2,932,115 research outputs found
The quantum N-body problem with a minimal length
The quantum -body problem is studied in the context of nonrelativistic
quantum mechanics with a one-dimensional deformed Heisenberg algebra of the
form , leading to the existence of a
minimal observable length . For a generic pairwise interaction
potential, analytical formulas are obtained that allow to estimate the
ground-state energy of the -body system by finding the ground-state energy
of a corresponding two-body problem. It is first shown that, in the harmonic
oscillator case, the -dependent term grows faster with than the
-independent one. Then, it is argued that such a behavior should be
observed also with generic potentials and for -dimensional systems. In
consequence, quantum -body bound states might be interesting places to look
at nontrivial manifestations of a minimal length since, the more particles are
present, the more the system deviates from standard quantum mechanical
predictions.Comment: To appear in PR
On the isometric embedding problem for length metric spaces
We prove that every proper -dimensional length metric space admits an
"approximate isometric embedding" into Lorentzian space .
By an "approximate isometric embedding" we mean an embedding which preserves
the energy functional on a prescribed set of geodesics connecting a dense set
of points.Comment: 40 pages, 10 figure
The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length
Theories of classification distinguish classes with some good structure
theorem from those for which none is possible. Some classes (dense linear
orders, for instance) are non-classifiable in general, but are classifiable
when we consider only countable members. This paper explores such a notion for
classes of computable structures by working out a sequence of examples.
We follow recent work by Goncharov and Knight in using the degree of the
isomorphism problem for a class to distinguish classifiable classes from
non-classifiable. In this paper, we calculate the degree of the isomorphism
problem for Abelian -groups of bounded Ulm length. The result is a sequence
of classes whose isomorphism problems are cofinal in the hyperarithmetical
hierarchy. In the process, new back-and-forth relations on such groups are
calculated.Comment: 15 page
The few-body problem for trapped bosons with large scattering length
We calculate energy levels of two and three bosons trapped in a harmonic
oscillator potential with oscillator length . The atoms are
assumed to interact through a short-range potential with a scattering length
, and the short-distance behavior of the three-body wave function is
characterized by a parameter . For large positive ,
the energies of states which, in the absence of the trap, correspond to three
free atoms approach values independent of and . For other states
the dependence of the energy is strong, but the energy is independent
of for .Comment: 4 pages, 3 figure
Stable marriage with ties and bounded length preference lists
We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some δ>1 (even if each woman's list is of length at most 4)
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
- …