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I am trying to understand how one uses the theory of quadratic forms to prove self adjointness of various operators.

Here are some example questions on the SE proving self adjointness in this way.

(1) quantum harmonic oscillator operator

(2) associated Legendre operator

They have provided sources (for example Kato's book) which develop the theory of quadratic forms and lead to the theorems that are claimed to be used to show self adjointness of the above linked operators.

What I do not understand however, is that the theory presented in Kato's book is done over a complex Hilbert space.

In examples (1) and (2) the Hilbert space is $L^2(-1,1)$ which is real. How is it they can use the theory developed in Kato to show self adjointness?

Is there a way to show that the above real scenario can be done as a special case of the complex scenario?

valcofadden
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  • Why do you say it is a real Hilbert space? In general in quantum mechanics Hilbert spaces are considered complex. Are you interested anyway in seeing them as real Hilbert spaces ? – Alan Garbarz Jan 18 '22 at 22:39
  • I would like to use the results from Kato's book to show self adjointness of the above operators. IIRC in both cases, the domain of the operators, say $D(T)$ can be defined as real function spaces. If I were to take the Hilbert space $L^2(\mathbb{C})$ instead of $L^2(\mathbb{R})$, I don't think $D(T)$ would remain dense inside its Hilbert space, which is required for $T^*$ to exist. Also, with the application I have in mind, $D(T)$ would contain functions taking exclusively real values. – valcofadden Jan 19 '22 at 14:38
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    Ok. I would write those spaces as $H_1=L^2((-1,1),\mathbb{R})$ and $H_2=L^2((-1,1),\mathbb{C})$ just to be as clear as possible. The former has a real inner product $(f,g)1=\int{-1}^1 dx f(x) g(x)$ and the latter a complex inner product $(f,g)2=\int{-1}^1 dx \overline{f(x)} g(x)$. Note that both inner products coincide for real functions. I am not sure, but possibly you can think of the inclusion map as a way to consider the real Hilbert space as a sub Hilbert space of the complex one and apply Kato's machinery. – Alan Garbarz Jan 19 '22 at 14:55
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    Sorry, I am not liking my advice now: $H_1$ is real and $H_2$ is complex as vector spaces so it does not make sense to consider $H_1$ a subspace of $H_2$ unless first you take the real structure of $H_2$ (consider the real part of its inner product to obtain a real Hilbert space) – Alan Garbarz Jan 19 '22 at 15:05

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$\newcommand{\IC}{\mathbb C}$

As discussed in the comments, $L^2(\Omega)$ can mean either the space of real- or of complex-valued square-integrable functions, and in quantum mechanics, you really need complex Hilbert spaces for results like Stone's theorem.

For quadratic forms however, real Hilbert spaces work just as well. If you have a quadratic form $q$ on a real Hilbert space $H$, define a quadratic form $q^{\IC}$ on the complexification $H^\IC$ by $$ q^\IC(f+ig)=q(f)+q(g) $$ for $f,g\in H$. It is not hard to see that the relevant properties of $q$ (dense domain, closability, closedness) carry over to $q^\IC$. Moreover, if $q$ is densely defined and closed and $A$ the positive self-adjoint operator associated with $q^\IC$, then the property $q^\IC(\bar u)=q^\IC(u)$ translates to $u\in D(A)$ iff $\bar u\in D(A)$ and $A\bar u=\overline {Au}$. In particular, $A$ maps $D(A)\cap H$ into $H$, and one can show that $A|_{D(A)\cap H}$ is a positive self-adjoint (real-linear) operator on $H$ associated with $q$.

MaoWao
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