# “Any physical law which can be expressed as a variational principle describes a self-adjoint operator”

The title is a Wikipedia quote but has no citation associated to it on Wikipedia except a reference to Cornelius Lanczos. I would like to clarify the statement, and explore the relevant concepts in depth. Does anyone know what results the Wikipedia page might be referring to? Can anyone offer insights as to why one might expect a principle such as

"Any physical law which can be expressed as a variational principle describes a self-adjoint operator"

to hold and what it means?

• I can see this maybe happening in quantum mechanics, where a variational principle would describe a hermitian operator, i.e., an observable. But seeing that the source says "citation needed", I'm not sure if the wikipedia entry is very accurate. – Sidd Nov 9 '16 at 1:13
• Any nonlinear equation obtained by a variational principle (e.g. nonlinear Schroedinger or Klein-Gordon) is a counterexample to the above, since a self-adjoint operator is a linear operator. – yuggib Nov 9 '16 at 9:50

So Lanczos only claims the converse, that self-adjointness implies a variational formulation. This is essentially right. Of course, if one wants a variational principle to involve actual optimization rather than just extremals the spectrum also needs to be bounded from below, but that is usually the case in "physical" problems. If $A$ is the operator the variational functional will then be $f(x)=(Ax,x)$ with some constraints on $x$, e.g. normalization and/or boundary conditions.