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Chametz found in one’s home after Pesach [and he did not do Mechiras Chametz]:

Found Chametz immediately after Pesach:^[1] In the event that [one did not sell his Chametz to a gentile over Pesach and he] found Chametz in his home immediately after Pesach, and it is certain to have been owned over Pesach, then the Chametz is prohibited in benefit. This applies even if one properly searched his home for Chametz on the night of the 14^th as the Sages instituted, and [afterwards] nullified all the Chametz that remained in his possession that he did not find in his Bedika, nevertheless it remains forbidden in benefit.^[2] This applies even if he has witnesses that he nullified all his Chametz or made it legally ownerless by saying that it is actually Hefker, before Pesach.

Found Chametz sometime after Pesach:^[3] If the Chametz was only found some time after the conclusion of Pesach, with enough time for the Chametz to have fallen there after Pesach, then it is disputed as to whether one may benefit from this questionable Chametz, as explained in D.^[4] Practically, one is to be stringent not to eat the Chametz, unless it’s a case of great loss [and the Chametz cannot be sold]. One may however benefit [i.e. sell] the Chametz according to all opinions.^[5]

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[1] Admur ibid; Michaber 448:5 regarding Bittul; Tur in name of Yerushalmi and Rosh; Maharam Chalvah; Peri Chadash 448:5; Elya Raba; P”M 448 A”A 8; Noda Beyehuda; Chasam Sofer; Makor Chaim; See M”B 448:25 Shaar Hatziyon 448:68

Other opinions: Some Poskim rule that if one did Bedika and Bittul to his Chametz before Pesach, then it is permitted in benefit after Pesach even for the person himself. [Opinion brought in Admur that is quoted in Sheiris Yehuda O.C. 14; Chok Yaakov; Mateh Yehuda; Magen Haelef; Rashbatz 199; See M”B ibid and Shaar Hatziyon 448:69]

^[2] The reason: When a Jew does Bedikas Chametz before Pesach as the sages enacted and then does Bittul to any Chametz that he did not find, then even if after Pesach he finds Chametz in his home, he has not transgressed even a Rabbinical prohibition of Baal Yiraeh and Baal Yimatzeh on this Chametz that is found in his possession after Pesach. Nevertheless, if the Chametz is found immediately after Pesach then the Chametz is prohibited [even in] benefit, even if he has witnesses that he nullified all his Chametz or made it legally ownerless by saying that it is actually Hefker, before Pesach. The reason for this is because if we allow one to benefit from such Chametz then there is suspicion that every person will intentionally not destroy his Chametz and then after Pesach will claim that he made it hefker before Pesach in order that he be allowed to benefit from it. [Admur ibid]

[3] Admur 448:30

[4] Some Poskim rule that the above prohibition on the Chametz only applies by Chametz found immediately after Pesach, being that it was definitely around by Pesach. However, if it was found after Pesach with enough time for the Chametz to have fallen there after Pesach, and thus we do not even know for certain that it was around on Pesach, then one may be lenient and even permit the Chametz to be eaten, as by every Rabbinical prohibition, when there is doubt we are allowed to be lenient. However, other opinions rule that any Chametz which has even mere doubt as to if it was around on Pesach is forbidden. [Admur ibid]

[5] Admur ibid