## Banach Journal of Mathematical Analysis

### Phillips symmetric operators and their extensions

#### Abstract

This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on $\mathbb{C}_{+}$. We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator $i\frac{d}{dx}+\alpha\delta(x-y)$ leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential $\gamma\in{L_{2}(\mathbb{R})}$.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 995-1016.

Dates
Accepted: 18 March 2018
First available in Project Euclid: 4 September 2018

https://projecteuclid.org/euclid.bjma/1536048015

Digital Object Identifier
doi:10.1215/17358787-2018-0009

Mathematical Reviews number (MathSciNet)
MR3858758

Zentralblatt MATH identifier
06946300

Subjects
Primary: 47B25: Symmetric and selfadjoint operators (unbounded)
Secondary: 47A10: Spectrum, resolvent

#### Citation

Kuzhel, Sergii; Nizhnik, Leonid. Phillips symmetric operators and their extensions. Banach J. Math. Anal. 12 (2018), no. 4, 995--1016. doi:10.1215/17358787-2018-0009. https://projecteuclid.org/euclid.bjma/1536048015

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