Friedrichs expansion

The Friedrichs extension (after Kurt Friedrichs ) is a mathematical construction according to which certain tightly-defined linear operators in Hilbert spaces can be extended to self-adjoint operators .

Semi-constrained operators

We consider a linear operator that is defined on a dense subspace of a Hilbert space . This subspace is called the domain of definition and is denoted by. Under certain circumstances, which are discussed in this article, one can extend the operator to one on a comprehensive subspace so that the extended operator is self-adjoint. ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle D(A)}$ ${\displaystyle A}$ ${\displaystyle D(A)}$

A densely-defined operator is called half-bounded if there is a real number such that for all . Obviously, positive operators are half-bounded and half-bounded operators are symmetric , because by definition all are real. ${\displaystyle A}$ ${\displaystyle c}$ ${\displaystyle \langle A\xi ,\xi \rangle \geq c\|\xi \|^{2}}$ ${\displaystyle \xi \in D(A)}$ ${\displaystyle \langle A\xi ,\xi \rangle }$

In quantum mechanics occurring operators are often semi-limited, which is then available for about a lower energy -Schranke. The question then naturally arises whether such an operator has a self-adjoint extension, which is then a quantum mechanical observable . ${\displaystyle c}$

The concept of the semi-constrained operator was first introduced by Aurel Wintner . Kurt Friedrichs later developed the theory of semi-bounded operators.

Energetic space

Let be a semi-bounded operator with for all and be a real number with . Be ${\displaystyle A}$ ${\displaystyle \langle A\xi ,\xi \rangle \geq c\|\xi \|^{2}}$ ${\displaystyle \xi \in D(A)}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda +c>0}$

${\displaystyle [\xi ,\eta ]_{\lambda }:=\langle A\xi ,\eta \rangle +\lambda \langle \xi ,\eta \rangle }$for . ${\displaystyle \xi ,\eta \in D(A)}$

Then a positively definite form is on and one can therefore define the norm on . is usually not a complete space with this standard ; this leads to the following construction. ${\displaystyle [\cdot ,\cdot ]_{\lambda }}$ ${\displaystyle D(A)}$ ${\displaystyle \|\xi \|_{\lambda }:={\sqrt {[\xi ,\xi ]_{\lambda }}}}$ ${\displaystyle D(A)}$ ${\displaystyle D(A)}$

${\displaystyle H_{\lambda }:=\{\xi \in H;{\rm {Es\,gibt\,eine\,Folge}}\,(\xi _{n})_{n}\,{\rm {in}}\,D(A)\,{\rm {mit}}\|\xi _{n}-\xi \|{\stackrel {n\to \infty }{\longrightarrow }}0\,{\rm {und}}\,\|\xi _{n}-\xi _{m}\|_{\lambda }{\stackrel {n,m\to \infty }{\longrightarrow }}0\}}$.

Note that the first limit condition refers to the Hilbert space norm on . A sequence in the definition of is called an approximating sequence for . Obviously , for one can choose the constant sequence as the approximating sequence . One can now prove the following statements: ${\displaystyle H}$ ${\displaystyle (\xi _{n})_{n}}$ ${\displaystyle H_{\lambda }}$ ${\displaystyle \xi \in H_{\lambda }}$ ${\displaystyle D(A)\subset H_{\lambda }}$ ${\displaystyle \xi \in D(A)}$ ${\displaystyle \xi _{n}=\xi }$

• If with approximating sequences and , then the limit exists and continues the form defined on. ${\displaystyle \xi ,\eta \in H_{\lambda }}$ ${\displaystyle (\xi _{n})_{n}}$ ${\displaystyle (\eta _{n})_{n}}$ ${\displaystyle [\xi ,\eta ]_{\lambda }:=\lim _{n\to \infty }[\xi _{n},\eta _{n}]_{\lambda }}$ ${\displaystyle D(A)}$
• ${\displaystyle H_{\lambda }}$is a Hilbert space with the positive definite form . ${\displaystyle [\cdot ,\cdot ]_{\lambda }}$
• If there is also a real number with , then sets which are defined by or norms are equivalent . ${\displaystyle \mu }$ ${\displaystyle \mu +c>0}$ ${\displaystyle H_{\lambda }=H_{\mu }\subset H}$ ${\displaystyle [\cdot ,\cdot ]_{\lambda }}$ ${\displaystyle [\cdot ,\cdot ]_{\mu }}$

So the space depends only on and not on the particular ; it is therefore designated with and is called the energetic space of . ${\displaystyle H_{\lambda }}$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle H_{A}}$ ${\displaystyle A}$

Friedrichs expansion

Be a semi-bounded operator. Then is symmetric, that is, it holds where is the adjoint operator . One defines ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A\subset A^{*}}$ ${\displaystyle A^{*}}$

${\displaystyle A_{F}\xi :=A^{*}\xi }$for , ${\displaystyle \xi \in D(A_{F}):=H_{A}\cap D(A^{*})}$

so is a self-adjoint operator that expands. is called the Friedrichs extension of . ${\displaystyle A_{F}}$ ${\displaystyle A}$ ${\displaystyle A_{F}}$ ${\displaystyle A}$

Note that in general neither nor is self adjoint. Only through the above clever choice of the domain of definition one obtains a self-adjoint operator located between and , which is the restriction of on this subspace. It is therefore ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle A^{*}}$ ${\displaystyle A\subset A_{F}\subset A^{*}}$

swell

• Hans Triebel : Higher Analysis , Verlag Harri Deutsch, 1980.

Individual evidence

1. ↑ Franz Rellich : Semi-restricted differential operators of higher order  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. (PDF; 702 kB), 1954, accessed June 17, 2011