tailieunhanh - Đề tài " Sum rules for Jacobi matrices and their applications to spectral theory "

We discuss the proof of and systematic application of Case’s sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J − J0 is Hilbert-Schmidt, and a proof of Nevai’s conjecture that the Szeg˝ condition o holds if J − J0 is trace class. | Annals of Mathematics Sum rules for Jacobi matrices and their applications to spectral theory By Rowan Killip and Barry Simon Annals of Mathematics 158 2003 253 321 Sum rules for Jacobi matrices and their applications to spectral theory By Rowan Killip and BARRy Simon Abstract We discuss the proof of and systematic application of Case s sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J J0 is Hilbert-Schmidt and a proof of Nevai s conjecture that the Szego condition holds if J J0 is trace class. 1. Introduction In this paper we will look at the spectral theory of Jacobi matrices that is infinite tridiagonal matrices b1 a1 0 0 a1 b2 a2 0 J 0 a2 b3 a3 . with aj 0 and bj e R. We suppose that the entries of J are bounded that is supn an supn bn TO so that J defines a bounded self-adjoint operator on 2 z 2 1 2 . . Let ỗj be the obvious vector in E z that is with components ỗjn which are 1 if n j and 0 if n j. The spectral measure we associate to J is the one given by the spectral theorem for the vector ỗ1. That is the measure X defined by m E 51 J E -1ỗ1 i J x E The first named author was supported in part by NSF grant DMS-9729992. The second named author was supported in part by NSF grant DMS-9707661. 254 ROWAN KILLIP AND BARRY SIMON There is a one-to-one correspondence between bounded Jacobi matrices and unit measures whose support is both compact and contains an infinite number of points. As we have described one goes from J to 1 by the spectral theorem. One way to find J given 1 is via orthogonal polynomials. Applying the Gram-Schmidt process to xn n 0 one gets orthonormal polynomials Pn x Knxn with Kn 0 and J Pn x Pm x d1 x bnm. These polynomials obey a three-term recurrence xPn x ara 1Pn 1 x bn 1Pn x anPn-1 x where an bn are the Jacobi matrix coefficients of

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