T. A. Suslina

Threshold Effects near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics

In L 2 \(\left( {{\mathbb{R}^d}} \right), \) we consider vector periodic DO A admitting a factorization A = X*X, where X is a homogeneous DO of first order. Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral decomposition of A in a small neighborhood of zero are called threshold effects at λ = O. An example of a threshold effect is the behavior of a DO in the small period limit Another example is related to the negative discrete spectrum of the operator A- α V, α> 0, where V(x) ≥ 0 and V(x) → 0 as |x|→ ∞. The “effective characteristics”, namely, the homogenized medium, the effective mass, the effective Hamiltonian, etc. arise in these problems. We propose a general approach to these problems based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. A great deal of considerations is done in abstract terms.

The periodic Dirac operator is absolutely continuous

We consider the periodic Dirac operatorD inL2(ℝd). The magnetic potentialA and the electric potentialV are periodic. Ford=2 the absolute continuity ofD is established forA,V∈Lr, loc,r>2; the proof is based on the estimates, obtained by the authors earlier [BSu2] for the periodic magnetic Schrödinger operatorM. Ford≥3 our considerations are based on the estimates forM, obtained in [So] forA∈C2d+3. Under the same condition onA, forV∈C, the absolute continuity ofD, d≥3, is proved. ForA=0 the arguments of the paper give a new (and much simpler) proof of the main result of [D].

Two-Dimensional Periodic Pauli Operator. The Effective Masses at the Lower Edge of the Spectrum

We calculate the tensor of effective masses for the two-dimensional periodic Pauli operator. The explicit representation for this tensor is given in terms of the magnetic field. It is proved that the tensor of effective masses is circular symmetric and always proportional to the unit matrix. We also consider the generalized Pauli operator with a variable metric. In the appendix we study the periodic elliptic operators of the second order and discuss the behaviour of the first band function near its minimum point.

Homogenization of initial boundary value problems for parabolic systems with periodic coefficients

Let be a bounded domain of class . In the Hilbert space , we consider matrix elliptic second-order differential operators and with the Dirichlet or Neumann boundary condition on , respectively. Here is the small parameter. The coefficients of the operators are periodic and depend on . The behaviour of the operator , , for small is studied. It is shown that, for fixed , the operator converges in the -operator norm to , as . Here is the effective operator with constant coefficients. For the norm of the difference of the operators and , a sharp order estimate (of order ) is obtained. Also, we find approximation for the exponential in the -norm with error estimate of order ; in this approximation, a corrector is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.

Two-parametric error estimates in homogenization of second-order elliptic systems in ℝd

In , we consider a selfadjoint operator , , given by the differential expression , where is the first-order differential operator, and are matrix-valued functions in periodic with respect to some lattice . It is assumed that g is bounded and positive definite, while and Q are, in general, unbounded. We study the generalized resolvent , where is a -periodic, bounded and positive definite matrix-valued function, and is a complex-valued parameter. Approximations for the generalized resolvent in the - and -norms with two-parametric error estimates (with respect to the parameters and ) are obtained.

Homogenization of higher-order parabolic systems in a bounded domain

ABSTRACT Let be a bounded domain of class . In , we consider matrix elliptic differential operators and of order 2p ( ) with the Dirichlet or Neumann boundary conditions, respectively. The coefficients of and are periodic and depend on , . The behavior of the operator , , for small is studied. It is shown that, for fixed , the operator converges in the -operator norm to , as . Here is the effective operator with constant coefficients. We obtain a sharp order estimate . Also, we find approximation for in the -norm with error estimate of order . The results are applied to homogenization of the solutions of initial boundary value problems for parabolic systems.