科研之友
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摘要
A string is a pair (L, m) where L epsilon [0, infinity] 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 as its mass density. To each string a differential operator acting in the space L-2(m) is associated. Namely, the Krein-Feller differential operator -DmDx; its eigenvalue equation can be written, e.g., as %26lt;br%26gt;f %26apos;(x) + z (0)integral(L) f(y) dm(y) = 0, x epsilon R, f %26apos;(0-) = 0. %26lt;br%26gt;A positive Borel measure tau on is called a (canonical) spectral measure of the string , if there exists an appropriately normalized Fourier transform of onto L (2)(tau). In order that a given positive Borel measure tau is a spectral measure of some string, it is necessary that: (1) integral(R) d tau(lambda)/1+vertical bar lambda vertical bar %26lt; infinity. (2) Either tau subset of [0, infinity), or tau is discrete and has exactly one point mass in (-infinity, 0). It is a deep result, going back to Krein in the 1950%26apos;s, that each measure with integral(R) d tau(lambda)/1+vertical bar lambda vertical bar %26lt; infinity and supp tau subset of [0, infinity) is a spectral measure of some string, and that this string is uniquely determined by tau. The question remained open, which conditions characterize whether a measure tau with supp tau not subset of [0, infinity) 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.
- 出版日期2012-7
