Traces of differential operators

Trace Formula on Transmission Eigenvalues of the Sturm–Liouville Problem

“…where λ is the spectral parameter and real number 0 n , j = 1, d, n ∈ Z, be the spectrum of problem (1) and (2), then for sufficiently large |n|…”

“…For the scalar Sturm-Liouville problems, there is an enormous literature on estimates of large eigenvalues and regularized trace formulae which may often be computed explicitly in terms of the coefficients of operators and boundary conditions. A detailed list of publications related to the present aspect can be found in [2].…”

“…The trace formulae can be used for an approximate calculation of the first eigenvalues of an operator [2] and in order to establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of an operator [3].…”

An inverse problem for a differential pencil using nodal points as data

“…Remark 2.1: The bibliography on the trace formulas is very extensive and we refer to the list of works in [9,11,16,17,19,20]. The trace for the SturmLiouville problem with non-separated boundary conditions can be obtained from formula (2.5).…”

Trace Formulae for Matrix Integro-Differential Operators

“…A trace formula of a differential operator has many applications in the inverse problem, in the numerical calculation of eigenvalues, in c 2012 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com the theory of integrable systems, etc. Sadovnichii and Podol'skii [11] stated several sharp methods to trace formulae of second-order operators, high-order operators as well as partial differential operators. Using Rouché's theorem for operator-valued functions in [12], we can suitably locate the eigenvalues of L(Q, M; h, H) and find a precise description for the formula of the square root of the large eigenvalues up to the o( 1 n )-term, which are similar to the results in [1,2]:…”