Hi, I discussed this with some friends but we could not figure out a proof. Usually when considering bound states of the Schrödinger equation of a given potential V(x) one assumes that the wave function converges to zero for large x. One could argue that this is due to the requirement that the wave function is square integrable, i.e. an Element of L². But mathematically this is not necessary. One can construct wave functions with peaks, where the width of the peaks shrinks to zero, the distance between two peaks increases and the height of the peaks increases for growing x. That means that the wave function does not converge to zero, but nevertheless it remains square integrable (provided that the width decreases fast enough). This may seem artificial and the potential need not make sense physically, but it is perfectly valid mathematically. What I have not used is the fact that we are talking about a bound state, i.e. it has an eigenvalue E<0 in the discrete spectrum of the Hamiltonian. Is there an argument using the discrete spectrum that forces the wave function to decay faster than 1/x or something like that? Or is it possible that something like these pathologically defined wave functions could be bound states of some strange Hamiltonian?