Citation
Zlatoš, Andrej (2003) Sum Rules and the Szegö Condition for Jacobi Matrices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DBVEVF23. https://resolver.caltech.edu/CaltechETD:etd05222003114151
Abstract
We consider Jacobi matrices J with real b_n on the diagonal, positive a_n on the next two diagonals, and with u'(x) the density of the absolutely continuous part of the spectral measure. In particular, we are interested in compact perturbations of the free matrix J_0, that is, such that the a_n go to 1 and b_n go to 0. We study the Case sum rules for such matrices. These are trace formulae relating sums involving the a_n's and b_n's on one side and certain quantities in terms of the spectral measure on the other. We establish situations where the sum rules are valid, extending results of Case and KillipSimon.
The matrix J is said to satisfy the Szego condition whenever the integral
int_{0}^{pi} log [u'(2 cos t)] dt,
which appears in the sum rules, is finite. Applications of our results include an extension of Shohat's classification of certain Jacobi matrices satisfying the Szego condition to cases with an infinite point spectrum, and a proof that if n(a_n  1) go to a, nb_n go to b, and 2a < b, then the Szego condition fails. Related to this, we resolve a conjecture by Askey on the Szego condition for Jacobi matrices which are Coulomb perturbations of J_0. More generally, we prove that if
a_n = 1 + a/n^c + O(n^{1eps}) and b_n = b/n^c + O(n^{1eps})
with 0 < γ ≤ 1 and eps > 0, then the Szego condition is satisfied if and only if 2a ≥b
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  jacobi matrices; spectral theory; sum rules; szego condition  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Awards:  Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2003. Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2002.  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  12 May 2003  
NonCaltech Author Email:  zlatos (AT) ucsd.edu  
Record Number:  CaltechETD:etd05222003114151  
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd05222003114151  
DOI:  10.7907/DBVEVF23  
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  1936  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  22 May 2003  
Last Modified:  12 Feb 2021 01:38 
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