M. Sh. Birman

Remarks on the spectral shift function

Novel compounds resulting from the reaction of hindered phenols, such as 3,5-di-t-butyl-4-hydroxybenxyl alcohol, with various aryl amines or carbazole are effective oxidation inhibitors for lubricants.

Asymptotic behavior of the spectrum of differential equations

This survey is devoted to an exposition of results on the asymptotics of the discrete spectrum of self-adjoint differential operators, mainly partial differential operators.

Spectral shift function, amazing and multifaceted

Several formula representations for the I. M. Lifshits — M. G. Kreîn spectral shift function (SSF) are discussed and intercompared. It is pointed out that the equivalence of these representations is not apparent, and different properties of the SSF are revealed by different formulas. The presentation is informal and contains no proofs.

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].

Operator integration, perturbations, and commutators

AbstractUnder mild assumption, integral representations of the form(*)

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

f(A_1 ) \cdot \mathfrak{J} - \mathfrak{J} \cdot f(A_1 ) = \int {\int {\frac{{f(\mu ) - f(\lambda )}}{{\mu - \lambda }}} } dE_1 (\mu )(A_1 \mathfrak{J} - \mathfrak{J}A_0 )dE_0 (\mu ),

The maxwell operator in domains with edges

are justified. Here Ak, k=0, 1, is a self-adjoint operator in a Hilbert space Hk, is an operator from H0 H1; in general, all the operators are unbounded; Ek is the spectral measure of the operator Ak. On the basis of the representation (*), estimates of the s-numbers of the operator