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H. Woracek:
"An addendum to M.G.~Krein's inverse spectral theorem for strings";
in: "ASC Report 29/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

A string is a pair (L,m) where L 2 [0,1] and m is a positive, possibly
unbounded, Borel measure supported on [0,L]; we think of L as the length
of the string and of m as its mass density. To each string a boundary value
problem is associated, namely
f0(x) + z Z 1
0
f(y) dm(y), x 2 R, f0(0−) = 0 .
A positive Borel measure on R is called a (canonical) spectral measure
of the string S[L,m], if there exists an appropriately normalized Fourier
transform of L2(m) onto L2( ).
In order that a given positive Borel measure is a spectral measure
of some string, it is necessary that:
- RR
d ( )
1+| | < 1.
- Either supp [0,1), or is discrete and has exactly one point
mass in (−1, 0).
It is a deep result, going back to M.G.Kre˘ın in the 1950´s, that each
measure with RR
d ( )
1+| | < 1 and supp [0,1) is a spectral measure
of some string, and that this string is uniquely determined by . The
question remained open, which conditions characterize whether a measure
with supp 6 [0,1) is a spectral measure of some string. In the present
paper, we answer this question. Interestingly, the solution is much more
involved than the first guess might suggest.

string, spectral measure, direct and inverse spectral theorem

http://www.asc.tuwien.ac.at/preprint/2010/asc29x2010.pdf