Yep! In the search field above the library window, press the "FT" button on the very left. You have to create the index the first time you use the feature. It takes a while and from there on new books are indexed automatically.
FT search has word and phrase search, boolean operators and NEAR search abilities. And there is a really cool match list, giving some context of the match before you actually go to it in the PDF file.
I've had good luck using Tesseract [0] for scanned PDFs. If you're not CLI-inclined, there are several GUIs for it available [1]. I have had good luck downloading scanned PDFs from archive.org and running them through Tesseract.
Did not know about Calibre for this - I was relying on opening each search and searching it individually.
If you want even faster search across different formats, you can try ripgrep-all ( https://github.com/phiresky/ripgrep-all ). It can search across epub, docx, pdf, zip, mp4 etc. If you are handy with the tool, you can write custom adaptor to search across images using OCR with tesseract.
FT search has word and phrase search, boolean operators and NEAR search abilities. And there is a really cool match list, giving some context of the match before you actually go to it in the PDF file.
You cannot search across libraries, though.