相对论薛定谔算符在紧支撑势下基态的体行为
Bulk behaviour of ground states for relativistic Schrödinger operators with compactly supported potentials
摘要：
我们提出了一个有质量和无质量的薛定谔算子与一个势阱的基态的概率表示，在该势阱中的行为被描述的矩母函数的第一个出口时间从井，和外部行为的拉普拉斯变换的首入时到井。这使得分析他们的行为在短期至中期范围内从原点。在第一部分中，我们得到精确的估计这两个泛函的稳定和相对论稳定的过程。接下来，通过结合标度性质和热核估计，我们得到了明确的本地率的基态的给定家庭的非局部薛定谔算子内部和外部的井。我们还展示了这种方法如何扩展到完全支持衰减的潜力。通过对势阱边缘附近的分析，我们进一步证明了基态的变化规律，这种规律定性地依赖于非定域算符的分数幂。
英文摘要：
We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment generating function of the first exit time from the well, and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.
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