Vol 200, No 1 (2019)

Schur polynomials admit a somewhat mysterious deformation to Macdonald and Kerov polynomials, which do not have a direct group theory interpretation but do preserve most of the important properties of Schur functions. Nevertheless, the family of Schur–Macdonald functions is not sufficiently large: for various applications today, we need their not-yet-known analogues labeled by plane partitions, i.e., three-dimensional Young diagrams. Recently, a concrete way to obtain this generalization was proposed, and miraculous coincidences were described, raising hopes that it can lead in the right direction. But even in that case, much work is needed to convert the idea of generalized 3-Schur functions into a justified and effectively working theory. In particular, we can expect that Macdonald functions (and even all Kerov functions, given some luck) enter this theory on an equal footing with ordinary Schur functions. In detail, we describe how this works for Macdonald polynomials when the vector-valued times, which are associated with plane partitions and are arguments of the 3-Schur functions, are projected onto the ordinary scalar times under nonzero angles that depend on the Macdonald parameters q and t. We show that the cut-and-join operators smoothly interpolate between different limit cases. Most of the examples are restricted to level 2.

Let h be a complex commutative subalgebra of the n×n matrices M_n(ℂ). In the algebra MPsd of matrix pseudodifferential operators in the derivation ∂, we previously considered deformations of h[∂] and of its Lie subalgebra h[∂]_>0 consisting of elements without a constant term. It turned out that the different evolution equations for the generators of these two deformed Lie algebras are compatible sets of Lax equations and determine the corresponding h-hierarchy and its strict version. Here, with each hierarchy, we associate an MPsd-module representing perturbations of a vector related to the trivial solution of each hierarchy. In each module, we describe so-called matrix wave functions, which lead directly to solutions of their Lax equations. We next present a connection between the matrix wave functions of the h-hierarchy and those of its strict version; this connection is used to construct solutions of the latter. The geometric data used to construct the wave functions of the strict h-hierarchy are a plane in the Grassmannian Gr(H), a set of n linearly independent vectors {w_i} in W, and suitable invertible maps δ: S¹ → h, where S₁ is the unit circle in ℂ*. In particular, we show that the action of a corresponding flow group can be lifted from W to the other data and that this lift leaves the constructed solutions of the strict h-hierarchy invariant. For n > 1, it can happen that we have different solutions of the strict h-hierarchy for fixed W and {w_i}. We show that they are related by conjugation with invertible matrix differential operators.

We investigate boundary value problems for a singular integral equation of the convolution type with a power-law nonlinearity. Such problems arise in the theory of p-adic open-closed strings. We prove constructive theorems on the existence of nontrivial solutions and also prove a uniqueness theorem in a certain weight class of functions. We study asymptotic properties of the constructed solutions and obtain the Vladimirov theorem on tachyon Gelds for open-closed strings as a particular case of the proved results.

For an infinite Kitaev chain with an impurity described by a deltalike potential, we analytically prove that two overlapping Majorana bound states in a topologically trivial phase in the case of a small superconducting gap exist under the condition V₀ = 2Δ, where V₀ is the value of the potential and Δ is the superconducting order parameter. For a semi-infinite Kitaev chain with an impurity in the case of a small gap, we prove that there are two overlapping Majorana bound states in the trivial phase and one Majorana bound state in the topological phase and that the Majorana bound state in the latter case is stable under changes in the model parameters. We find explicit analytic expressions for the corresponding wave functions in all cases.

We transform the Hubbard Hamiltonian for electrons of ns bands of crystal atoms into the electron-hole Hamiltonian using Shiba operators. We use this Hamiltonian to study the behavior of the electron-hole system as a function of its internal and external parameters and show that in contrast to a one-component electron system, there are no conditions for the appearance of structural phase transitions in this two-component system.