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Ok, let's take this scenario: Tom and Mary work in the same company. Tom really likes Mary and he wants to love Mary and has very strong romantic affection toward Mary, but Mary always refuses him and has never given any attention to him.
Over time, Tom starts to get depressed since he thinks about his love for Mary all the time, but Mary doesn't care about him. He cannot get Mary's image out of his mind and that makes him depressed.
So, I don't think the word lovesick is suitable since
lovesick = in love, or missing the person one loves, so much that one is unable to act normally
So, lovesick could be two-sided love, but in the above story, there is only one-sided love.
I found the term unrequited love or one-sided love on Wikipedia. However, that term does not express the sickness of the one-sided love.
So, how do I express the idea that a person got sick because they love someone very much but does not receive any attention from that someone?