Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant (Q1107037)

[For Part III, see Ann. Phys. 158, 415-420 (1984; Zbl 0596.35028).] The author discusses the asymptotics of the eigenvalues of the Schrödinger operator \(-\Delta +\Delta (V+W)\) in the quasiclassical limit (\(\lambda\) \(\to \infty)\). This is the fourth in a series of papers and continue the second one \((W=0)\). V is a nonnegative potential which has exactly two zeros (remarks on the multi well situation are made in the last section of the present paper) at a,b, and W is vanishing near a and b (apart from some additional assumptions on the potentials). The main result is, that W, that may be considered as a perturbation of V, does not disturb the eigenvalue splitting. However it may influence the ``concentration'' of the ground state in the wells shifting it. The results are related to \textit{B. Helffer} and \textit{J. Sjöstrand} [Commun. Partial Differ. Equations 9, 337-408 (1984; Zbl 0546.35053) and Ann. Inst. Henri Poincaré, Phys. Théor. 42, 127-212 (1985; Zbl 0595.35031)].