The author considers the nonlinear Schrödinger equation with an additional quadratic potential \[ iu_t+u_{xx}-x^2u=F(u), \quad x\in \mathbb{R}, \;t\geq 0, \] where \(F(u)\) is a cubic, possible nonlocal, nonlinearity satisfying some reasonable conditions. The main results can be summarized as follows: In each eigenvalue of the harmonic oscillator there bifurcates an unbounded branch of nonlinear bound states in the sense of the global bifurcation theorem of Rabinowitz which give rise to the existence of infinitely many nonlinear modes. This theory provides a strict theoretical proof of the existence of a symmetric bi-soliton which was found by numerical simulations. Under slightly more restrictive conditions on the nonlinearity, the bifurcating solutions can be characterized as critical points of the corresponding energy functional. Furthermore, stability and decay properties of the solutions are discussed.