TAILIEUCHUNG - Trace formulae for Schr¨odinger systems on graphs

For Schr¨odinger systems on metric graphs with δ-type conditions at the central vertex, firstly, we obtain precise description for the square root of the large eigenvalue up to the o(1/n)-term. Secondly, the regularized trace formulae for Schr¨odinger systems are calculated with some techniques in classical analysis. Finally, these formulae are used to obtain a result of inverse problem in the spirit of Ambarzumyan. | Turk J Math 34 (2010) , 181 – 196. ¨ ITAK ˙ c TUB doi: Trace formulae for Schr¨ odinger systems on graphs Chuan-Fu Yang, Zhen-You Huang and Xiao-Ping Yang Abstract For Schr¨ odinger systems on metric graphs with δ -type conditions at the central vertex, firstly, we obtain precise description for the square root of the large eigenvalue up to the o(1/n) -term. Secondly, the regularized trace formulae for Schr¨ odinger systems are calculated with some techniques in classical analysis. Finally, these formulae are used to obtain a result of inverse problem in the spirit of Ambarzumyan. Key Words: Schr¨ odinger systems, metric graph, δ -type conditions, trace formula, Ambarzumyan-type theorem 1. Introduction In a finite-dimensional space, an operator has a finite trace. But in an infinite-dimensional space, ordinary differential operators do not necessarily have finite trace (the sum of all eigenvalues). But Gelfand and Levitan [15] observed that the sum n (λn − μn ) often makes sense, where {λn } and {μn } are the eigenvalues of the “perturbed problem” and “unperturbed problem”, respectively. The sum n (λn − μn ) is called a regularized trace. Gelfand and Levitan first obtained an identity of trace for the Schr¨ odinger operator [15]. We describe briefly here the result. Let λj , j = 0, 1, · · · , be eigenvalues of the eigenvalue problem −y (x) + q(x)y(x) = λy(x), y (0) = y (π) = 0. Then there is the following identity of trace: ∞ n=0 [λn − n2 − 1 π π q(x)dx] = 0 1 1 [q(π) + q(0)] − 4 2π π q(x)dx. 0 The trace identity of a differential operator deeply reveals spectral structure of the differential operator and has important applications in the numerical calculation of eigenvalues, inverse problem, theory of solitons, theory of integrable system [22, 41]. However, the calculation of every eigenvalue for the differential operator is very difficult. The most important application of the trace formulae is in solving inverse problems .

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