The Krein–von Neumann extension revisited

Abstract

We revisit the Krein–von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein–von Neumann extension for singular, general (i.e., three-coefficient) Sturm–Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein–von Neumann extension of the strictly positive minimal Sturm–Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.

Keywords:

• Krein–von Neumann extension
• singular Sturm–Liouville operators
• Bessel and Jacobi-type differential operators

AMS Subject Classifications:

• Primary: 34B09
• 34B24
• 34C10
• 34L40
• Secondary: 34B20
• 34B30

Acknowledgments

We gratefully acknowledge discussions with Jussi Behrndt. We are indebted to Boris Belinskiy for kindly organizing the special session, ‘Modern Applied Analysis’ at the AMS Sectional Meeting at the University of Tennessee at Chattanooga, October 10–11, 2020, and for organizing the associated special issue in Applicable Analysis.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See also Theorems 2 and 5–7 in the English summary on page 492.

2 His construction appears in the proof of Theorem 42 on pages 102–103.