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On the spectrum of second-order differential operators with complex coefficients. (English) Zbl 0944.34018

An extension of the Weyl limit-point, limit-circle classification for the Sturm-Liouville equation with a complex-valued potential on \([a,b)\), where \(-\infty<a<b\leq\infty\) and \(a\) and \(b\) are the endpoints regular and singular, respectively, was obtained by A. R. Sims [J. Math. Mech., Vol. 6, 247-285 (1957; Zbl 0077.29201)]. The authors establish an analogue of the Sims theory to the equation \[ -(py')'+qy=\lambda wy, \] where \(p\) and \(q\) are complex-valued, and \(w\) is a positive weight function. An \(m\)-function is constructed and a relationship between its properties and the spectrum of corresponding \(m\)-accretive operators is analysed.

MSC:

34B24 Sturm-Liouville theory
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain

Citations:

Zbl 0077.29201
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