Summary: For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeros of the potentials. The results generalize some recent work of A. Ambrosetti, A. Malchiodi
and W.-M. Ni
[C. R., Math., Acad. Sci. Paris 335, No. 2, 145–150 (2002; Zbl 1072.35068
)] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in the paper cited above.