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A Schrödinger operator on the axis is considered; its localized potential is the sum of a small potential and certain potentials with contracting supports, which can increase unboundedly when their supports are contracted. Sufficient conditions are presented for the absence (or existence) of eigenvalues for such an operator. In the case where eigenvalues exist, their asymptotic expansion is constructed. Introduction The study of one-dimensional Schrödinger operators with small potential has a long history. In [1, Chapter III, §22], it was proved that, in the case of a small rectangular potential well, the Schrödinger operator has a single eigenvalue, and the asymptotic expansion of this eigenvalue with respect to a small parameter was constructed. The operator (0.1) − d 2 dx2 + δW (x), 0 < δ 1, where W (x) is an integrable real function that decays sufficiently rapidly at infinity, was studied in [2, 3, 4, 5]. By the Birman–Schwinger method, it was proved that if (0.2) 〈W 〉 := ∫ ∞ −∞ W (x) dx > 0, then the operator (0.1) has no eigenvalues; if (0.3) 〈W 〉 ≤ 0, then the operator has a single eigenvalue λδ with the asymptotic expansions (0.4) λδ = −δ 〈W 〉 4 (1 +O(δ)) if 〈W 〉 < 0 and (0.5) λδ = −δ 1 4 Å ∫ ∞ −∞ Å ∫ x −∞ W (t) dt ã2 dx ã2 (1 +O(δ)) in the critical case where 〈W 〉 = 0. 2010 Mathematics Subject Classification. Primary 35J10, Secondary 35P20.

Schrödinger operator on the axis with potentials depending on two parameters