Is it obligatory to say "Assalamu Alaykum" upon entering the house, even if it is empty, and is there any evidence for that?

Greeting upon entering the house is a recommended act, not an obligatory one. It is recommended to greet whether there is someone in the house or not.

The recommendation to greet if there are people in the house is supported by the Prophet's (peace and blessings be upon him) saying: "When a man enters his house, let him say: 'O Allah, I ask You for the best of entry and the best of exit. In the name of Allah we enter, and in the name of Allah we exit, and upon Allah, our Lord, we rely,' then let him greet his family." And his saying to Anas: "O my son, when you enter upon your family, greet them, it will be a blessing upon you and upon the people of your house."

As for when there is no one in the house, it is supported by the general meaning of Allah's Almighty saying: (So when you enter houses, greet yourselves with a greeting from Allah, blessed and good), and by the actions of the Companions, such as Ibn Umar and Mujahid (may Allah be pleased with them). Mujahid said: "If you enter a house where there is no one, say: 'Peace be upon us and upon the righteous servants of Allah.'"