropic invariant manifolds with the intention to 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 concept and its purposes to quantum information and quantum box thought, in addition to the nonlinear conception (involving instantons), may be thought of in a moment quantity.

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**Sample text**

It is sufficient to prove that the matrix with elements a(x - Q(T),QTj(T)) aTi I X=Q(T) ' i,j = 1, ... , k, is nondegenerate. We have By the assumptions of our lemma, the matrix with elements (QTi' QTj) is nondegenerate. 6 is proved. 42) has an obvious geometrical meaning, namely, the vector (x - Q( T( x))) is normal to the surface {x E jRn : x = Q( T)} at the point y = Q(T(X)). 40), and the parameters (Tl(X, t), ... 42). We shall use this definition of action in order to generalize the formulas obtained to the complex case.

Without loss of generality we can consider the case n = 2 and k = 1. First we calculate the derivatives 8cI>j8t, 8cI>j8T, and 8cI>j8a. We have 8cI> at = . (P, Q + iJ) + (P, v) - 1 . -1 + 2(v,BC v) - H(P, Q, t) 1 2(v,BC + (BC- -1'-1 CC v). 1 v, iJ) §4. Phase and action on Lagrangian manifolds with real germ By using the fact that the functions P7') Qr and Va, variational system, we obtain U 49 a are solution of the ap) I _ (au I ap) - dt aa' aT a=O - aa a=O' aT 13 _ ~(au = (_Hpqaau I a a=O = Hqp (a au I a a=O -HqqaaV I ,-HqpPr-HqqQr) a a=O ,Pr ) - Hqq (aav I ,Qr) = -HqpB - HqqC a a=O and quite similarly 6 = HppB+HpqC (here the arguments P(T, t), Q(T, t), and t of the functions Hpp , Hpq , etc.

12. The function S(x) 1 = So + (P,x - Q) + 2(x - Q,BC-1(x - Q)), where B = oW/OQ, C action on (AO, rn). 4. ;,q consider a zerodimensional Lagrangian manifold (a point) AO = {p = 0, q = O} and the complex germs rl = {w± = ±iQ, Z± = Q}. It is easy to see that the axioms (rl) , (r2) hold for (A ri) and 0 Z± /00'. = 1. Let us calculate the action on (AO,rl). We get S±(x) = st ±ix2 /2, where st are certain real constants. °, This example shows that the imaginary part of an action can be negative. 13.