Hydrogen-Like Schrödinger Operators at Low Energies

Consider a Schrödinger operator on an asymptotically Euclidean manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy’s second 2nd-microlocal approach, “the Lagrangian approach,” we analyze – uniformly, all the way down to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E equals 0"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">E=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> – the output of the limiting resolvent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-parenthesis upper E plus-or-minus i Baseline 0 right-parenthesis equals limit Underscript epsilon right-arrow 0 Superscript plus Baseline Endscripts upper R left-parenthesis upper E plus-or-minus i epsilon right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo> ± </mml:mo> <mml:mi>i</mml:mi> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ϵ </mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:munder> <mml:mi>R</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo> ± </mml:mo> <mml:mi>i</mml:mi> <mml:mi> ϵ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R(E\pm i 0) = \lim _{\epsilon \to 0^+} R(E\pm i \epsilon )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E equals 0"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">E=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In fact, we will construct a compactification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma 1 right-bracket Subscript upper E Baseline times upper X"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mi>E</mml:mi> </mml:msub> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,1]_E\times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">r\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime. So, in summary, we develop the low-energy scattering theory of attractive Coulombic potentials in the time-independent formalism, in the process studying delicate behavior at low energy and large scales.