  In this work, we study inverse resonance problems for the Schr\"odinger
operator on the real line with the potential supported in $[0,1]$. In general,
all eigenvalues and resonances can not uniquely determine the potential. (i) It
is shown that if the potential is known a priori on $[0,1/2]$, then the unique
recovery of the potential on the whole interval from all eigenvalues and
resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then
for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing
for the unique determination of the potential, and for the case $a<1/2$, all
eigenvalues and resonances plus a part of so-called sign-set can uniquely
determine the potential. (iii) It is also shown that all eigenvalues and
resonances, together with a set of logarithmic derivative values of
eigenfunctions and wave-functions at $1/2$, can uniquely determine the
potential.
