Infinite number of eigenvalues of $$2{\times}2$$ operator matrices: Asymptotic discrete spectrum
- PDF / 638,399 Bytes
- 21 Pages / 612 x 792 pts (letter) Page_size
- 9 Downloads / 190 Views
INFINITE NUMBER OF EIGENVALUES OF 2×2 OPERATOR MATRICES: ASYMPTOTIC DISCRETE SPECTRUM T. H. Rasulov∗ and E. B. Dilmurodov∗ We study an unbounded 2×2 operator matrix A in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the number of eigenvalues of A. We consider a 2×2 operator matrix Aμ , where μ > 0 is the coupling constant, associated with the Hamiltonian of a system with at most three particles on the lattice Z3 . We find the critical value μ0 of the coupling constant μ for which Aμ0 has an infinite number of eigenvalues. These eigenvalues accumulate at the lower and upper bounds of the essential spectrum. We obtain an asymptotic formula for the number of such eigenvalues in both the left and right parts of the essential spectrum.
Keywords: operator matrix, coupling constant, dispersion function, Fock space, creation operator, annihilation operator, Birman–Schwinger principle, essential spectrum, discrete spectrum, asymptotics DOI: 10.1134/S0040577920120028
1. Introduction Let H1 and H2 be Hilbert spaces. Any linear bounded operator A acting in the direct sum H := H1 ⊕H2 is known to be represented as A11 A12 , (1) A := A21 A22 where the matrix elements Aij are linear bounded operators from Hj to Hi , i.e., Aij ∈ L(Hj , Hi ), i, j ∈ {1, 2}. The operator A is self-adjoint if and only if A11 = A∗11 , A22 = A∗22 , and A21 = A∗12 . Such matrices, i.e., matrices whose elements are linear operators in Banach or Hilbert spaces, are usually called 2×2 operator matrices [1]. One of the main classes of such matrices is represented by Hamiltonians of a system with a nonconserved bounded number of particles on a continuous space or on a lattice. Such systems usually occur in solid-state physics problems [2], quantum field theory [3], and statistical physics [4]. The discrete spectrum of operator matrices is currently intensively studied in linear operator theory. One of the important questions in such an operator spectral analysis is the question of an infinite number of eigenvalues located to the left of the lower bound and to the right of the upper bound of the essential spectrum (such an effect for the left edge is called the Efimov effect; see, e.g., [5]–[7]). The Efimov effect [8] is one of the most remarkable and interesting features of a three-body system in contrast to a two-body system. It was first discovered for a three-particle continuous Schr¨odinger operator. This effect occurs when there are weakly bound states with a binding energy close to zero for at least two pairwise interactions of the three-particle system. The existence of the Efimov effect in the continuous case was rigorously proved ∗
Bukhara State University, Bukhara, Uzbekistan; Bukhara Department, Romanovsky Mathematics Institute, Bukhara, Uzbekistan, e-mail: [email protected] (corresponding author), [email protected]. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 205, No. 3, pp. 368–390, November, 2020. Received March 4, 2020. Revised April 23, 2020. Accepted July 22, 2020. 1564
c 2020 Pleiades Publishing, L
Data Loading...
