This work presents probability inequalities for sums of independent, random, selfadjoint matrices. The results frame simple, easily verifiable hypotheses on the summands, and they yield strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of rectangular matrices follow as an immediate corollary, and similar techniques yiel...

We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and PT -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low t...

A trace formula is proved for pairs of selfadjoint operators that are close to each other in a certain sense. An important role is played by a function analytic in the open upper half-plane and with positive imaginary part there. This function, called the characteristic function of the pair, coincides with Krĕın’s Q-function in the case where the selfadjoint operators are canonical extensions o...

Abstract An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes fact that such operator strictly less than supremum symbol. Consequences time-frequency localization are also given.

Krein space operator-theoretic methods are used to prove that the operator (sgn x) d is similar to a selfadjoint operator in the Hilbert space L2 (R) . Let L be a symmetric ordinary differential expression. Spectral properties of the operators associated with the weighted eigenvalue problem Lu = )wu have been studied extensively. When w is positive, this problem leads to a selfadjoint problem i...

We consider the Dirac equation with a magnetic-solenoid field (the superposition of a solenoid field and a collinear uniform magnetic field). Using von Neumann’s theory of the selfadjoint extensions of symmetric operators, we construct selfadjoint Dirac Hamiltonians for the case of the Aharonov-Bohm solenoid in 2 + 1 and 3 + 1 dimensions. Each extension of the family is specified by certain asy...

We propose to build in this paper a combinatorial invariant, called the ”spectral monodromy” from the spectrum of a single (non-selfadjoint) h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting selfadjoint h-pseudodifferential operators, given ...