The Spectral Theory of Second Order Two-Point Differential Operators II. Asymptotic Expansions and the Characteristic Determinant

✍ Scribed by J. Locker

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 443 KB
• Volume: 114
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1994.1151

✦ Synopsis

In this second paper of a four-part series, we construct the characteristic determinant of a two-point differential operator (L) in (L^{2}[0,1]), where (L) is determined by (\ell=-D^{2}+q) and by independent boundary values (B_{1}, B_{2}). For the solutions (u(\cdot ; \rho)) and (v(\cdot ; \rho)) of the differential equation (\rho^{2} u+u^{\prime \prime}-q u=0) with (u(1 ; \rho)=e^{i p}), (u^{\prime}(1 ; \rho)=i \rho e^{i \rho}) and (v(0 ; \rho)=1, v^{\prime}(0 ; \rho)=-i \rho), asymptotic formulas are established on a half plane (\operatorname{Im} \rho \geqslant-d). Then (u(\cdot ; \rho)) and (v(\cdot ; \rho)) lead to the characteristic determinant (\Delta(\rho)) of (L), which also satisfies an asymptotic formula on (\operatorname{Im} \rho \geqslant-d). 1994 Academic Press, Inc