what is it called: box potential with one infinite wall The finite square well and the infinite square well problem are well known, however is there a reason that there is almost no reference to the one sided infinite square well? 

Consider a particle with mass $m$ moving in the one dimensional potential $V(x)= \infty$ for $x<0$ , $V(x) = -V_{0}$ for $0\le x \le L$; $V(x) = 0$ for $x>L$
  i) Can you scale this problem so that all units drop out ?
  ii) Can you find the boundstate eigenenergies and associated wavefunctions as a function of the parameter $\lambda$?
  iii) Can you find the free eigenstates which are characterized by the eigenenergies $E\ge 0$? 

I searched Griffiths Quantum Mechanics, but it didn't have any clue to how to solve this. 
 Can anybody tell me the proper formal name, so I can look it up, or tell me a reason why none such does exist?  
 A: The one-sided infinite square well eigenfunctions are all the odd numbered eigenfunctions of the finite square well twice as wide, by reflection symmetry. The odd-parity solutions obey the boundary conditions for the infinite square well, so this is exactly the same problem as the symmetric finite square well.
A: I think I have heard this potential called the semi-infinite square well. The name makes sense, at least; if I were going to pick a name for it that's probably what I would choose.
In any case, whether the potential has a name or not should have no effect on your ability to solve the corresponding Schrödinger equation. :-P
A: In ye olde nuclear physics (i.e. from the time of the liquid drop model and the semi-empirical mass formula) these were often called "hard core" potentials. The term is not exclusive to a rectangular well, however: the defining feature is the (effectively) infinite potential at low radius.
Very useful things, too.
A: http://chemistry.illinoisstate.edu/standard/che460/handouts/460PinHalfWell.pdf
Refer this pdf for this problem (detailed solution of this problem)
