The damped string problem revisited

Abstract

We revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it, employing methods familiar from supersymmetric quantum mechanics. We also derive completeness and Riesz basis results (with parentheses) for the associated root functions under less smoothness assumptions on the coefficients than usual, using operator theoretic methods (rather than detailed eigenvalue and root function asymptotics) only.

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

“…Recent results on damped strings can be found in [90], who use a Dirac operator approach rather than separation of variables. The main results are trace formulas and completeness of eigenvectors and associated vectors.…”

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).…”

“…Spectral problems for the differential pencil (1.1) with separated boundary conditions, periodic or antiperiodic boundary conditions were investigated in [2,4,5,11,24,25], etc. The inverse nodal problem for the differential pencil (1.1) with separated boundary conditions was studied in [13] (with an excessive assumption that the function p(x) was known a priori) and [23], in [4] for the case of the Robin boundary conditions, and in [5] for the Dirichlet boundary conditions.…”

Trace formulae for differential pencils with spectral parameter dependent boundary conditions

“…Differential equations with a nonlinear dependence on the spectral parameter frequently appear in mathematics as well as in applications. Some aspects of spectral problems for second-order differential pencils were studied ( [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and other papers). Such problems play an important role in mathematics and have many applications in natural sciences and engineering.…”