English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators$H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the operator-valueddifferential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with$V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assumeregularity of the left endpoint $a$ and the limit point case at the rightendpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^*\in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at theleft endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with$u$ lying in the domain of the underlying maximal operator $H_{\max}$ in$L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish theexistence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the correspondingWeyl-Titchmarsh $m$-function $m_{\alpha}$ and its Herglotz property, anddetermine the structure of the Green's function of $H_{\alpha}$.  Developing Weyl-Titchmarsh theory requires control over certain(operator-valued) solutions of appropriate initial value problems. Thus, weconsider existence and uniqueness of solutions of 2nd-order differentialequations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on}\, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following generalassumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval,$x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurableoperator-valued function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$,and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbertspace. We also study the analog of this initial value problem with $y$ and $f$replaced by operator-valued functions $Y, F \in \cB(\cH)$.  Our hypotheses on the local behavior of $V$ appear to be the most generalones to date.

Initial Value Problems and Weyl--Titchmarsh Theory for Schr\"odinger Operators with Operator-Valued Potentials