Spectral Determinants and an Ambarzumian Type Theorem on Graphs
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Integral Equations and Operator Theory
Spectral Determinants and an Ambarzumian Type Theorem on Graphs M´arton Kiss Abstract. We consider an inverse problem for Schr¨ odinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be described by the roots of a polynomial. We verify that one of the roots is equal to the mean value of the potential and apply it to 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 has to be zero. Mathematics Subject Classification. Primary 34A55, 34B20, 34B24, 34B45; Secondary 34L40, 47A75. Keywords. Ambarzumian, Inverse problems, Inverse eigenvalue problem, Differential equations on graphs, Quantum graphs, Schr¨ odinger operators, Matrix Tree Theorem.
1. Introduction Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering [3]. Ambarzumian’s theorem in inverse spectral theory refers to a setting when a differential operator can be reconstructed from at most one spectrum due to the presence of a constant eigenfunction. The original theorem from 1929 states for q ∈ C[0, π] that if the eigenvalues of −y + q(x)y = λy (1.1) y (0) = y (1) = 0 are λn = n2 π 2 (n ≥ 0), then q = 0 [2]. Neumann boundary conditions are crucial here for otherwise the statement is not true. Eigenvalues other than 1 zero are used only through eigenvalue asymptotics to get 0 q = 0; hence a subsequence λr = r2 π 2 +o(1) of them is sufficient to reach the same conclusion even if q ∈ L1 (0, π). On finite intervals inverse eigenvalue problems have a This work was supported by the Hungarian NKFIH Grant SNN-125119.
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vast literature. In general we mention Horv´ ath [18] and the fundamental paper of Borg [7] as well as the works referenced by and referencing these. For Ambarzumian’s theorem a recent stability result is found in Horv´ ath [19]. Boman et al. [6] proved that an interval with zero potential can be identified by its spectrum even if we only know a priori that the underlying space is a graph with appropriate boundary conditions. Let us turn to the list of extensions when the underlying space is a fixed graph. On a tree with edges of equal length knowing the smallest eigenvalue 0 exactly and a specific part of the spectrum approximately is enough for recover to the zero potential (Carlson and Pivovarchik [8]). On a tree with different edge lengths this is still true (see Lemma 4.4 of Law and Yanagida [25]), however, the required set of eigenvalues is given by an existence proof. An ultimate generalization of Ambarzumian type scenarios with a fixed underlying space is found by Davies [12]. In an abstract framework which contains arbitrary graphs with arbitrary edge lengths as particular special cases he was able to recover to the zero potential by determining the mean value of the potential from heat trace asymptotic
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