The main argument boils down to subject matter and prophecies. In academic circles if a book prophesies about some event unambiguously then it must have been written after the event. Isaiah prophesied both about the destruction of Assyria and the babylonian captivity and the destruction of Babylon, so it must have been written after all three of these events, and the easiest notion is to break it up into three books, with Isaiah part 1 (chapter 1 - 39) focusing on Assyria as the enemy, Isaiah part 2 (40 - 55) focusing on Babylon, and Isaiah part 3 (56-66) has the Persians as enemies. Also if we divide Isaiah this way, then part 1 focuses more on judgement, part 2 on redemption, etc. E.g. not all parts have the same tone and subject matter.
But if you make this split you run into problems with similar passages and literary devices used across the parts, so then you say that there must have been an editing process in which parts of Isaiah 1 were re-written or edited to be in harmony with Isaiah part 2.
In this way, differences between sections of the text are evidence of different books, but harmony between portions of text is proof that the text needs to be broken up even more, etc.
Here is an excerpt from the preface of Hermeneia's commentary to Deutero-Isaiah (Isaiah part 2)
There are a number of reasons for distinguishing between DtIsa and
Isaiah, but the different period of the historical events that form
the background is particularly important. In Isaiah, Israel’s enemies
are the Assyrians. Their empire was destroyed by the Neo-Babylonians
and the Medes at the end of the seventh century B.C.E. For DtIsa, the
antagonist is Babylon. The exile (598/597 B.C.E.) and the fall of
Jerusalem (587/586 B.C.E.) are the preceding events presupposed. In
Isa 44:28* and 45:1* the Persian king Cyrus II (559–530 B.C.E.) is
mentioned. In addition to this disparity in historical background,
there are literary differences in style and genre. The thrust of
Isaiah’s theology is primarily judgment; DtIsa proclaims salvation and
a new beginning. It must be said, however, that the tripartite
division of the book of Isaiah into Isaiah I (chaps. 1–39), Isaiah II
or Deutero-Isaiah (chaps. 40–55) and Isaiah III or Trito-Isaiah
(chaps. 56–66), is no longer unquestionably accepted in research. The
question of how the different parts of the book hang together is
beginning to be raised once more. As H.-J. Kraus rightly puts it, this
is probably not just a “bookbinding problem.” What all three parts
have in common is the theme of Zion/Jerusalem. Common to them also is
the entitling and concept of Israel’s God as “the Holy One”8 and
“king.” There are a number of possible ways of linking the different
parts of the book of Isaiah, and these have continually been
discussed. C. C. Torrey, for example, assumed that Isaiah 34–35 and
40–66 were connected. C. R. Seitz has recently investigated chaps.
36–37, seeing them as a compositional link between Isaiah I and
Deutero-Isaiah. There are relationships in points of detail too.
Isaiah 40 would seem to presuppose Isaiah 6 (the chapter describing
the installation of the prophet Isaiah) or its tradition. The
presupposition in both passages is the concept of the divine council.
The “oracles concerning Babylon” in Isa 13:1–14:23* and 21:1–10* play
a part in the description of Babylon’s downfall in 46:1–2* and
47:1–15*.13 In DtIsa messianic expectations are transferred to Cyrus
(45:1*). The literary problem of the composition of Isaiah I must
undoubtedly be seen in a more differentiated way than Duhm’s
formulation allows. This goes so far as to assume that the combination
of Isaiah I and II had a subsequent influence on the text and
structure of Isaiah I.14 For example, I too believe that Isaiah’s
installation has been moved from the beginning of the prophet’s
biography (which is its proper place, according to the genre) to its
present position in Isaiah 6. Thus the “hardness of heart” (6:10*)
also no longer comes at the beginning; instead we have Isaiah chap. 1,
the people’s guilt and its consequences, and Isaiah chap. 2, the
pilgrimage of the nations to Zion. Trito-Isaiah presents a special
problem. I myself no longer consider the sequence Deutero-Isaiah
(40–55)—Trito-Isaiah (56–66) to be self-evident. Of course the two
have much in common in language and tradition. But the differences
should also be noted. Isaiah 60–62 presents a description of Yahweh’s
return to Zion that makes a more original, primary impression than
52:7–10*, for example, whose theme is the same. In Trito-Isaiah the
tone toward foreigners is clearly more hostile. The relationship to
the cult is also different. And an assimilation of penitential and
confessional liturgical fragments can be detected. Here further
discussion is required.
Baltzer, K. (2001). Deutero-Isaiah: a commentary on Isaiah 40–55. (P.
Machinist, Ed.) (p. 1). Minneapolis, MN: Fortress Press.
I would look at all three Hermeneia commentaries to get a good overview, they have excellent bibliographies and will provide relatively modern academic viewpoints in their introductory sections to each book.