The notion of bounded states in quantum mechanics and their characterization with operators 
Is there any case of potential $V$, such that the continuity of the operator
  $H=c\ \Delta+V$
  is not spoiled? 

And I don't know any non-differnetial operator examples for continous spectra. I guess I don't know the full spectrum of operators (ba dum tss!).
My question comes about as I wonder about the justification of the term "bounded state", especially in relatoion to the fairly straight forward classical concept.
 A: There are many potentials V such that there are no bounded states : For instance all positive potentials which converge to zero at infinity (at least if they are not to pathological).
In this case there might be resonances with finite lifetime, but eventually the particle will allways escape to infinity.
A: The trivial case is where V=const.  Any potentials that offer boundaries, I believe will always create non-continuous spectra due to the nature of differential equations and boundary conditions.
A: Examples of operators with continuous spectrum that are not differential operators in the position representation are the components of the position operator. 
Adding to the free Hamiltonian a repulsive potential doesn't change the continuous spectrum, except for shifting the ground state energy. It also introduces no resonances.
In order to have a bound state, the minimal potential at infinity must be larger than the minimal finite potential. (But because of quantum effects, this is only a necessary condition.)
The reason a bound state is called bound is that the probablility of moving far apart decays exponentially with the distance. 
A: Each time the discrete spectrum is not empty, you have bound states. More precisely these bound states are the eigenfunctions corresponding to the discrete spectrum ( proved mathematically by S.Agmon's theory of exponential decaying solutions of second-order elliptic equations [review] ).
Each operator $H$ of your type that has compact resolvent has purely discrete spectrum, so there is an orthonormal basis of bound states. To see some further characterization of operators $-\Delta + V$ with purely discrete spectrum see this B.Simon paper.
