Despite all the efforts of the Obama administration, the prospects for a peace agreement between Israelis and Palestinians have seldom seemed dimmer. “The core issues are unbridgeable,” said Shlomo Avineri, a former director general of the Israeli foreign office who is open to compromise between Arabs and Israelis. For Dr. Ghassan Khatib, a spokesman for the Palestinian Authority who is also in favor of a two-state solution, “nothing has moved forward except illegal settlements’’ in the past 20 years.