The infrared limit of the SRG evolution and Levinson's theorem

On a finite momentum grid with N integration points <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and weights <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></math> ( <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></math> ) the Similarity Renormalization Group (SRG) with a given generator G unitarily evolves an initial interaction with a cutoff λ on energy differences, steadily driving the starting Hamiltonian in momentum space <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></math> to a diagonal form in the infrared limit ( <math altimg="si5.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo stretchy="false">→</mo><mn>0</mn></math> ), <math altimg="si6.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></msubsup><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></math> , where <math altimg="si7.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></math> is a permutation of the eigenvalues <math altimg="si8.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math> which depends on G . Levinson's theorem establishes a relation between phase-shifts <math altimg="si9.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo stretchy="false">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> and the number of bound-states, <math altimg="si10.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msub></math> , and reads <math altimg="si11.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo stretchy="false">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>−</mo><mi>δ</mi><mo stretchy="false">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>π</mi></math> . We show that unitarily equivalent Hamiltonians on the grid generate reaction matrices which are compatible with Levinson's theorem but are phase-inequivalent along the SRG trajectory. An isospectral definition of the phase-shift in terms of an energy-shift is possible but requires in addition a proper ordering of states on a momentum grid such as to fulfill Levinson's theorem. We show how the SRG with different generators G induces different isospectral flows in the presence of bound-states, leading to distinct orderings in the infrared limit. While the Wilson generator induces an ascending ordering incompatible with Levinson's theorem, the Wegner generator provides a much better ordering, although not the optimal one. We illustrate the discussion with the nucleon–nucleon ( NN ) interaction in the <math altimg="si12.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow><none/><mprescripts/><none/><mrow><mn>1</mn></mrow></mmultiscripts></math> and <math altimg="si13.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow><none/><mprescripts/><none/><mrow><mn>3</mn></mrow></mmultiscripts></math> channels