This paper is dedicated to the study of the problem of existence of homoclinic solutions to a Schrödinger equation of the form \[ x''-V(t)x+x^3=0,\eqno{(1)} \] where \(V:\mathbb R\to\mathbb R\) is a \(L^1\)-function. The authors assume that there exist \(00\) and for any integer \(M\geq 1\), there exists \(T^\ast=T^\ast(M)>0\), such that if \[ \begin{aligned} & t_1- t_0\geq S^\ast,\\ & t_{j+1}-t_j\geq T^\ast,\quad j=1,\dots,2K-1,\\ & t_{2K+1}- t_{2K}\geq S^\ast, \end{aligned} \] then there exist \(M^{2K-1}\) geometrically distinct globally defined positive solutions \(u\) of (1) such that \[ \lim\limits_{t\to+\infty}(u(t),u'(t))=(0,0). \]