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  "body": "\n## Summary\n\nPerturbation theory of linear operators studies how the properties of an operator $T$ change under replacement by $T + V$.\nHere $V$ is small in a suitable sense.\nThe main questions are whether closedness, self-adjointness, the [[spectrum|BMKATJ802A]] and the eigenvalues survive the change, and how they move.\nThe basic notion of smallness for unbounded operators is relative boundedness.\nThe theory underlies much of mathematical quantum mechanics, where a Hamiltonian is a free part plus an interaction.\n\n## Explanations\n\n### L1 human\n\nOften one understands a simple operator well, such as the Laplacian, and wants to study a more complicated one obtained by adding a term.\nPerturbation theory says when the new operator still behaves like the old one.\nFor matrices, this is the study of how [[eigenvalues move|VR89YHC0TX]] when the entries change a little.\nFor differential operators, the added term may be unbounded, so one measures it against the original operator instead of against the identity.\nFor example, a bounded real-valued potential $V$ added to the Laplacian on $\\R^n$ never destroys self-adjointness.\n",
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