A perturbation technique which is superior to the weak-coupling WKB method

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Abstract
We formulate a perturbation theory for the Schrödinger equation which we believe makes the weak-coupling WKB method in many practical cases obsolete. In particular we show that the entire solution consists of various pieces which are of two distinct types: one type in terms of parabolic cylinder functions (valid near an extremum of the potential), and another type in terms of certain exponential functions (valid in regions away from an extremum). Both types are similarly constructed and can be matched in regions of common validity. Below the turning point of an appropriately constructed function the argument of the wave function is real, and one and the same eigenvalue expansion is obtained together with both types of solutions. Above such a point the argument is complex, and the wave function is formulated in terms of an auxiliary parameter determined from the secular equation. Finally it is shown that the systematics of our approach also permits the generation of the exponential type of solution by the second-quantization procedure in analogy to the well-known method used for the harmonic oscillator. In the subsequent paper the large-order behavior of our solutions is derived.

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