Minimizing Schrödinger eigenvalues for confining potentials

Abstract We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator −Δ + V in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${L}^{2}({\mathbb{R}}^{d})$ when the integral ∫ e − tV d x is given for some t &gt; 0. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality.