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 $L^1$-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