A Kreĭn string is (essentially) a pair ?[?,?] where 0≤∞ and ?:[0,?)→[0,∞) is nondecreasing. Each string gives rise to an operator model, the Kreĭn-Feller differential operator −???? acting in the space ?2(??) . This operator has a selfadjoint realization which is nonnegative. Provided that ?+lim?→??(?)<∞ , this realization has discrete spectrum and, when (??) denotes the sequence of positive eigenvalues arranged increasingly, then lim???‾‾‾√=1?∫0??′(?)‾‾‾‾‾√??.
We show that for a class of strings defined by a weaker growth restriction the spectrum is discrete, the integral on the right side is still finite, and the asymptotic behaviour of the eigenvalues is determined by the above formula.
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