On certain high-order partial differential expressions with δ-potential

Abstract

The paper is devoted to the study of the formal differential expressions (−Δ) ^l u+a δ u for arbitrary l∈ℕ and arbitrary dimension of the space ℝ ^d . Approximations of the singular part by means of a family of rank-one operators are constructed and resolvent convergence of this family is investigated. It is demonstrated, that the construction of self-adjoint operators in the space L ² (ℝ ^d ), corresponding to this expression, is connected with the problem of multiplication of distributions.

Keywords:

• multiplication of distributions
• operators with δ-potential
• self-adjoint extension
• resolvent convergence
• resonance
• rank-one perturbation

AMS Subject Classifications :

• 46F99
• 35P99
• 81Q051
• 44A05

Acknowledgements

The author thanks to the reviewer for several helpful remarks. This work was supported by the Ministry of Science and High Education (Poland), grant N 201 382634.