Abstract: We discuss simple quantum mechanical systems exhibiting unusual self-adjoint extensions. Our basic example is the one-dimensional Schrodinger operator with the potential -x^4: we show that such a system is quantum-mechanically incomplete in the terminology of Reed and Simon, having a four-parameter family of self-adjoint extensions. We characterize them in terms of generalized boundary values and prove that their spectra are purely discrete, of multiplicity one or two depending on the extension chosen. Furthermore, we discuss extension to this behavior to motion on a star graph. Finally, we show how the corresponding Feynman path integral describing the time evolution of such a system can be constructed.