Vol 186, No 1 (2016)

We consider a method for solving the problem of quantum tunneling through repulsive potential barriers for a composite system consisting of several identical particles coupled via pair oscillator-type potentials in the oscillator symmetrized-coordinate representation. We confirm the efficiency of the proposed approach by calculating complex energy values and analyzing metastable states of composite systems of three, four, and five identical particles on a line, which leads to the effect of quantum transparency of the repulsive barriers.

We study the asymptotic behavior of the wave function of the system of three Coulomb particles in the united-atom limit in the adiabatic representation of the three-body problem. This result is used to calculate the nuclear widths of muonic-molecule energy levels. We discuss features of the approach with regard to excited states of the muonic molecule ttµ with a nonzero orbital angular momentum.

Let A be a self-adjoint operator in a separable Hilbert space. We assume that the spectrum of A consists of two isolated components σ₀ and σ₁ and the set σ₁ is in a finite gap of the set σ₁. It is known that if V is a bounded additive self-adjoint perturbation of A that is off-diagonal with respect to the partition spec(A) = σ₀ ∪ σ₁, then for \(\left\| V \right\| < \sqrt 2 d\), where d = dist(σ₀, σ₁), the spectrum of the perturbed operator L = A+V consists of two isolated parts ω₀ and ω₁, which appear as perturbations of the respective spectral sets s0 and s1. Furthermore, we have the sharp upper bound ||EA(σ₀) - EL(ω₀)|| ≤ sin (arctan(||V||/d)) on the difference of the spectral projections E_A(σ₀)) and E_L(ω₀)) corresponding to the spectral sets σ₀ and ω₀ of the operators A and L. We give a new proof of this bound in the case where ||V|| < d.

We prove that radial wave functions of a charged quantum particle moving in a two-dimensional plane of the three-dimensional coordinate space and scattering by a Coulomb center at rest in the same plane are governed by the Coulomb equation with a half-integer index. We investigate the structure of these functions and consider three physically interesting limits: the non-Coulomb limit and high- and low-energy limits. We explicate the basic differences between two- and three-dimensional Coulomb scattering.

We study the wave function of a system of three particles in a continuum. The Faddeev equations are used to explicitly identify the singularities of the wave function in the momentum space. We obtain the asymptotic behavior of the wave function in the configuration space by calculating the asymptotic behavior of the Fourier transform of the wave function in the momentum space. Our attention is focused on configurations in which two particles are at a relatively small distance from each other while the third particle is significantly remote from the center of mass of the pair. We show that the coordinate asymptotic form of the wave function for such a configuration contains scattered waves of a new type in addition to the standard terms. We use the obtained exact data concerning the coordinate asymptotic form of the wave function to critically analyze the multiplicative ansatz used in several works to describe systems of three particles in a continuum.