Proof of the Landau-Pekar formula for the effective mass of the polaron at strong coupling

Abstract We study the Fröhlich polaron in the regime of strong coupling and prove the asymptotically sharp lower bound on the effective mass $m_{\mathrm{eff}}(\alpha )\geq \alpha ^{4} m_{\mathrm{LP}}-C\alpha ^{4- \epsilon }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>eff</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≥</mml:mo> <mml:msup> <mml:mi>α</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>LP</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> </mml:math> , where $m_{\mathrm{LP}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>LP</mml:mi> </mml:msub> </mml:math> is the constant predicted by L. D. Landau and S. I. Pekar [13] in 1948 for the effective mass of the classical polaron. Together with the corresponding upper bound, recently verified by R. Seiringer and the author, we confirm the validity of the celebrated Landau-Pekar formula for the effective mass of the Fröhlich polaron $\underset{\alpha \rightarrow \infty }{\lim }\alpha ^{-4}m_{ \mathrm{eff}}(\alpha )=m_{\mathrm{LP}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:munder> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:msup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>eff</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>LP</mml:mi> </mml:msub> </mml:math> , which has been conjectured by H. Spohn [27] in 1987.