7 research outputs found
Mixed Data in Inverse Spectral Problems for the Schr\"{o}dinger Operators
We consider the Schr\"{o}dinger operator on a finite interval with an
-potential. We prove that the potential can be uniquely recovered from one
spectrum and subsets of another spectrum and point masses of the spectral
measure (or norming constants) corresponding to the first spectrum. We also
solve this Borg-Marchenko-type problem under some conditions on two spectra,
when missing part of the second spectrum and known point masses of the spectral
measure have different index sets.Comment: 33 pages, 1 figur
Spectral bounds for periodic Jacobi matrices
We consider periodic Jacobi operators and obtain upper and lower estimates on
the sizes of the spectral bands. Our proofs are based on estimates on the
logarithmic capacities and connections between the Chebyshev polynomials and
logarithmic capacity of compact subsets of the real line
Uniqueness theorems for meromorphic inner functions
We prove some uniqueness problems for meromorphic inner functions on the
upper half-plane. In these problems we consider spectral data depending
partially or fully on the spectrum, derivative values at the spectrum, Clark
measure or the spectrum of the negative of a meromorphic inner function
Inverse Problems for Jacobi Operators with Mixed Spectral Data
We consider semi-infinite Jacobi matrices with discrete spectrum. We prove
that the Jacobi operator can be uniquely recovered from one spectrum and
subsets of another spectrum and norming constants corresponding to the first
spectrum. We also solve this Borg-Marchenko-type problem under some conditions
on two spectra, when missing part of the second spectrum and known norming
constants have different index sets
Widom Factors
Ankara : The Department of Mathematics and The Graduate School of Engineering and Science of Bilkent University, 2014.Thesis (Master's) -- Bilkent University, 2014.Includes bibliographical references leaves 41-43.In this thesis we recall classical results on Chebyshev polynomials and logarithmic
capacity. Given a non-polar compact set K, we define the n-th Widom
factor Wn(K) as the ratio of the sup-norm of the n-th Chebyshev polynomial on
K to the n-th degree of its logarithmic capacity. We consider results on estimations
of Widom factors. By means of weakly equilibrium Cantor-type sets, K(γ),
we prove new results on behavior of the sequence (Wn(K))∞
n=1.By K. Schiefermayr[1], Wn(K) ≥ 2 for any non-polar compact K ⊂ R. We
prove that the theoretical lower bound 2 for compact sets on the real line can be
achieved by W2s (K(γ)) as fast as we wish.
By G. Szeg˝o[2], rate of the sequence (Wn(K))∞
n=1 is slower than exponential growth. We show that there are sets with unbounded (Wn(K))∞
n=1 and moreoverfor each sequence (Mn)∞
n=1 of subexponential growth there is a Cantor-type set
which Widom factors exceed Mn for infinitely many n.
By N.I. Achieser[3][4], limit of the sequence (Wn(K))∞
n=1 does not exist in the
case K consists of two disjoint intervals. In general the sequence (Wn(K))∞
n=1 may behave highly irregular. We illustrate this behavior by constructing a Cantor-type
set K such that one subsequence of (Wn(K))∞
n=1 converges as fast as we wish to
the theoretical lower bound 2, whereas another subsequence exceeds any sequence
(Mn)∞ n=1 of subexponential growth given beforehandHatinoğlu, BurakM.S
Ambarzumian-type problems for discrete Schr\"{o}dinger operators
We discuss the problem of unique determination of the finite free discrete
Schr\"{o}dinger operator from its spectrum, also known as Ambarzumian problem,
with various boundary conditions, namely any real constant boundary condition
at zero and Floquet boundary conditions of any angle. Then we prove the
following Ambarzumian-type mixed inverse spectral problem: Diagonal entries
except the first and second ones and a set of two consecutive eigenvalues
uniquely determine the finite free discrete Schr\"{o}dinger operator