The frame
On this site the Iron Age page ends in 586 BCE and the Persian period page begins in 539 BCE. This page fills the gap, and its frame is Judah and Benjamin only. Both boundaries appear in the Met's timeline for the eastern Mediterranean and Syria, on which the site's Levant period pages draw: there Iron Age II is dated about 1000-586 BCE, and Persian rule under the Achaemenid dynasty about 539-331 BCE. Under the key events of 587/586 BCE the timeline states that Jerusalem, the capital of the kingdom of Judah, fell to the Babylonian king Nebuchadnezzar II, who deported a significant part of the population to Babylon. Source 1.
Boundaries and overlaps
586 is a Judahite date, not a boundary for the whole region. For the region as a whole the same timeline has a separate line, "Babylonian rule, ca. 606-539 B.C.", and its overview says that with the fall of the Assyrian empire at the end of the seventh century BCE the Neo-Babylonian empire claimed the region. The years between about 606 and 586, which the timeline already counts as Babylonian rule, therefore remain on the Iron Age page. The same overview speaks of "only fifty years of rule by the Babylonians" before the region was incorporated into the Achaemenid Persian empire: a round figure, closer to 586-539 than to the roughly 67 years of the regional line, and the page does not reconcile the two. The end is marked by the key events of 539-525 BCE: Cyrus the Great releases captive peoples, including the Jews, and allows them to return to their homelands. Source 1.
Phases that cross the boundaries
Within Judah and Benjamin, too, local phases do not stop at the page's boundaries. According to Oded Lipschits, Tell en-Nasbeh, that is Mizpah, served as the capital (bîrah) for 141 years, from 586 BCE until the time of Nehemiah (445 BCE), that is, for decades into the Persian period. Source 3. Liora Freud describes the pottery of "Babylonian Stratum 9/10" in the City of David as a transitional assemblage of the 6th century BCE: vessels made before the 586 destruction continued in use, and completely new types appear only toward the end of the century, at the beginning of the Persian period. Source 2.
On this site
Two articles are placed in this period. "Jerusalem's pottery: a transitional assemblage of the Babylonian period" asks whether Jerusalem was entirely abandoned after 586, on the evidence of bowls that are neither typical of the Iron Age nor typically Persian. "Mizpah and Benjamin: the seat of rule under Babylon" separates excavation data, a demographic model and an administrative interpretation in the question of whether the seat of rule moved from Jerusalem to Mizpah. Together they show both sides of the question in the title of Lipschits's chapter, Judah as an "Empty Land": destruction and the decline of urban life, alongside settlement continuity in Benjamin. Source 3. No museum object on the site is assigned to this period yet.