<p>In this paper, we study the following magnetic Schrödinger operator in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_473_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>: <Equation ID="Equ26"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_473_Article_Equ26.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="150" /> </MediaObject> <EquationSource Format="TEX">$H=(i \nabla +A)^2- \tilde{V},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>H</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mover accent="true"> <mi>V</mi> <mo stretchy="false">~</mo> </mover> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_473_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\tilde{V}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>V</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> is non-negative potential supported over the tube built along a curve which is a local deformation of a straight one, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_473_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$B:=\textrm{curl}(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>:</mo> <mo>=</mo> <mtext>curl</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a non-zero and local (i.e., a compact supported) magnetic field. We prove that the magnetic field does not alter the essential spectrum of this system and establish a sufficient condition for the discrete spectrum to be empty.</p>

Three-dimensional magnetic Schrödinger operator with the potential supported in a tube

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发表日期：

2025年9月17日

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Diana Barseghyan,
Juan Bory-Reyes,
Baruch Schneider

摘要

In this paper, we study the following magnetic Schrödinger operator in $\mathbb {R}^3$ $R^{3}$ : $H=(i \nabla +A)^2- \tilde{V},$ $H = {(i \nabla + A)}^{2} - \tilde{V},$ where $\tilde{V}$ $\tilde{V}$ is non-negative potential supported over the tube built along a curve which is a local deformation of a straight one, and $B:=\textrm{curl}(A)$ $B : = curl (A)$ is a non-zero and local (i.e., a compact supported) magnetic field. We prove that the magnetic field does not alter the essential spectrum of this system and establish a sufficient condition for the discrete spectrum to be empty.