TY - JOUR

T1 - Construction of self-adjoint differential operators with prescribed spectral properties

AU - Behrndt, Jussi

AU - Khrabustovskyi, Andrii

N1 - Funding Information:
The authors are most grateful to the anonymous referee for many useful comments which improved the manuscript considerably. In particular, the referee kindly provided us with an elegant proof of Proposition A.3 and Corollary A.4. This article is based upon work from COST Action CA18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu. J. B. gratefully acknowledges support for the Distinguished Visiting Austrian Chair at Stanford University by the Europe Center and the Freeman Spogli Institute for International Studies. A. K. is supported by the Austrian Science Fund (FWF) under Project No. M 2310-N32.
Funding Information:
The authors are most grateful to the anonymous referee for many useful comments which improved the manuscript considerably. In particular, the referee kindly provided us with an elegant proof of Proposition A.3 and Corollary A.4 . This article is based upon work from COST Action CA18232 MAT‐DYN‐NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu. J. B. gratefully acknowledges support for the Distinguished Visiting Austrian Chair at Stanford University by the Europe Center and the Freeman Spogli Institute for International Studies. A. K. is supported by the Austrian Science Fund (FWF) under Project No. M 2310‐N32.
Publisher Copyright:
© 2022 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.

PY - 2022/6

Y1 - 2022/6

N2 - In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

AB - In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

KW - boundary condition

KW - differential operator

KW - discrete spectrum

KW - essential spectrum

KW - Neumann Laplacian

KW - Schrödinger operator

KW - singular potential

UR - http://www.scopus.com/inward/record.url?scp=85129456788&partnerID=8YFLogxK

U2 - 10.1002/mana.201900491

DO - 10.1002/mana.201900491

M3 - Article

AN - SCOPUS:85129456788

SN - 0025-584X

VL - 295

SP - 1063

EP - 1095

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

IS - 6

ER -