Quantum graphs with singular two-particle interactions

We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions. In order to find such Hamiltonians closed and semi-bounded quadratic forms are constructed, from which the associated self-adjoint operators are extracted. We provide a general characterisation of such operators and, furthermore, produce certain classes of examples. We then consider identical particles and project to the bosonic and fermionic subspaces. Finally, we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate Weyl asymptotic law

On monoids of endomorphisms of a cycle graph

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities

KK-theory of Leavitt path algebras

Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N'_E=(n_{ij})\in\Z^{(E_0\times E_0)}$, n_{ij}=#\{ arrows from $i$ to $j\}$. Write $N^t_E$ and 1 for the matrices \in \Z^{(E_0\times E_0\setminus\Sink(E))} which result from $N'^t_E$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_\Z(E)\otimes R$. We show that if $R$ is either a Noetherian regular ring or a stable $C^*$-algebra, then there is an exact sequence ($n\in\Z$) K_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow} K_n(R)^{(E_0)}\to K_n(L_R(E))\to K_{n-1}(R)^{(E_0\setminus\Sink(E))} We also show that for general $R$, the obstruction for having a sequence as above is measured by twisted nil-$K$-groups. If we replace $K$-theory by homotopy algebraic $K$-theory, the obstructions dissapear, and we get, for every ring $R$, a long exact sequence KH_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow}KH_n(R)^{(E_0)}\to KH_n(L_R(E))\to KH_{n-1}(R)^{(E_0\setminus\Sink(E))} We also compare, for a $C^*$-algebra \fA, the algebraic $K$-theory of L_\fA(E) with the topological $K$-theory of the Cuntz-Krieger algebra C^*_\fA(E). We show that the map K_n(L_\fA(E))\to K^{\top}_n(C^*_\fA(E)) is an isomorphism if \fA is stable and $n\in\Z$, and also if \fA=\C, $n\ge 0$, $E$ is finite with no sinks, and $\det(1-N_E^t)\ne 0$.Comment: 30 page

Area theorem and smoothness of compact Cauchy horizons

We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every compactly generated Cauchy horizon is smooth and compact. We explore the consequences of this result for time machines, topology change, black holes and cosmic censorship. For instance, it is shown that compact Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable dominant energy condition wherever there is some source content.Comment: 44 pages. v2: added Sect. 2.4 on the propagation of singularities and a second version of the area theorem (Theor. 14) which quantifies the area increase due to the jump se

Topological Sectors and a Dichotomy in Conformal Field Theory

Let A be a local conformal net of factors on the circle with the split property. We provide a topological construction of soliton representations of the tensor product of n copies of A, that restrict to true representations of subnet inviant under cyclic permutations (cyclic orbifold). We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the the cyclic orbifold has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the mu-index of A finite then A turns out to be strongly additive. By [KLM], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.Comment: 50 pages, Latex; minor correction