2013/07/23 by Israel Kac, Kac, Israel, Vyacheslav Pivovarchik +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1307.6171
18 pages
arxiv created 2013/07/23 · arxiv updated 2013/07/24
In [V. Barcilon Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. \bf 93 (1983) 222-234] two boundary value problems were considered generated by the differential equation of a string y′′+λp(x)y=0, 0≤ x ≤ L<+∞ \eqno(*) with continuous real function p(x) (density of the string) and the boundary conditions y(0)=y(L)=0 the first problem and y′(0)=y(L)=0 the second one. In the above paper the following formula was stated p(0)=1L2μ1\mathop∏n=1∞λn2μn μn+1 \eqno(**) where \λk\k=1∞ is the spectrum of the first boundary value problem and \μk\k=1∞ of the second one. Rigorous proof of (**) was given in [C.-L. Shen On the Barcilon formula for the string equation with a piecewise continuous density function. Inverse Problems \bf 21, (2005) 635--655] under more restrictive conditions of piecewise continuity of p′(x). In this paper (**) was deduced using p(0)=limλ→ +∞(ϕ(L,-λ)λ^12ψ(L,-λ))2 \eqno(***) where ϕ(x,λ) is the solution of (*) which satisfies the boundary conditions ϕ(0)-1=ϕ′(0)=0 and ψ(x,λ) is the solution of (*) which satisfies ψ(0)=ψ′(0)-1=0. In our paper we prove that (***) is true for the so-called M.G. Krein's string which may have any nondecreasing mass distribution function M(x) with finite nonzero M′(0). Also we show that (**) is true for a wide class of strings including those for which M(x) is a singular function, i.e. M′(x)=p(x)\mathop=a.e.0.