论文标题
在具有潜力的卷积操作员的光谱上
On spectra of convolution operators with potentials
论文作者
论文摘要
本文着重于$ l_2（\ mathds r^d）$中有界的自我接合运算符的光谱属性是具有可集成卷积内核的卷积运算符的总和，并且是通过连续电位收敛到Infinity的零元素的乘法运算符。我们研究了该操作员的基本和离散光谱。结果表明，总和的基本频谱是卷积操作员基本光谱的结合和电势的图像。然后，我们为存在离散频谱的存在提供了许多足够的条件，并为离散特征值的数量获得下限和上限。特别关注运营商拥有多个离散频谱的许多点的情况。我们还将这项工作中考虑的操作员的光谱特性与古典Schrödinger运营商的光谱特性进行了比较。
This paper focuses on the spectral properties of a bounded self-adjoint operator in $L_2(\mathds R^d)$ being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower and upper bounds for the number of discrete eigenvalues. Special attention is paid to the case of operators possessing countably many points of the discrete spectrum. We also compare the spectral properties of the operators considered in this work with those of classical Schrödinger operators.
