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arXiv · 2101.01001

On the domains of Bessel operators

Jan Dereziński·Vladimir Georgescu
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Abstract

We consider the Schr\"odinger operator on the halfline with the potential $(m^2-\frac14)\frac1{x^2}$, often called the Bessel operator. We assume that $m$ is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $|\Re(m)|<1$ and of its unique closed realization for $\Re(m)>1$ coincide with the minimal second order Sobolev space. On the other hand, if $\Re(m)=1$ the minimal second order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.

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Jan Dereziński, Vladimir Georgescu. 2021-01-04. On the domains of Bessel operators. https://doi.org/10.1007/s00023-021-01058-9

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