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Award ID contains: 2052519

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  1. (, Canadian Mathematical Bulletin)
    Abstract We consider 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. Moreover, we consider applications of these uniqueness results to inverse spectral theory of canonical Hamiltonian systems and obtain generalizations of the Borg-Levinson two-spectra theorem for canonical Hamiltonian systems and unique determination of a Hamiltonian from its spectral measure under some conditions. 
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    Free, publicly-accessible full text available March 1, 2026
  2. ; ; (, Communications in Mathematical Physics)
    Abstract We obtain a perturbative proof of localization for quasiperiodic operators on$$\ell ^2(\mathbb Z^d)$$ ℓ 2 ( Z d ) with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also prove that the spectra of such operators contain infinitely many gaps. 
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  3. ; ; (, Journal of Mathematical Physics)
    We use Bézout’s theorem and Bernstein–Khovanskii–Kushnirenko theorem to analyze the level sets of the extrema of the spectral band functions of discrete periodic Schrödinger operators on Z2. These approaches improve upon previous results of Liu and Filonov–Kachkovskiy. 
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    Free, publicly-accessible full text available June 1, 2026
  4. ; ; (, International Mathematics Research Notices)
    We show that, for one-dimensional discrete Schrödinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known results of Gordon and del Rio–Makarov–Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials. 
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    Free, publicly-accessible full text available April 1, 2026
  5. ; (, SIAM Journal on Applied Algebra and Geometry)
    Free, publicly-accessible full text available March 31, 2026
  6. ; (, Zurnal matematiceskoj fiziki, analiza, geometrii)
    Full Text Available 
  7. ; (, Communications in Mathematical Physics)
    We consider discrete periodic operator on Z^d with respect to lattices of full rank. We describe the class of lattices for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed d-2. 
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  8. ; ; (, Journal of Mathematical Physics)
    We consider a discrete non-linear Schrödinger equation on Z and show that, after adding a small potential localized in the time-frequency space, one can construct a three-parametric family of non-decaying spacetime quasiperiodic solutions to this equation. The proof is based on the Craig–Wayne–Bourgain method combined with recent techniques of dealing with Anderson localization for two-dimensional quasiperiodic operators with degenerate frequencies. 
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