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We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plusa compactly supported potential $q$ on the half-line. We prove the followingresults: 1) a forbidden domain for the resonances is specified, 2) asymptoticsof the resonance-counting function is determined, 3) in each nondegenerate gap$\g_n$ for $n$ large enough there is exactly an eigenvalue or an antiboundstate, 4) the asymptotics of eigenvalues and antibound states are determined athigh 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 antiboundstates, 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 virtualstates, then for any sequence $(\s)_1^\iy, \s_n\in \{0,1\}$, there exists apotential $q$ such that $H$ has $\s_n$ bound states and $1-\s_n$ antiboundstates in each gap open $\g_n$ for $n$ large enough.

Schr\"odinger operator with periodic plus compactly supported potentials on the half-line