On the Domains of Bessel Operators
Jan Dereziński, Vladimir Georgescu
Abstract
Abstract We consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow></mml:math> , 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 $$|\mathrm{Re}(m)|<1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mi>Re</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> and of its unique closed realization for $$\mathrm{Re}(m)>1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Re</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Re</mml:mi><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> 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.