<p>We study the Fröhlich polaron in the regime of strong coupling and prove the asymptotically sharp lower bound on the effective mass <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi mathvariant="normal">eff</mi> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msup> <mi>α</mi> <mn>4</mn> </msup> <msub> <mi>m</mi> <mi mathvariant="normal">LP</mi> </msub> <mo>−</mo> <mi>C</mi> <msup> <mi>α</mi> <mrow> <mn>4</mn> <mo>−</mo> <mi>ϵ</mi> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$m_{\mathrm{eff}}(\alpha )\geq \alpha ^{4} m_{\mathrm{LP}}-C\alpha ^{4- \epsilon }$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi mathvariant="normal">LP</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$m_{\mathrm{LP}}$</EquationSource> </InlineEquation> is the constant predicted by L. D. Landau and S. I. Pekar [<CitationRef CitationID="CR13">13</CitationRef>] 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 <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mi>α</mi> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msub> <mi>m</mi> <mi mathvariant="normal">eff</mi> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>m</mi> <mi mathvariant="normal">LP</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\underset{\alpha \rightarrow \infty }{\lim }\alpha ^{-4}m_{ \mathrm{eff}}(\alpha )=m_{\mathrm{LP}}$</EquationSource> </InlineEquation>, which has been conjectured by H. Spohn [<CitationRef CitationID="CR27">27</CitationRef>] in 1987.</p>

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

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2026年7月28日

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Morris Brooks

摘要

We study the Fröhlich polaron in the regime of strong coupling and prove the asymptotically sharp lower bound on the effective mass $m_{eff} (α) \geq α^{4} m_{LP} - C α^{4 - ϵ}$ $m_{\mathrm{eff}}(\alpha )\geq \alpha ^{4} m_{\mathrm{LP}}-C\alpha ^{4- \epsilon }$ , where $m_{LP}$ $m_{\mathrm{LP}}$ 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 $lim_{α \to \infty} α^{- 4} m_{eff} (α) = m_{LP}$ $\underset{\alpha \rightarrow \infty }{\lim }\alpha ^{-4}m_{ \mathrm{eff}}(\alpha )=m_{\mathrm{LP}}$ , which has been conjectured by H. Spohn [27] in 1987.