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  "body": "\n## Summary\n\nThe deficiency indices of a symmetric operator are two cardinal numbers that measure how far it is from being self-adjoint.\nThey are the dimensions of the solution spaces of $A^* u = i u$ and $A^* u = -i u$, where $A^*$ is the adjoint.\nBy von Neumann's theory, they decide whether the operator has self-adjoint extensions and describe all of them.\n\n## Explanations\n\n### L1 human\n\nA symmetric operator, such as a derivative acting on functions that vanish near the boundary, may fail to be self-adjoint because its domain is too small.\nThe deficiency indices count the missing directions, in two ways that correspond to the two numbers $i$ and $-i$.\nIf both counts are zero, nothing is missing and the closure is already self-adjoint.\nIf they are equal, the operator can be made self-adjoint by adding the missing directions in a balanced way; if they differ, it cannot.\nFor $-i \\, d/dx$ on functions vanishing near the ends of $[0,1]$, both indices are $1$, matching a one-parameter family of boundary conditions.\n",
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