微弯层中的束缚态
Bound States in Mildly Curved Layers
摘要：
最近的研究表明，一个非相对论性量子粒子被约束在一个定地完备单连通非紧曲面上的一个定宽硬壁层上，只要曲面不是平面，它就可以有束缚态。本文研究了这些束缚态的弱耦合渐近性，即当表面是一个微曲平面时的情形。在适当的假设下，表面曲率的规律性和衰减，我们推导出基态本征值展开的主导顺序。该论点是基于平面硬壁层中薛定谔算子的Birman-Schwinger分析。
英文摘要：
It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e. the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schroedinger operators in a planar hard-wall layer.
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