Abstract
Conditions characterizing the bound states (B.S.) of a one-dimensional well arc very well known (1.2) as well as extended conditions in higher dimensions in some particular co-ordinate systems (3.4) (mainly spherical co-ordinates in R3). Another extension however consists in characterizing, in one dimension, the B.S. of the well repeated two times (double well) and the band structure of the well repeated an infinite number of times (periodic well). The technique used for the proofs in the three kinds of wells are rather different and range from integral-equation technique to L.C.A.0. approach, W.K.J.B. approximations, phase method . . . . . This variety of methods impedes studying the natural problem which consists in comparing the point or band spectrum of the three wells. Let us therefore, using the phase approach, present a unified way connecting the single, double or periodic wells. We study only the zero-energy case or equivalently the (( critical strength ,) of the potential introducing B.S. or bands in the wells. The fundamental potential is a symmetrical one:
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