  We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus
a compactly supported potential $q$ on the half-line. We prove the following
results: 1) a forbidden domain for the resonances is specified, 2) asymptotics
of the resonance-counting function is determined, 3) in each nondegenerate gap
$\g_n$ for $n$ large enough there is exactly an eigenvalue or an antibound
state, 4) the asymptotics of eigenvalues and antibound states are determined at
high energy, 5) the number of eigenvalues plus antibound states is odd $\ge 1$
in each gap,
  6) between any two eigenvalues there is an odd number $\ge 1$ of antibound
states, 7) for any potential $q$ and for any sequences $(\s_n)_{1}^\iy, \s_n\in
\{0,1\}$ and $(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$, there exists a potential
$p$ such that each gap length $|\g_n|=\vk_n, n\ge 1$ and $H$ has exactly $\s_n$
eigenvalues and $1-\s_n$ antibound state in each gap $\g_n\ne \es$ for $n$
large enough, 8) if unperturbed operator (at $q=0$) has infinitely many virtual
states, then for any sequence $(\s)_1^\iy, \s_n\in \{0,1\}$, there exists a
potential $q$ such that $H$ has $\s_n$ bound states and $1-\s_n$ antibound
states in each gap open $\g_n$ for $n$ large enough.
