Krein's identity and trace formulas for half-line Schrodinger operators

We consider self-adjoint extensions of the minimal operator generated by the differential expression \cL = - d²/dx²+V on the half-line $[0,\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\"odinger-type and form a one-parameter family formally given by H_α=-d²/dx²+V, α\in[0,π), with the boundary condition \sin(α)f'(0) = \cos(α)f(0) at $x=0$. We derive a formula that relates the resolvent operator of H_α to the resolvent operator of H₀ in terms of the Jost solution corresponding to the underlying differential equation $\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of H_α and the free operator H_α⁽⁰⁾ with $V\equiv 0$ in terms of the parameter α and the Jost function for $\cL u=zu$.

Grantor dc:publisher

University of Tennessee at Chattanooga

Author dc:creator

• Sofo, Philip C.

Contributors dc:contributor

• Nichols, Roger A.
• Barioli, Francesco; Belinskiy, Boris P.; van der Merwe, Lucas C.
• College of Arts and Sciences

• Differential operators
• Schrödinger operator
• Operator theory
• Kreĭn spaces
• Differential equations

Licence

• https://rightsstatements.org/page/InC/1.0/?language=en

Language dc:language

English, eng

Repository record dc:identifier

https://scholar.utc.edu/theses/495

OAI identifier oai:identifier

oai:scholar.utc.edu:theses-1649