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Zitat von Bernhard
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Stattdessen:
https://en.m.wikipedia.org/wiki/Self-adjoint_operator
Subtleties of the unbounded case
In many applications, we are led to consider operators that are unbounded; examples include the position, momentum, and Hamiltonian operators in quantum mechanics, as well as many differential operators. In the unbounded case, there are a number of subtle technical issues that have to be dealt with. In particular, there is a
crucial distinction between operators that are merely symmetric (defined in this section) and those that are
self-adjoint (defined in the next section). In the case of differential operators defined on bounded domains, these technical issues have to do with making an appropriate choice of boundary conditions ...
In the physics literature, the term Hermitian is used in place of the term symmetric. It should be noted, however, that the
physics literature generally glosses over the distinction between operators that are merely symmetric and operators that are actually self-adjoint ...
Although the notion of a symmetric operator is easy to understand, it is not the "right" notion in the unbounded case. Specifically, the spectral theorem applies only to operators that are self-adjoint and not to operators that are merely symmetric. In particular, although the eigenvalues of a symmetric operator are necessarily real, a symmetric operator need not have any eigenvectors, let alone an orthonormal basis of them.