6,487 research outputs found

### On the spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator $\hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat \alpha_j + \hat
V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3,$ is proved given that either $A\in
C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)$, 2q > n-2, or the Fourier series of the
vector potential $A:R^n\to R^n$ is absolutely convergent. Here, $\hat
V^{(s)}=(\hat V^{(s)})^*$ are continuous matrix functions and \hat V^{(s)}\hat
\alpha_j=(-1}^s\hat \alpha_j\hat V^{(s)} for all anticommuting Hermitian
matrices $\hat \alpha_j$, $\hat \alpha_j^2=\hat I$, s=0,1.Comment: 17 page

### Motivic Milnor fibre for nondegenerate function germs on toric singularities

We study function germs on toric varieties which are nondegenerate for their
Newton diagram. We express their motivic Milnor fibre in terms of their Newton
diagram. We extend a formula for the motivic nearby fibre to the case of a
toroidal degeneration. We illustrate this by some examples.Comment: 14 page

### Exact relations for quantum-mechanical few-body and many-body problems with short-range interactions in two and three dimensions

We derive relations between various observables for N particles with
zero-range or short-range interactions, in continuous space or on a lattice, in
two or three dimensions, in an arbitrary external potential. Some of our
results generalise known relations between large-momentum behavior of the
momentum distribution, short-distance behavior of the pair correlation function
and of the one-body density matrix, derivative of the energy with respect to
the scattering length or to time, and the norm of the regular part of the
wavefunction; in the case of finite-range interactions, the interaction energy
is also related to dE/da. The expression relating the energy to a functional of
the momentum distribution is also generalised, and is found to break down for
Efimov states with zero-range interactions, due to a subleading oscillating
tail in the momentum distribution. We also obtain new expressions for the
derivative of the energy of a universal state with respect to the effective
range, the derivative of the energy of an efimovian state with respect to the
three-body parameter, and the second order derivative of the energy with
respect to the inverse (or the logarithm in the two-dimensional case) of the
scattering length. The latter is negative at fixed entropy. We use exact
relations to compute corrections to exactly solvable three-body problems and
find agreement with available numerics. For the unitary gas, we compare exact
relations to existing fixed-node Monte-Carlo data, and we test, with existing
Quantum Monte Carlo results on different finite range models, our prediction
that the leading deviation of the critical temperature from its zero range
value is linear in the interaction effective range r_e with a model independent
numerical coefficient.Comment: 51 pages, 5 figures. Split into three articles: Phys. Rev. A 83,
063614 (2011) [arXiv:1103.5157]; Phys. Rev. A 86, 013626 (2012)
[arXiv:1204.3204]; Phys. Rev. A 86, 053633 (2012) [ arXiv:1210.1784

### Global in Time Solutions to Kolmogorov-Feller Pseudodifferential Equations with Small Parameter

The goal in this paper is to demonstrate a new method for constructing
global-in-time approximate (asymptotic) solutions of (pseudodifferential)
parabolic equations with a small parameter. We show that, in the leading term,
such a solution can be constructed by using characteristics, more precisely, by
using solutions of the corresponding Hamiltonian system and without using any
integral representation. For completeness, we also briefly describe the
well-known scheme developed by V.P.Maslov for constructing global-in-time
solutions.Comment: 27 page

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