Higher dimensional nonclassical eigenvalue asymptotics

Nonclassical Eigenvalue Asymptotics *

1 he leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {(x, y)l 1x1 " / yl < 11 and for 4 + lxlD I yj4 all of which are non-Weyl because of infinite phase space volumes are computed. Along the way, a general inequality on quantum partition functions coriputed in a kind of Born-Oppenheimer approximation is proved.

Nonclassical Eigenvalue Asymptotics for Operators of Schrödinger

which depends on the volume u)n of the unit sphere in R n and the beta function. Assuming /3 < 2 we see that integral (2) becomes divergent if V (x) vanishes to a sufficiently high order. The simplest such potential is V(x,y) = \x\\y\P o n R n + R m . The Weyl (volume counting) principle, when applied to the corresponding Schrödinger operator — A-hV(x), fails to predict discrete spectrum below ...

Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian

The Riemann Zeros and Eigenvalue Asymptotics

Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the “Riemann dynamics” should be chaotic and ha...

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide i.e. a domain of the form Ωθ := rθω × R where ω is a connected open and bounded subset of R2 and rθ = rθ(x3) is a rotation by the angle θ(x3) depending on the longitudinal variable x3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in Ωθ. We suppose that the derivative θ̇ of the rotation angle can be written as θ̇(x3) = β − ...