<p>We study the BCS critical temperature on half-spaces in dimensions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2088_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$d=1,2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2088_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, at least at weak coupling in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2088_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$d=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and weak coupling and small chemical potential in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2088_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$d=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.</p>

BCS Critical Temperature on Half-Spaces

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2025年3月2日

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Barbara Roos,
Robert Seiringer

摘要

We study the BCS critical temperature on half-spaces in dimensions $d=1,2,3$ $d = 1, 2, 3$ with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on $\mathbb {R}^d$ $R^{d}$ , at least at weak coupling in $d=1,2$ $d = 1, 2$ and weak coupling and small chemical potential in $d=3$ $d = 3$ . Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.