Let H be a self-adjoint operator on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and generate a bounded holomorphic semigroup of angle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frac{\pi }{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>. In this paper, we study the following questions. First, if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$e^{-zH}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>z</mi> <mi>H</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is an integral operator with integral kernel K(z, x, y), then, under what circumstances, K(z, x, y) has a boundary limit and the limit becomes the integral kernel of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$e^{-itH}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>t</mi> <mi>H</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>. On the other hand, if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$e^{-itH}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>t</mi> <mi>H</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is an integral operator with integral kernel K(it, x, y), whether <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ e^{-zH} $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>z</mi> <mi>H</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is also an integral operator and how to construct the integral kernel from K(it, x, y), if it exists. Finally, we apply the results to study the joint continuity of the integral kernel of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$e^{-itH}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>t</mi> <mi>H</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H= -\Delta + V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$V\in L^{2}(\mathbb {R}^\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>ν</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that the Fourier transform of V has compact support.

Boundary limit of the integral kernels of some holomorphic semigroups

期刊论文
发表日期：

2026年5月27日

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Haimeng Luo,
Shiliang Zhao

摘要

Let H be a self-adjoint operator on $L^2(X)$ $L^{2} (X)$ and generate a bounded holomorphic semigroup of angle $\frac{\pi }{2}$ $\frac{π}{2}$ . In this paper, we study the following questions. First, if $e^{-zH}$ $e^{- z H}$ is an integral operator with integral kernel K(z, x, y), then, under what circumstances, K(z, x, y) has a boundary limit and the limit becomes the integral kernel of $e^{-itH}$ $e^{- i t H}$ . On the other hand, if $e^{-itH}$ $e^{- i t H}$ is an integral operator with integral kernel K(it, x, y), whether $ e^{-zH} $ $e^{- z H}$ is also an integral operator and how to construct the integral kernel from K(it, x, y), if it exists. Finally, we apply the results to study the joint continuity of the integral kernel of $e^{-itH}$ $e^{- i t H}$ where $H= -\Delta + V$ $H = - Δ + V$ and $V\in L^{2}(\mathbb {R}^\nu )$ $V \in L^{2} (R^{ν})$ such that the Fourier transform of V has compact support.