Two-loop electron self-energy with accelerated partial-wave expansion

Calculations of the two-loop electron self-energy for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>n</a:mi><a:mo>=</a:mo><a:mn>1</a:mn></a:mrow></a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>n</b:mi><b:mo>=</b:mo><b:mn>2</b:mn></b:mrow></b:math> states of hydrogen-like ions are reported, performed to all orders in the nuclear binding strength parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>Z</c:mi><c:mi>α</c:mi></c:mrow></c:math> (where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>Z</d:mi></d:math> is the nuclear charge number and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>α</e:mi></e:math> is the fine structure constant). The presented approach features an accelerated convergence of the partial-wave expansion and allows calculations to be accomplished for nuclear charges lower than previously possible and with a higher numerical accuracy.