We briefly review a recursive construction of hbar-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ħ-expansion of the operator X = X_0 + hbar X_1 + hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = exp(X/hbar). The wave function Psi associated with W turns out to have the WKB (Wentzel-Kramers-Brillouin) form Psi=exp(S/hbar), and the coefficients S_n of the ħ-expansion S = S_0 + hbar S_1 + hbar^2 S_2 + ... are also determined by a set of recurrence relations. We use this WKB form to show that the associated tau function has an ħ-expansion of the form log tau = hbar^{-2} (F_0 + hbar F_1 + hbar^2 F_2 + ...).

We discuss correlators for models of minimal gravity and propose an algorithm for calculating invariant relations from formulas for residues that can be compared with the expansion coefficients for the partition function in the Liouville theory. For (2, 2K-1) models, we explicitly obtain a factor corresponding to conversion from the semiclassical hierarchy basis to the Liouville theory basis and also test a hypothesis about the pattern of the spectral curve using a direct calculation

We propose an operator method for calculating the semiclassical asymptotic form of the energy splitting value in the general case of tunneling between symmetric orbits in the phase space. We use this approach in the case of a particle on a circle to obtain the asymptotic form of the energy tunneling splitting related to the over-barrier reflection from the potential. As an example, we consider the quantum pendulum in the rotor regime.

Direct and inverse problems for the Hirota difference equation are considered. Jost solutions and scattering data are introduced and their properties are presented. In a special case Darboux transformation is shown to enable description of the evolution with respect to discrete time and a recursion procedure for consequent construction of the Jost solution at arbitrary time, if the initial value is given. Some properties of the soliton solutions are discussed.

Using the bilinear formalism, we consider multicomponent and matrix Kadomtsev-Petviashvili hierarchies. The main tool is the bilinear identity for the tau function realized as a vacuum expectation value of a Clifford group element composed of multicomponent fermionic operators. We also construct the Baker– Akhiezer functions and obtain auxiliary linear equations that they satisfy.

We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional (Dirac, Weyl, Majorana, and Majorana–Weyl) spinors in the Clifford algebra formalism. We give the explicit form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of arbitrary space dimensions and signatures, using the notion of the Clifford algebra additional signature to describe conjugations. We show that the additional signature can take only certain values despite its dependence on the matrix representation.

We consider a model of a real massive scalar field defined as homogeneous on a d-dimensional sphere such that the sphere radius, time scale, and scalar field are related by the equations of the general theory of relativity. We quantize this system with three degrees of freedom, define the observables, and find dynamical mean values of observables in the regime where the scalar field mass is much less than the Planck mass.

We prove that line solitons of the two-dimensional hyperbolic nonlinear Schrodinger equation are unstable with respect to transverse perturbations of arbitrarily small periods, i.e., short waves. The analysis is based on the construction of Jost functions for the continuous spectrum of Schrodinger operators, the Sommerfeld radiation conditions, and the Lyapunov– Schmidt decomposition. Precise asymptotic expressions for the instability growth rate are derived in the limit of short periods.

We consider the one-dimensional stationary Schr¨odinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the well-known formulas to the case of mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schr¨odinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically prohibited region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.