<p>We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in ℝ^<i>n</i>, <i>n</i> ⩾ 4, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted <i>L</i>²-spaces when <i>n</i> ⩾ 5 is odd. For potentials decaying like <i>O</i>(∣<i>x</i>∣^−<i>ρ</i>) for some <i>ρ</i> &gt; <i>n</i> − 1, we obtain, for all dimensions <i>n</i> ⩾ 4, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({(4{\pi}t)}^{-{n \over 2}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mrow> <mi>π</mi> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mrow> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </math></EquationSource> </InlineEquation> corresponding to the decay for the heat equation. These results complete those obtained in [16,22] for dimensions <i>n</i> = 1 and <i>n</i> = 3. The same questions for <i>n</i> =2 are still open.</p>