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On the domains of minimal and maximal operators for regularisable singular differential expressions

  • D. Race (a1)

Certain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space , for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.

This is contrasted with a known property of the corresponding expression domains. Whereas for an expression M, the operator domains contain only functions y for which both y and My lie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary for My to exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.

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2J. P. Boyd . Sturm-Liouville problems with an interior pole. J. Math. Phys. 22 (1981), 15751590.

5W. N. Everitt , J. Gunson and A. Zettl . Some comments on Sturm-Liouville eigenvalue problems with interior singularities. J. App. Math and Phys. (ZAMP) 38 (1987), 813838.

9W. N. Everitt and A. Zettl . Products of differential expressions without smoothness assumptions. Quaestiones Math. 3 (1978), 6782.

11M. S. Homer . Boundary value problems for the Laplace tidal wave equation. Proc. Roy. Soc. London Ser. A 428 (1990), 157180.

13A. Zettl . Formally self-adjoint quasi-differential operators. Rocky Mountain J. Math. 5 (1975), 453474.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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