An Ambarzumian type theorem on graphs with odd cycles
UDC 517.9 We consider an inverse problem for Schrödinger operators on a connected equilateral graph $G$ with standard matching conditions.  The graph $G$ consists of at least two odd cycles glued together at a common vertex.  We prove an Ambarzumia...
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| author | Kiss, M. Kiss, M. |
| author_facet | Kiss, M. Kiss, M. |
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| description | UDC 517.9
We consider an inverse problem for Schrödinger operators on a connected equilateral graph $G$ with standard matching conditions.  The graph $G$ consists of at least two odd cycles glued together at a common vertex.  We prove an Ambarzumian-type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential must be equal to zero. |
| doi_str_mv | 10.37863/umzh.v74i12.6734 |
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DOI: 10.37863/umzh.v74i12.6734
UDC 517.9
M. Kiss1 (Inst. Math. Budapest Univ. Technology and Economics, Hungary)
AN AMBARZUMIAN TYPE THEOREM ON GRAPHS WITH ODD CYCLES 2
ТЕОРЕМА ТИПУ АМБАРЦУМЯНА ДЛЯ ГРАФIВ З НЕПАРНИМИ ЦИКЛАМИ
We consider an inverse problem for Schrödinger operators on a connected equilateral graph G with standard matching
conditions. The graph G consists of at least two odd cycles glued together at a common vertex. We prove an Ambarzumian-
type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential must be
equal to zero.
Розглянуто обернену задачу для операторiв Шредiнгера на зв’язному рiвносторонньому графi G зi стандартними
умовами узгодження. Граф G складається принаймнi з двох непарних циклiв, що склеєнi в спiльнiй вершинi.
Доведено результат типу Амбарцумяна, тобто якщо певна частина спектра така ж сама, як i у випадку нульового
потенцiалу, то потенцiал повинен бути нульовим.
1. Introduction. The addressed problem originates from a work of Ambarzumian [2] on reconstruc-
tion of a differential operator from its eigenvalues. Another source of the problem is the so-called
quantum graphs, i.e., differential operators on graphs [3, 21, 26]. From the classical theory of Sturm –
Liouville equations we refer to [7, 17], for special Ambarzumian type inverse problems see [10, 18].
Previous results for graphs are [6, 8, 12, 20, 22, 25]. Both in forward and in inverse problems
on graphs a usual ingredient is the calculation of spectral determinants (or alternatively functional
determinants or characteristic functions) [1, 5, 9, 11, 13 – 16, 19, 23, 24, 27]. For a more detailed
discussion of these results see the introduction in [20].
2. Results and discussion. Let r \geq 2 and consider r cycle graphs C1, C2, . . . , Cr with odd
cycle lengths n1, n2, . . . , nr (nj = 1 is also possible). Let the vertices of Cj be vj0, . . . , vjnj = vj0,
and let us form the graph G as the union of Cj ’s, identifying the vertex vj0 for all j. We shall say
that G is a graph consisting of r \geq 2 odd cycles glued together at a common vertex. The edge of G
between vj k - 1 and vjk is sometimes denoted by ejk ; however, when the particular location of the
edges are not important, we shall refer to them as e1, e2, . . . , e| E| .
Choosing an arbitrary orientation, we parametrize each edge with x \in [0, 1], and consider a
Schrödinger operator with potential qj(x) \in L1(0, 1) on the edge ej and with Neumann (or Kirch-
hoff) boundary conditions (sometimes called standard matching conditions), i.e., solutions are re-
quired to be continuous at the vertices and, in the local coordinate pointing outward, the sum of
derivatives is zero. More formally, consider the eigenvalue problem
- y\prime \prime + qj(x)y = \lambda y (2.1)
on ej for all j with the conditions
yj(\kappa j) = yk(\kappa k) (2.2)
if ej and ek are incident edges attached to a vertex v where \kappa = 0 for outgoing edges, \kappa = 1 for
incoming edges (and can be both 0 or 1 for loops); and in every vertex v
1 e-mail: mkiss@math.bme.hu.
2 This work was supported by the Hungarian NKFIH (Grant SNN-125119).
c\bigcirc M. KISS, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12 1679
1680 M. KISS\sum
ej \mathrm{l}\mathrm{e}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{s}v
y\prime j(0) =
\sum
ej\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}v
y\prime j(1) (2.3)
(loops are counted on both sides).
Theorem 2.1. Consider the eigenvalue problem (2.1) – (2.3). Let G be a graph consisting of
r \geq 2 odd cycles glued together at a common vertex. If \lambda = 0 is the smallest eigenvalue and
for infinitely many k \in \BbbZ + there are r - 1 eigenvalues (counting multiplicities) such that \lambda =
= (2k + 1)2\pi 2 + o(1), then q = 0 a.e. on G.
If the lengths of the odd cycles are all 1, i.e., the cycles are all loops, then the statement reduces
to that of Theorem 2.1 in [28], which states the following:
Suppose G is a flower-like graph, i.e., a single vertex attached r loops of length 1. For k =
= 1, 2, . . . , let mk be a sequence of integers with \mathrm{l}\mathrm{i}\mathrm{m}mk = +\infty . If eigenvalues are nonnegative,
\lambda k = (2mk + 1)2\pi 2 are eigenvalues with multiplicities (r - 1), where mk is a strictly ascending
infinite sequence of positive integers, then qj(x) = 0 a.e. on [0, 1], for each j = 1, 2, . . . , r. We
have to require r \geq 2 for the consequence to hold.
3. Calculation of the spectral determinant. Denote by cj(x, \lambda ) the solution of (2.1) which
satisfies the conditions cj(0, \lambda ) - 1 = c\prime j(0, \lambda ) = 0 and by sj(x, \lambda ) the solution of (2.1) which satisfies
the conditions sj(0, \lambda ) = s\prime j(0, \lambda ) - 1 = 0. Each yj(x, \lambda ) may be written as a linear combination
yj(x, \lambda ) = Aj(\lambda )cj(x, \lambda ) +
\surd
\lambda Bj(\lambda )sj(x, \lambda ).
Then yj(0, \lambda ) = Aj(\lambda ) is the same on each outgoing edge; hence, as in [20], we index the functions
A(\lambda ) by vertices, and then
yj(x, \lambda ) = Av(\lambda )cj(x, \lambda ) +
\surd
\lambda Bj(\lambda )sj(x, \lambda ),
if ej starts from v. If the eigenfunctions are normalized, i.e.,
\sum
j
\| yj(x, \lambda )\| 22 = 1, then Av(\lambda ) =
= Bj(\lambda ) = O(1) [8, 20, 28]. The coefficients Av and Bj form a
\bigl(
| V | + | E|
\bigr)
-dimensional vector,
which satisfies | V | Kirchhoff conditions at the vertices and | E| continuity conditions at the incoming
ends of edges, namely, for all v \in V (G),\sum
ej : ...\rightarrow v
1\surd
\lambda
Avj (\lambda )c
\prime
j(1, \lambda ) +Bj(\lambda )s
\prime
j(1, \lambda )\underbrace{} \underbrace{}
1\surd
\lambda
y\prime j(1,\lambda )
-
\sum
ej : v\rightarrow ...
Bj(\lambda )\underbrace{} \underbrace{}
1\surd
\lambda
y\prime j(0,\lambda )
= 0,
where in the first sum vj denotes the starting point of ej ; and, for all ej \in E(G),
Au(\lambda )cj(1, \lambda ) +
\surd
\lambda Bj(\lambda )sj(1, \lambda ) - Av(\lambda ) = 0,
if ej points from u to v (see equations (2.3) and (2.4) in [20]).
The matrix M of this homogeneous linear system of equations has a special structure. For the
convenience of the reader we repeat its description from [20]. Namely, M =
\biggl[
A B
C D
\biggr]
, where
A is a | V | by | V | matrix, avu =
1\surd
\lambda
\sum
c\prime j(1, \lambda ), the sum is taken on edges pointing from u
to v; in the zero-potential case A is - \mathrm{s}\mathrm{i}\mathrm{n}
\surd
\lambda times the (transpose of the) directed adjacency matrix
of G;
B and C are like incidence matrices;
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
AN AMBARZUMIAN TYPE THEOREM ON GRAPHS WITH ODD CYCLES 1681
bvj =
\left\{
s\prime j(1, \lambda ), if ej ends in v,
- 1 if ej starts from v,
s\prime j(1, \lambda ) - 1, if ej is a loop in v,
0, otherwise
and
cjv =
\left\{
- 1, if ej ends in v,
cj(1, \lambda ), if ej starts from v,
- 1 + cj(1, \lambda ), if ej is a loop in v,
0, otherwise;
D is an | E| by | E| diagonal matrix, djj =
\surd
\lambda sj(1, \lambda ).
The determinant of the matrix M is the so-called spectral determinant of the problem (2.1) – (2.3).
Example. Consider a flower-like graph, i.e., a single vertex with r loops. Then
M = M1 =
\left[
1\surd
\lambda
r\sum
k=1
c\prime k(1, \lambda ) s\prime 1(1, \lambda ) - 1 . . . s\prime r(1, \lambda ) - 1
- 1 + c1(1, \lambda )
\surd
\lambda s1(1, \lambda ) . . . 0
...
...
. . .
...
- 1 + cr(1, \lambda ) 0 . . .
\surd
\lambda sr(1, \lambda )
\right]
with determinant
\mathrm{d}\mathrm{e}\mathrm{t}M1 = \lambda
r - 1
2
\left( r\sum
k=1
c\prime k(1, \lambda )
r\prod
j=1
sj(1, \lambda ) -
r\sum
k=1
(s\prime k(1, \lambda ) - 1)( - 1 + ck(1, \lambda ))
\prod
j \not =k
sj(1, \lambda )
\right)
corresponding to formula (2.9) in [28].
For \lambda = (2k+1)2\pi 2+ d+ o(1) the elements of M have the following asymptotics, independent
of k (see [8], equation (2.3), or [22], Lemma 3.1):
1\surd
\lambda
c\prime j(1, \lambda ) =
1
2
\surd
\lambda
\left( d -
1\int
0
qj
\right) + o
\biggl(
1\surd
\lambda
\biggr)
, (3.1)
s\prime j(1, \lambda ) = - 1 + o
\biggl(
1\surd
\lambda
\biggr)
, (3.2)
cj(1, \lambda ) = - 1 + o
\biggl(
1\surd
\lambda
\biggr)
, (3.3)
\surd
\lambda sj(1, \lambda ) =
1
2
\surd
\lambda
\left( 1\int
0
qj - d
\right) + o
\biggl(
1\surd
\lambda
\biggr)
. (3.4)
Remark. Using these asymptotics, we get
\mathrm{d}\mathrm{e}\mathrm{t}M1 = - 4
r\sum
k=1
\prod
j \not =k
\surd
\lambda sj(1, \lambda ) + o(\lambda - r
2 ).
This is a special case of (4.1) and of (4.2) below.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1682 M. KISS
4. Proofs of main result.
Lemma 4.1. The total multiplicities of the eigenvalues \lambda = (2k + 1)2\pi 2 + O(1) are exactly
| E| - | V | .
Proof. If q = 0, \mathrm{d}\mathrm{e}\mathrm{t}M is a polynomial of \mathrm{c}\mathrm{o}\mathrm{s}
\surd
\lambda and \mathrm{s}\mathrm{i}\mathrm{n}
\surd
\lambda , hence its zeros are 2\pi -periodic in\surd
\lambda . \lambda = (2k+1)2\pi 2+O(1) with periodicity implies
\surd
\lambda = (2k+1)\pi exactly. For \lambda = (2k+1)2\pi 2
A and D are zero matrices, thus the rank of M is 2| V | , and its nullspace is exactly (| E| - | V | )-
dimensional. Let us write for a moment \lambda = \lambda (q) to denote the dependency of the eigenvalues on
the potential. If we arrange the eigenvalues in an increasing sequence \lambda 1, . . . , \lambda n, . . . , then there is
a constant c depending only on the graph and the L1-norm of the potential and not depending on n,
the particular index of the eigenvalue, such that | \lambda n(q) - \lambda n(0)| \leq c for all n [20]. Hence, the total
multiplicity of eigenvalues \lambda = (2k + 1)2\pi 2 +O(1) is the same for all q \in L1.
Lemma 4.2. The determinant of M for \lambda = (2k + 1)2\pi 2 +O(1) is O(\lambda - 1
2
(| E| - | V | )).
Proof. Each term in the Leibniz formula for the determinant must contain at least (| E| - | V | )
factors from A and D having a magnitude of O
\biggl(
1\surd
\lambda
\biggr)
.
Lemma 4.3. Assume that a graph has the same number of edges as vertices. Then the deter-
minant of its (unoriented) incidence matrix is zero, except if there is no even cycles in the graph
and every component contains exactly one (odd) cycle. In that case the determinant is \pm 2\kappa where \kappa
denotes the number of components.
Proof. If the graph contains an even cycle, then the corresponding rows are dependent. If a
component contains no cycles, then the corresponding columns are dependent. The number of cycles
(including loops) is equal to the number of components, hence, if a component contains more than
one cycles or loops, then there must be another component without cycles. In all of these cases
the determinant of the incidence matrix is zero. Otherwise it is enough to prove the statement for
connected graphs, as the incidence matrix is a direct sum of that of the components. It is true for odd
cycles as well as for a single vertex with a loop. If the graph is not a single cycle or a single loop,
there is at least one vertex with only one incident edge. Removing this vertex (and its edge) from
the graph does not change the absolute value of the determinant of the incidence matrix. This can be
repeated until we reach a single cycle or a single loop.
Lemma 4.4. If \lambda = (2k + 1)2\pi 2 + O(1), then the determinant of a | V | \times | V | submatrix of C
(and of B) is \pm 2\kappa + o
\biggl(
1\surd
\lambda
\biggr)
if the indices of the rows in C (the columns in B) correspond to
the edges of a subgraph which has no even cycles and every component contains exactly one (odd)
cycle. Otherwise the determinant is o
\biggl(
1\surd
\lambda
\biggr)
.
Proof. Leaving out the o
\biggl(
1\surd
\lambda
\biggr)
terms from the submatrix we make only o
\biggl(
1\surd
\lambda
\biggr)
error in its
determinant. What we get is the negative of an incidence matrix of a subgraph with | V | vertices and
| V | edges. Then the statement follows from the previous lemma.
Theorem 4.1. If \lambda = (2k + 1)2\pi 2 +O(1), then
\mathrm{d}\mathrm{e}\mathrm{t}M = ( - 1)| V |
\sum
\tau
4\kappa (\tau )
\prod
ej /\in \tau
\surd
\lambda sj(1, \lambda ) +O(\lambda - | E| - | V | +1
2 ), (4.1)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
AN AMBARZUMIAN TYPE THEOREM ON GRAPHS WITH ODD CYCLES 1683
where the sum is taken for such subgraphs \tau of G which have | V | vertices and their incidence matrix
is nonsingular (i.e., \tau has no even cycles, has \kappa components, each of which contains exactly one
(odd) cycle).
Proof. The main terms in the Leibniz formula for the determinant are those which contain
exactly (| E| - | V | ) elements from D. The product of a fixed set of (| E| - | V | ) elements in D is
weighted by the determinant of the respective minor, with all other elements of D substituted by
zero. The remaining rows in C and columns in B look like an unordered incidence matrix of the
graph \tau spanned by the remaining | V | edges. Then the determinant of the minor is ( - 1)| V | times
the square of the determinant of the incidence matrix of \tau .
Corollary 4.1. If \lambda = (2k + 1)2\pi 2 + O(1) and the graph G consists of r odd cycles of length
n1, . . . , nr, glued together at a common vertex, then
\mathrm{d}\mathrm{e}\mathrm{t}M = - 4
r\sum
i=1
\prod
j \not =i
nj\sum
l=1
\surd
\lambda s(1, \lambda , qjl) +O(\lambda - r
2 ), (4.2)
where qjl is the potential on the lth edge of the j th cycle.
Proof. The incidence matrix of a subgraph of G is nonsingular if and only if we leave out one
edge from every but one cycle. Note also that | V | is odd and | E| - | V | = r - 1. Then the statement
follows from (4.1).
Substituting the asymptotics (3.1) – (3.4), we get the following corollary.
Corollary 4.2. If \lambda = (2k+1)2\pi 2+ d+ o(1) and the graph G consists of r odd cycles of length
n1, . . . , nr, glued together at a common vertex, then
\mathrm{d}\mathrm{e}\mathrm{t}M = - 4
\biggl(
- 1
2
\surd
\lambda
\biggr) r - 1
p(d) + o(\lambda - r - 1
2 ),
where
p(d) =
r\sum
i=1
\prod
j \not =i
\left( njd -
nj\sum
l=1
1\int
0
qjl
\right) . (4.3)
Lemma 4.5. Under the assumptions of Theorem 2.1, p(d) =
\sum r
i=1
\prod
j \not =i
njd
r - 1.
Proof. \lambda is an eigenvalue of the eigenvalue problem (2.1) – (2.3) if and only if \mathrm{d}\mathrm{e}\mathrm{t}M(\lambda ) = 0.
Let the distinct roots of p(d) be d1, . . . , dl. By the previous corollary for \lambda = (2k + 1)2\pi 2 + O(1)
the distinct roots of \mathrm{d}\mathrm{e}\mathrm{t}M(\lambda ) are exactly of the form \lambda = (2k + 1)2\pi 2 + dj + o(1), 1 \leq j \leq l. By
Lemma 4.1 the total multiplicity of these eigenvalues is | E| - | V | . In Theorem 2.1 we assumed that
there are the same number of eigenvalues such that \lambda = (2k + 1)2\pi 2 + o(1), hence dj = 0 for all j
and then p(d) is a constant multiple of dr - 1. The principal coefficient is given by (4.3).
Proof of Theorem 2.1. Let us introduce Qj =
\sum nj
l=1
\int 1
0
qjl.
For a fixed m substituting d =
Qm
nm
to (4.3), we get
r\sum
i=1
1
ni
\biggl(
Qm
nm
\biggr) r - 1
=
1\prod r
j=1
nj
p
\biggl(
Qm
nm
\biggr)
=
1
nm
\prod
j \not =m
\biggl(
Qm
nm
- Qj
nj
\biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1684 M. KISS
Introducing hj =
Qj
nj
, we have
r\sum
i=1
1
ni
hr - 1
m =
1
nm
\prod
j \not =m
(hm - hj), m = 1, 2, . . . , r.
We can assume h1 \geq h2 \geq . . . \geq hr. Then, for m = 2, the left-hand side is nonpositive, hence
h2 \leq 0. Similarly, hr - 1 \geq 0. Hence, for m = 1,
1
n1
hr - 2
1 (h1 - hr) =
r\sum
i=1
1
ni
hr - 1
1 .
If hr = 0, then h1 = 0 as nj ’s are positive. Similarly, h1 = 0 implies hr = 0. If neither of them is
zero, then
1
n1
(h1 - hr) =
\sum r
i=1
1
ni
h1 and
1
nr
(hr - h1) =
\sum r
i=1
1
ni
hr. Subtracting, we get
r - 1\sum
i=2
1
ni
(h1 - hr) = 0.
As nj ’s are positive, if r > 2 then h1 = h2 = . . . = hr = 0, while for r = 2, h1n1 + h2n2 = 0. In
both cases,
\sum
j
Qj =
\sum
ejl\in G
1\int
0
qjl =
\int
G
q = 0.
Let us denote the operator of the eigenvalue problem (2.1) – (2.3) by L.
\langle \varphi ,L\varphi \rangle
\langle \varphi ,\varphi \rangle
\geq \lambda 0 = 0 and
equality holds if and only if \varphi is an eigenfunction of L. It follows that the constant 1 must be an
eigenfunction corresponding to the eigenvalue 0. Substituting this to (2.1) gives q(x) = 0.
Theorem 2.1 is proved.
5. Glossary. A walk W in a graph is an alternating sequence of vertices and edges, say
X0, e1, . . . , el, Xl where ei = Xi - 1Xi, 0 < i < l. The length of W is l. This walk W is called
a trail if all its edges are distinct. A path is a walk with distinct vertices. A trail whose end
vertices coincide (a closed trail) is called a circuit. To be precise, a circuit is a closed trail without
distinguished endvertices and direction, so that, for example, two triangles sharing a single vertex
give rise to precisely two circuits with six edges. If a walk W = X0, e1, . . . , el, Xl is such that
l \geq 3, X0 = Xl, and the vertices Xi, 0 < i < l, are distinct from each other and X0, then W is said
to be a cycle [4, p. 5].
The incidence matrix of a graph has a row for each vertex and a column for each edge, and is
defined as
R = (rij), rij =
\left\{
0, if ej is not incident to vi,
1, if ej is not a loop and incident to vi,
2, if ej is a loop at vi.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
AN AMBARZUMIAN TYPE THEOREM ON GRAPHS WITH ODD CYCLES 1685
References
1. E. Akkermans, A. Comtet, J. Desbois, G. Montambaux, Ch. Texier, Spectral determinant on quantum graphs, Annal.
Phys., 284, № 1, 10 – 51 (2000).
2. V. Ambarzumian, Über eine Frage der Eigenwerttheorie, Z. Phys., 53, 690 – 695 (1929).
3. G. Berkolaiko, P. Kuchment, Introduction to quantum graphs, Math. Surveys and Monogr., Amer. Math. Soc. (2013).
4. B. Bollobás, Modern graph theory, Springer, New York (1998).
5. J. Bolte, S. Egger, R. Rueckriemen, Heat-kernel and resolvent asymptotics for Schrödinger operators on metric
graphs, Appl. Math. Res. Express, 2015, № 1, 129 – 165 (2014).
6. J. Boman, P. Kurasov, R. Suhr, Schrödinger operators on graphs and geometry II. Spectral estimates for L1 -potentials
and an Ambartsumian theorem, Integral Equat. and Operator Theory, 90, № 3 (2018).
7. G. Borg, Eine Umkehrung der Sturm – Liouvilleschen Eigenwertaufgabe, Acta Math., 78, 1 – 96 (1946).
8. R. Carlson, V. Pivovarchik, Ambarzumian’s theorem for trees, Electron. J. Different. Equat., 2007, № 142, 1 – 9
(2007).
9. R. Carlson, V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A: Math. and
Theor., 41, № 14, Article 145202 (2008).
10. Y. H. Cheng, Tui-En Wang, Chun-Jen Wu, A note on eigenvalue asymptotics for Hill’s equation, Appl. Math. Lett.,
23, № 9, 1013 – 1015 (2010).
11. S. Currie, B. A. Watson, Eigenvalue asymptotics for differential operators on graphs, J. Comput. and Appl. Math.,
182, № 1, 13 – 31 (2005).
12. E. B. Davies, An inverse spectral theorem, J. Operator Theory, 69, № 1, 195 – 208 (2013).
13. Yves Colin De Verdière, Semi-classical measures on quantum graphs and the Gauss map of the determinant manifold,
Ann. Henri Poincaré, 16, 347 – 364 (2015).
14. J. Desbois, Spectral determinant of Schrödinger operators on graphs, J. Phys. A: Math. and Gen., 33, № 7 (2000).
15. L. Friedlander, Determinant of the Schrödinger operator on a metric graph, Contemp. Math., 415, 151 – 160 (2006).
16. J. M. Harrison, K. Kirsten, C. Texier, Spectral determinants and zeta functions of Schrödinger operators on metric
graphs, J. Phys. A: Math. and Theor., 45, № 12, Article 125206 (2012).
17. M. Horváth, Inverse spectral problems and closed exponential systems, Ann. Math., 162, № 2, 885 – 918 (2005).
18. M. Horváth, On the stability in Ambarzumian theorems, Inverse Problems, 31, № 2, Article 025008 (2015).
19. I. Kac, V. Pivovarchik, On multiplicity of a quantum graph spectrum, J. Phys. A: Math. and Theor., 44, № 10,
Article 105301 (2011).
20. M. Kiss, Spectral determinants and an Ambarzumian type theorem on graphs, Integral Equat. and Operator Theory,
92, 1 – 11 (2020).
21. P. Kuchment, Quantum graphs: an introduction and a brief survey, Analysis on Graphs and its Applications, 291 – 314
(2008).
22. Chun-Kong Law, Eiji Yanagida, A solution to an Ambarzumyan problem on trees, Kodai Math. J., 35, № 2, 358 – 373
(2012).
23. M. Möller, V. Pivovarchik, Spectral theory of operator pencils, Hermite – Biehler functions, and their applications,
Oper. Theory: Adv. and Appl., Springer (2015).
24. K. Pankrashkin, Spectra of Schrödinger operators on equilateral quantum graphs, Lett. Math. Phys., 77, № 2,
139 – 154 (2006).
25. V. N. Pivovarchik, Ambarzumian’s theorem for a Sturm – Liouville boundary value problem on a star-shaped graph,
Funct. Anal. and Appl., 39, № 2, 148 – 151 (2005).
26. Yu. V. Pokornyi, A. V. Borovskikh, Differential equations on networks (geometric graphs), J. Math. Sci., 119, № 6,
691 – 718 (2004).
27. Ch. Texier, \zeta -Regularized spectral determinants on metric graphs, J. Phys. A: Math. and Theor., 43, № 42, Arti-
cle 425203 (2010).
28. Chuan-Fu Yang, Xiao-Chuan Xu, Ambarzumyan-type theorems on graphs with loops and double edges, J. Math.
Anal. and Appl., 444, № 2, 1348 – 1358 (2016).
Received 10.05.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
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| id | umjimathkievua-article-6734 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:29:55Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c7/a0dee784ff81ec2ec3fda1ded10775c7.pdf |
| spelling | umjimathkievua-article-67342023-01-23T14:02:46Z An Ambarzumian type theorem on graphs with odd cycles An Ambarzumian type theorem on graphs with odd cycles Kiss, M. Kiss, M. Ambarzumian, inverse problems, inverse eigenvalue problem, differential equations on graphs, quantum graphs, Schrodinger operators, odd cycles 2020. Math. Subject Classification: Primary 34A55, 34B20, 34B24, 34B45; Secondary 34L40, 47A75 UDC 519.1 UDC 517.9 We consider an inverse problem for Schrödinger operators on a connected equilateral graph $G$ with standard matching conditions.&nbsp;&nbsp;The graph $G$ consists of at least two odd cycles glued together at a common vertex.&nbsp;&nbsp;We prove an Ambarzumian-type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential must be equal to zero. УДК 517.9 Теорема типу Амбарцумяна для графів з непарними циклами&nbsp; Розглянуто обернену задачу для операторів Шредінгера на зв’язному рівносторонньому графі $G$ зі стандартними умовами узгодження.&nbsp;&nbsp;Граф $G$ складається принаймні з двох непарних циклів, що склеєні&nbsp; в спільній вершині.&nbsp;&nbsp;Доведено результат типу Амбарцумяна, тобто якщо певна частина спектра така ж сама, як і у випадку нульового потенціалу, то потенціал повинен бути нульовим. Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6734 10.37863/umzh.v74i12.6734 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1679 - 1685 Український математичний журнал; Том 74 № 12 (2022); 1679 - 1685 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6734/9344 Copyright (c) 2023 Márton Kiss |
| spellingShingle | Kiss, M. Kiss, M. An Ambarzumian type theorem on graphs with odd cycles |
| title | An Ambarzumian type theorem on graphs with odd cycles |
| title_alt | An Ambarzumian type theorem on graphs with odd cycles |
| title_full | An Ambarzumian type theorem on graphs with odd cycles |
| title_fullStr | An Ambarzumian type theorem on graphs with odd cycles |
| title_full_unstemmed | An Ambarzumian type theorem on graphs with odd cycles |
| title_short | An Ambarzumian type theorem on graphs with odd cycles |
| title_sort | ambarzumian type theorem on graphs with odd cycles |
| topic_facet | Ambarzumian inverse problems inverse eigenvalue problem differential equations on graphs quantum graphs Schrodinger operators odd cycles 2020. Math. Subject Classification: Primary 34A55 34B20 34B24 34B45; Secondary 34L40 47A75 UDC 519.1 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6734 |
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