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# On the number of negative eigenvalues of a multi-dimensional Schrodinger operator with point interactions

### Abstract

We prove that the number $N$ of negative eigenvalues of a Schr\"odinger operator $L$ with finitely many points of $\delta$-interactions on $\mathbb R^{d}$ (${d}\le3$) is equal to the number of negative eigenvalues of a certain class of matrix $M$ up to a constant. This $M$ is expressed in terms of distances between the interaction points and the intensities. As applications, we obtain sufficient and necessary conditions for $L$ to satisfy $N=m,n,n$ for ${d}=1,2,3$, respectively, and some estimates of the minimum and maximum of $N$ for fixed intensities. Here, we denote by $n$ and $m$ the numbers of interaction points and negative intensities, respectively.

### Article Information

 Title On the number of negative eigenvalues of a multi-dimensional Schrodinger operator with point interactions Source Methods Funct. Anal. Topology, Vol. 16 (2010), no. 4, 383-392 MathSciNet MR2777195 Copyright The Author(s) 2010 (CC BY-SA)

### Authors Information

Osamu Ogurisu
Division of Mathematical and Physical Sciences, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan

### Citation Example

Osamu Ogurisu, On the number of negative eigenvalues of a multi-dimensional Schrodinger operator with point interactions, Methods Funct. Anal. Topology 16 (2010), no. 4, 383-392.

### BibTex

@article {MFAT538,
AUTHOR = {Ogurisu, Osamu},
TITLE = {On the number of negative eigenvalues of a multi-dimensional  Schrodinger operator with point interactions},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {16},
YEAR = {2010},
NUMBER = {4},
PAGES = {383-392},
ISSN = {1029-3531},
MRNUMBER = {MR2777195},
URL = {http://mfat.imath.kiev.ua/article/?id=538},
}