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 1950īs, 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

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