We present full description of spectra for elastic beam Hamiltonian defined on periodic hexagonal lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar valued self-adjoint fourth-order operator equipped with a real periodic symmetric potential. Compared to the Schrödinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the graph by their dependence on angles at which edges are met. We show that for a special equal-angle lattice, known as graphene, dispersion relation has a similar structure as reported for Schrödinger operator on periodic hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We next discuss the role of the potential on reducibility of Fermi surface at uncountably many low-energy levels for this special lattice.Applying perturbation analysis, the developed theory is extended to derive dispersion relation for angle-perturbed Hamiltonian of lattices in a geometric-neighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, but up to some quantifiable accuracy, singular Dirac points exist at the same points as the graphene case.

We consider the Schrödinger 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.

Given a non-polar compact set K, we define the n-th Widom factor W n (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. By G. Szegő, the sequence (W n (K)) ∞ n=1 has subexponential growth. Our aim is to consider compact sets with maximal growth of the Widom factors. We show that for each sequence (M n ) ∞ n=1 of subexponential growth there is a Cantor-type set whose Widom's factors exceed M n . We also present a set K with highly irregular behavior of the Widom factors.

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the 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 Schrödinger operator.

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