Sturm–Liouville Problems with Transfer Condition Herglotz Dependent on the Eigenparameter: Eigenvalue Asymptotics

Autor: Sonja Currie, Bruce A. Watson, Casey Bartels
Rok vydání: 2021
Předmět:
Zdroj: Complex Analysis and Operator Theory. 15
ISSN: 1661-8262
1661-8254
DOI: 10.1007/s11785-021-01119-1
Popis: We consider a Sturm–Liouville equation $$\ell y:=-y'' + qy = \lambda y$$ on the intervals $$(-a,0)$$ and (0, b) with $$a,b>0$$ and $$q \in L^2(-\,a,b)$$ . Boundary conditions $$y(-\,a)\cos \alpha = y'(-\,a)\sin \alpha $$ , $$y(b)\cos \beta =y'(b)\sin \beta $$ , where $$\alpha \in [0,\pi )$$ and $$\beta \in (0,\pi ]$$ , are imposed, together with transmission conditions rationally-dependent on the eigenparameter via $$\begin{aligned} -\,y(0^+)\left( \lambda \eta -\xi -\sum \limits _{i=1}^{N} \frac{b_i^2}{\lambda -c_i}\right)&= y'(0^+) - y'(0^-),\\ y'(0^-)\left( \lambda \kappa +\zeta -\sum \limits _{j=1}^{M}\frac{a_j^2}{\lambda -d_j}\right)&= y(0^+) - y(0^-), \end{aligned}$$ with $$b_i, a_j>0$$ for $$i=1,\ldots ,N,$$ and $$j=1,\dots ,M$$ . Here we take $$\eta , \kappa \ge 0$$ and $$N,M\in {\mathbb N}_0$$ . In Bartels et al. (Integr Equ Oper Theory 90(3):1–20, 2018), it was shown that this boundary value problem can be posed as a self-adjoint operator eigenvalue problem in a suitable Hilbert space. In the present work we develop eigenvalue asymptotics for this class of problem.
Databáze: OpenAIRE
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