The classical Weyl formula expresses the leading term of the asymptotics of the counting function N(lambda, H) of the spectrum of a self-adjoint operator H in an invariant form: one can "hear" the volume of the subset of the cotangent bundle where the symbol of the operator H is less than lambda. In particular, it is applicable to Schrodinger operators with electric potentials growing at infini...

In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. We consider the quantum dynamics of a real massless scalar field coupled to gravity in this framework. A Hamiltonian operator for the scalar field can be well defined in the coupled diffeomorphism invariant Hilbert space, which is both ...

The diffusion operator HD = − 1 2 d dx a d dx − b d dx = − 1 2 exp(−2B) d dx a exp(2B) d dx , where B(x) = R x 0 b a (y)dy, defined either on R = (0,∞) with the Dirichlet boundary condition at x = 0, or on R, can be realized as a self-adjoint operator with respect to the density exp(2Q(x))dx. The operator is unitarily equivalent to the Schrödinger-type operator HS = − 1 2 d dx a d dx + Vb,a, wh...

Let A,B be nonzero positive semidefinite matrices. We prove that ‖AB‖ ‖A‖ ‖B‖ ≤ ‖A + B‖ ‖A‖+ ‖B‖ , ‖A ◦B‖ ‖A‖ ‖B‖ ≤ ‖A + B‖ ‖A‖+ ‖B‖ for any unitarily invariant norm with ‖diag(1, 0, . . . , 0)‖ ≥ 1. Some related inequalities are derived. AMS classification: 15A60, 15A45

Let f(t) be a nonnegative concave function on 0 ≤ t < ∞ with f(0) = 0, and letX,Y be n×nmatrices. Then it is known that ‖f(|X+Y |)‖1 ≤ ‖f(|X|)‖1+‖f(|Y |)‖1, where ‖ · ‖1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

We obtain a representation of unitarily invariant norm in terms of Ky Fan norms [1, p.35]. Indeed we obtain a more general result in the context of Eaton triple with reduced triple. Examples are given. 2000 Mathematics Subject Classification: 15A60, 65F35.

Let A − λB and C − λD be two normal matrix pencils or Hermitian matrix pencils or definite matrix pencils, and let AXD −BXC = S. Our main results in this part are estimates of a unitarily invariant norm of the solution X under some conditions on the spectra of two pencils. §