We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems (including characterization): in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). We prove that a potential is uniquely determined by zeros of reflection coefficients and there exist distinct potentials with the same resonances. We describe the set of "isoresonance potentials". Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for an other compactly supported potential, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonances counting function is determined, 4) these results are applied to canonical systems. Contents 25 7. Canonical systems 29 References 36

We consider the Schrödinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential → all eigenvalues is a real analytic isomorphism for some class of potentials.

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.

Let G n , n E N, denote the set of gaps of the Hill operator. We solve the following problems: 1) find the effective masses M^> 2) compare the effective mass M^ with the length of the gap G n , and with the height of the corresponding slit on the quasimomentum plane (both with fixed number n and their sums), 3) consider the problems 1), 2) for more general cases (the Dirac operator with periodic coefficients, the Schrodinger operator with a limit periodic potential). To obtain 1)-3) we use a conformal mapping corresponding to the quasimomentum of the Hill operator or the Dirac operator.

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