论文标题
schrödinger方程的耦合常数依赖性具有反方势
Coupling constant dependence for the Schrödinger equation with an inverse-square potential
论文作者
论文摘要
我们考虑一维schrödinger方程$ -f''+q_αf= ef $在正半轴上，潜在的$q_α（r）=（α-1/4）r^{ - 2} $。众所周知，值$α= 0 $在此问题中起着特殊的作用：Hamiltonian的正式差异表达式$ - \ partial^2_r +q_α（r）$的所有自我偶相实现都具有无限的许多特征值$ a <0 $，并且最多可以以$ agenvalue $ $α\ egeq 0 $ ageq 0 $。我们发现自相关边界条件和本征函数扩展的参数化，这些膨胀在$α$中是分析性的，尤其是在$α= 0 $的情况下并不单数。我们采用合适的单数titchmarsh- weyl $ m $ - 功能，明确地找到了所有自我伴侣汉密尔顿人的光谱措施，并证明了它们对$α$和边界条件的平稳依赖性。我们使用公式进行光谱措施，详细分析了通过点$α= 0 $的“相变”如何发生在特征值和汉密尔顿人的连续光谱中。
We consider the one-dimensional Schrödinger equation $-f''+q_αf = Ef$ on the positive half-axis with the potential $q_α(r)=(α-1/4)r^{-2}$. It is known that the value $α=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r + q_α(r)$ for the Hamiltonian have infinitely many eigenvalues for $α<0$ and at most one eigenvalue for $α\geq 0$. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in $α$ and, in particular, is not singular at $α= 0$. Employing suitable singular Titchmarsh--Weyl $m$-functions, we explicitly find the spectral measures for all self-adjoint Hamiltonians and prove their smooth dependence on $α$ and the boundary condition. Using the formulas for the spectral measures, we analyse in detail how the "phase transition" through the point $α=0$ occurs for both the eigenvalues and the continuous spectrum of the Hamiltonians.
