• Clausius
In summary, the conversation discusses a problem with a linear operator in non-relativistic quantum mechanics, specifically the determination of its domain of self-adjointness. The speaker mentions a book by Gerald Teschl, which covers the self-adjointness problem for different Hamiltonians and provides a link to lecture notes available for free. The speaker also suggests that the operator in question may have the same domain of self-adjointness as the radial part of the hydrogen atom operator.
Clausius
hi everybody, i'd like to discuss with you a problem occurred to me in the study of a central symmetry field in non rel. qm.
at a certain point, i get the linear operator $$P_{r}^{2}=\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}r$$, which is the square of radial momentum, and i want to determine the domain in which it is self-adjoint, i.e. the maximal subset $$D(P_{r}^{2})\in L^{2}(\mathbb{R}_{+},r^{2}\ \mathrm{d}r)$$ such that $$D(P_r^2)=D(P_r^2\dagger)$$ and the two coincide in the domain above.

Goerg Teschl (professor at TU in Vienna) wrote a book based on some lecture notes available somewhere on the internet for free. I think a link to these notes is right here on PF in the <Reference> session. You may search for the latest version of his notes. He has an extended coverage of the self-adjointess problem for different hamiltonians.

I think the $p_{r}^{2}$ operator has the same domain of self-adjointness as the radial part of the H-atom operator. But you may check on that.

Since it's Gerald Teschl, you may have trouble searching for Goerg Teschl, so here's the link ;)

## 1. What is the self-adjointness domain of P_r^2?

The self-adjointness domain of P_r^2 refers to the set of all possible values that the operator P_r^2 can act upon and still result in a self-adjoint operator. In other words, it is the set of all possible inputs that will produce a valid and meaningful output when the operator is applied.

## 2. Why is the self-adjointness domain important?

The self-adjointness domain is important because it determines the validity and usefulness of the operator P_r^2. If an input falls outside of the self-adjointness domain, the resulting output may not make sense or may not accurately represent the intended result. Therefore, understanding the self-adjointness domain is crucial in utilizing the operator effectively.

## 3. How is the self-adjointness domain of P_r^2 determined?

The self-adjointness domain of P_r^2 is determined by analyzing the properties and behavior of the operator. In particular, it is necessary to consider the type of operator (linear, non-linear, etc.), the mathematical properties of the operator, and the nature of the inputs that it can act upon. Using this information, the self-adjointness domain can be defined and identified.

## 4. What are the consequences of an input falling outside of the self-adjointness domain?

If an input falls outside of the self-adjointness domain of P_r^2, the resulting output may be invalid or meaningless. This can lead to incorrect conclusions or interpretations of data, which can have significant consequences in scientific research and analysis. Additionally, it may also cause issues with the stability and accuracy of numerical computations using the operator.

## 5. Are there any techniques for extending the self-adjointness domain of P_r^2?

Yes, there are techniques for extending the self-adjointness domain of P_r^2. One approach is to modify the operator or its properties in a way that expands its self-adjointness domain. Another method is to use mathematical transformations or techniques to map inputs outside of the original self-adjointness domain to valid inputs within the domain. However, these techniques should be used carefully and only after thorough understanding and analysis of the operator.

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