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.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
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![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![THERE)](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f833d059e4565ca5c84985c780b21f1f89f0b9)
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.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![c](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
![{\ displaystyle \ langle A \ xi, \ xi \ rangle \ geq c \ | \ xi \ | ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5af0ff9e5a023baae0ae6ed0fd361686a7ceb8)
![{\ displaystyle \ xi \ in D (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1dc4159f4e69aa5823ca700e7f65852f51e03c)
![{\ displaystyle \ langle A \ xi, \ xi \ rangle}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13606bfc27ac648677ccdd00077894f271b61493)
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 .
![c](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
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
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ langle A \ xi, \ xi \ rangle \ geq c \ | \ xi \ | ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5af0ff9e5a023baae0ae6ed0fd361686a7ceb8)
![{\ displaystyle \ xi \ in D (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1dc4159f4e69aa5823ca700e7f65852f51e03c)
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![{\ displaystyle \ lambda + c> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a08990145d98bad355b12c4b173507cec0f6b83)
for .
![{\ displaystyle \ xi, \ eta \ in D (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f15949dbc4a63697db999df2aee0cc3195edd556)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e75e113eefa5710aca1ea01bbe78b95040b260b)
![{\ displaystyle \ | \ xi \ | _ {\ lambda}: = {\ sqrt {[\ xi, \ xi] _ {\ lambda}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519ea704a7bec1218f6b2e66e717be437a8945c1)
![THERE)](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f833d059e4565ca5c84985c780b21f1f89f0b9)
![THERE)](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f833d059e4565ca5c84985c780b21f1f89f0b9)
.
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:
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![{\ displaystyle (\ xi _ {n}) _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d11c62d28e78ed357f2199b9aa913f9f071bc3b)
![{\ displaystyle H _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be3b96da50f480b5e88788d14a66462000d6083)
![{\ displaystyle \ xi \ in H _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/962aaf08f1401445cc3dd6d08d43b646f5e57352)
![{\ displaystyle D (A) \ subset H _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6113d1b72402fe17e8627e96ab0833818a50642)
![{\ displaystyle \ xi \ in D (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1dc4159f4e69aa5823ca700e7f65852f51e03c)
![{\ displaystyle \ xi _ {n} = \ xi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50437333240e15aaaa99b9cebfb8e4e43f66d267)
- If with approximating sequences and , then the limit exists and continues the form defined on.
![{\ displaystyle \ xi, \ eta \ in H _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10ab4dd0d8206f8b215f374d5b144f6a18e009ff)
![{\ displaystyle (\ xi _ {n}) _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d11c62d28e78ed357f2199b9aa913f9f071bc3b)
![{\ displaystyle (\ eta _ {n}) _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e387c8f36f6de90aad511f2ac504679847cc078)
![{\ displaystyle [\ xi, \ eta] _ {\ lambda}: = \ lim _ {n \ to \ infty} [\ xi _ {n}, \ eta _ {n}] _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/281db7dd9c71d8ce071ae3c56f8f47f13aad3119)
![THERE)](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f833d059e4565ca5c84985c780b21f1f89f0b9)
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is a Hilbert space with the positive definite form .![{\ displaystyle [\ cdot, \ cdot] _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e75e113eefa5710aca1ea01bbe78b95040b260b)
- If there is also a real number with , then sets which are defined by or norms are equivalent .
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![{\ displaystyle \ mu + c> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df300200d05c80f19f20f10bd125cf93611ab36)
![{\ displaystyle H _ {\ lambda} = H _ {\ mu} \ subset H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8364cbdaa8b1d2554d28ca45223f310ae8ec5fe)
![{\ displaystyle [\ cdot, \ cdot] _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e75e113eefa5710aca1ea01bbe78b95040b260b)
![{\ displaystyle [\ cdot, \ cdot] _ {\ mu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4df523ea7bc60e818184ec7d66a3a1041fff2f)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be3b96da50f480b5e88788d14a66462000d6083)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![HA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f561a051435337d6fd20d1b3a2cb945fe97f3b)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
Friedrichs expansion
Be a semi-bounded operator. Then is symmetric, that is, it holds where is the adjoint operator . One defines
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle A \ subset A ^ {*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a06b174229aaae2edc1f408608acebaee1332e)
![A ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece)
for ,
![{\ displaystyle \ xi \ in D (A_ {F}): = H_ {A} \ cap D (A ^ {*})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e47c886eacab2c424920681e5868a939348dfa5)
so is a self-adjoint operator that expands. is called the Friedrichs extension of .
![{\ displaystyle A_ {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f64867539242b9693f2154d734886d054b39f0d)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle A_ {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f64867539242b9693f2154d734886d054b39f0d)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
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![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece)
![A ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece)
swell
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Hans Triebel : Higher Analysis , Verlag Harri Deutsch, 1980.
Individual evidence
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↑ 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@1@ 2Template: Toter Link / imu2.zib.de