  The problem of construction of a simple one - dimensional Hamiltonian whose
spectrum coincides with the set of primes is considered. We note that
quasiclassically a Hamiltonian whose spectrum has the same counting function as
that of the primes in the leading order (i. e. integral logarithm) can be
constructed with the function V(x) whose inverse is asymptotically given by
x=li(V^(1/2)). Hamiltonians, whose spectra coincide with the first 1 - 1000
primes are constructed numerically and their fractal structure is revealed. The
possibility of using such a "prime-generating" Hamiltonian for investigation of
certain number-theoretical problems is discussed.
