BCS Critical Temperature on Half-Spaces

We study the BCS critical temperature on half-spaces in dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:math> with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mi>d</mml:mi></mml:msup> </mml:math> , at least at weak coupling in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> and weak coupling and small chemical potential in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:math> . Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.