INVERSE EIGENVALUE PROBLEMS FOR NONLOCAL STURM-LIOUVILLE OPERATORS
INVERSE EIGENVALUE PROBLEMS FOR NONLOCAL STURM-LIOUVILLE OPERATORS
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发表时间：
2012-03
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通讯作者：
L. Nizhnik
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作者：
L. Nizhnik
我们解决了一类具有奇异非局部势和非局部边界条件的Sturm-Liouville算子的逆谱问题。K（x, s）是一个厄米特对称核。如果K(x, s) = v(x)δ(s−x0)+δ(x−x0)¯v(s)，其中δ是Dirac函数，我们有一个具有非局部点势的Schringer算子，Ly≡- d2y (x) dx2 + v(x)y(x0)+δ(x−x0)(y,v)L2(1)。当考虑这样的算子时，如果微分表达式ly(x)= d2y (x) dx2 + v(x)y(x0),xx0,i nt, p (x)y(x0 - 0) =y(x0 +0)=y(x0), y(x0 - 0) - y(x0 +0)=(y,v)L2(1)，则可以避免使用Dirac的δ-函数。请注意，这种非局部算符不仅出现在量子力学中，也出现在其他领域，如扩散过程理论，参见(2)中的相关参考文献。研究了具有周期边值条件的有界区间上具有非局部点势的一维Schringer算子。由于其周期性，我们可以只考虑点非局部势在区间的端点处有支持的情况。然后我们有如下的非局部Sturm-Liouville特征值问题：
We solve the inverse spectral problem for a class of Sturm-Liouville operators with singular nonlocal potentials and nonlocal boundary conditions. K(x, s) is a Hermitian-symmetric kernel. If K(x, s )= v(x)δ(s − x0 )+ δ(x − x0)¯ v(s), where δ is Dirac's function, we have a Schringer operator with nonlocal point potential, Ly ≡− d 2 y(x) dx2 + v(x)y(x0)+δ(x − x0)(y,v)L2 (1). When considering such an operator, one can avoid using Dirac's δ-function if the differential expression ly(x )= d 2 y(x) dx2 + v(x)y(x0) ,xx0 ,i nt h e p o i n tx = x0 is supplemented with the boundary-value conditions y(x0 −0) = y(x0 +0 )=y(x0), y(x0 −0) − y(x0 +0 )=(y,v)L2 (1). Note that such nonlocal operators appear not only in quantum mechanics but in other areas such as the theory of diffusion processes, see the related references in (2). In this paper, we study a one-dimensional Schringer operator with nonlocal point potential on a bounded interval with periodic boundary-value conditions. Because of the periodicity, we can limit the considerations to only the case where the point nonlocal potential has its support at an endpoint of the interval. Then we have the following nonlocal Sturm-Liouville eigenvalue problem: