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We consider the Schrödinger operator H = −Δ − V( x), V > 0, acting in the space $ L^2 (\mathbb{R}^d) $ and study relations between the behavior of V at infinity and properties of the negative spectrum of H. Bibliography: 34 titles. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic formula for the number of negative eigenvalues of a class of two‐dimensional Schrödinger operators with small magnetic fields. This number increases as a coupling constant of the magnetic field tends to zero. [ABSTRACT FROM AUTHOR]

We study the spectral properties of Jacobi matrices. By using ``higher order'' trace formulae we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. Complicated expressions for traces of some operators can be magically simplified allowing us to apply induction arguments. Our theorems are generalizations of a recent result of R. Killip and B. Simon [17]. [ABSTRACT FROM AUTHOR]