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A criterion for compactness in Lp(ℝ) of the resolvent of the maximal general form Sturm–Liouville operator

Part of: Boundary value problems Ordinary differential operators

Published online by Cambridge University Press:  19 April 2016

N. A. Chernyavskaya  and
L. A. Shuster
N. A. Chernyavskaya
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel (nina@math.bgu.ac.il)
L. A. Shuster
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel (miriam@macs.biu.ac.il)
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Extract

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We consider the equation

where and

We assume that this equation is correctly solvable in Lp(ℝ). Under these assumptions, we study the problem of compactness of the resolvent of the maximal continuously invertible Sturm–Liouville operator . Here

In the case p = 2, for the compact operator , we obtain two-sided sharp-by-order estimates of the maximal eigenvalue.

Keywords

Sturm–Liouville operatorcompactness of resolventestimates of minimal eigenvalues

MSC classification

Secondary: 34B24: Sturm-Liouville theory 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 3 , June 2016 , pp. 483 - 508
Copyright
Copyright © Royal Society of Edinburgh 2016