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Comment on “The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs” [J. Math. Phys.52, 063512 (2011)]

### Abstract

In this comment we answer negatively to our conjecture concerning the deficiency indices. More precisely, given any non-negative integer n, there is locally finite graph on which the adjacency matrix has deficiency indices (n, n).

© 2013 AIP Publishing LLC

Received 24 July 2012
Accepted 06 April 2013
Published online 10 June 2013

Acknowledgments:
We would like to thank Matthias Keller for helpful discussions and Thierry Jecko for precious remarks.

/content/aip/journal/jmp/54/6/10.1063/1.4803899

1.

1. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Tome I–IV: Analysis of Operators (Academic Press, 1978), p. 396.

2.

2. R. Wojciechowski, “Stochastic completeness of graphs,” Ph.D. dissertation (City University of New York, 2008);

3.

3. B. Mohar and M. Omladič, The Spectrum of Infinite Graphs with Bounded Vertex Degrees (Teubner, Leipzig, 1985), pp. 122–125.

5.

5. S. Golénia and C. Schumacher, “The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs,” J. Math. Phys. 52, 063512 (2011).

http://dx.doi.org/10.1063/1.3596179
9.

9. J. M. Berezanskiı, Expansions in Eigenfunctions of Selfadjoint Operators (American Mathematical Society, Providence, RI, 1968), p. 809.

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### Abstract

In this comment we answer negatively to our conjecture concerning the deficiency indices. More precisely, given any non-negative integer n, there is locally finite graph on which the adjacency matrix has deficiency indices (n, n).

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