## Banach Journal of Mathematical Analysis

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

### Phillips symmetric operators and their extensions

Sergii Kuzhel and Leonid Nizhnik

#### 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 $\gamma \in {L}_{2}\left(\mathbb{R}\right)$.

#### Article information

**Source**

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

**Dates**

Received: 25 November 2017

Accepted: 18 March 2018

First available in Project Euclid: 4 September 2018

**Permanent link to this document**

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

**Keywords**

symmetric operator characteristic function wandering subspace bilateral shift Lebesgue spectrum

#### 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