We prove an analog of Levinson's theorem for scattering on a weighted (m + 1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each so-called half-bound state is counted as half a bound state).

1.
N.
Levinson
,
Phys. Rev.
75
,
1445
(
1949
).
2.
E.
Farhi
and
S.
Gutmann
,
Phys. Rev. A
58
,
915
(
1998
);
3.
E.
Farhi
,
J.
Goldstone
, and
S.
Gutmann
,
Theor. Comput.
4
,
169
(
2008
);
4.
A. M.
Childs
,
Phys. Rev. Lett.
102
,
180501
(
2009
);
[PubMed]
5.
K. M.
Case
and
M.
Kac
,
J. Math. Phys.
14
,
594
(
1973
).
6.
D. B.
Hinton
,
M.
Klaus
, and
J. K.
Shaw
,
SIAM J. Math. Anal.
22
,
754
(
1991
).
7.
M.
Varbanov
and
T. A.
Brun
,
Phys. Rev. A
80
,
052330
(
2009
);
8.
J.
Goldstone
, personal communication (
2008
).
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