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

Abstract 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> {-\Delta_{D}+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,\infty)^{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> {\operatorname{Re}V\leq 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>&lt;</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {\int_{\Omega}[\prod_{i=1}^{d}x_{i}]|V(x)|\,dx&lt;\infty} , 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> {T_{V}(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> <