Sharp Gaussian upper bounds for Schrödinger semigroups with singular potentials and mixed boundary conditions

This paper establishes sharp Gaussian upper bounds for Schrödinger semigroups generated by operators of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>D</m:mi> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> {-ΔD+V} on the positive orthant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msup> </m:math> {(0,∞)ᵈ} with Dirichlet boundary conditions, where V is a singular complex-valued potential. We prove that if V satisfies a dissipativity condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Re</m:mi> <m:mo>⁡</m:mo> <m:mi>V</m:mi> </m:mrow> <m:mo>≤</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {ReV≤ 0} , a weighted global Kato condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∏</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>d</m:mi> </m:msubsup> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">|</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {∫Ω[∏ᵢ₌₁ᵈxᵢ]|V(x)|\,dx<∞} , and a form smallness assumption, then the perturbed semigroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mi>V</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {TV(t)} admits an integral kernel satisfying the sharp bound <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msubsup> <m:mi>k</m:mi> <m:mi>t</m:mi> <m:mi>V</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo maxsize="260%" minsize="260%">[</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>∧</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∏</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <