Related papers: Scattering theory on graphs (2): the Friedel sum r…
We consider the scattering theory for the Schr\"odinger operator $-\Dc_x^2+V(x)$ on graphs made of one-dimensional wires connected to external leads. We derive two expressions for the scattering matrix on arbitrary graphs. One involves…
We consider graphs made of one-dimensional wires connected at vertices and on which may live a scalar potential. We are interested in a scattering situation where the graph is connected to infinite leads. We investigate relations between…
We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n=1. This theorem counts the number of bound states in terms of…
We illustrate the relation between the scattering phase appearing in the Friedel sum rule and the phase of the transmission amplitude for quantum scatterers connected to two one-dimensional leads. Transmission zero points cause abrupt phase…
The properties of scattering phases in quantum dots are analyzed with the help of lattice models. We first derive the expressions relating the different scattering phases and the dot Green functions. We analyze in detail the Friedel sum…
We prove an analog of Levinson's theorem for scattering on a weighted (m+1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal…
We consider graphs made of one-dimensional wires connected at vertices, and on which may live a scalar potential. We are interested in a scattering situation where such a network is connected to infinite leads. We study the correlations of…
We propose a simple method for resolution of co-spectrality of Schr\"odinger operators on metric graphs. Our approach consists of attaching a lead to them and comparing the $S$-functions of the corresponding scattering problems on these…
We investigate the propagation of a massless scalar field on a star graph, modeling the junction of $n$ quantum wires. The vertex of the graph is represented by a point-like impurity (defect), characterized by a one-body scattering matrix.…
Elastic scattering in a quantum wire has several novel features not seen in 1D, 2D or 3D. In this work we consider a single channel quantum wire as its application is inevitable in making devices based on quantum interference effects. We…
We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far.…
A generalized Friedel sum rule is derived for a quantum dot with internal orbital and spin degrees of freedom. The result is valid when all many-body correlations are taken into account and it links the phase shift of the scattered electron…
We investigate numerically the scattering of waves on discrete graphs. An efficient algorithm is developed to compute the reflection and transmission (spectral) coefficients. We then explore various configurations of input and output leads,…
In this paper we investigate the spectral and the scattering theory of Schr\"odinger operators acting on perturbed periodic discrete graphs. The perturbations considered are of two types: either a multiplication operator by a short-range or…
We derive and analyze the f-sum rule for a two-dimensional (2D) system of interacting electrons whose behavior is described by the Dirac equation. We apply the sum rule to analyze the spectral weight transfer in graphene within different…
Since the experimental observation of quantum mechanical scattering phase shift in mesoscopic systems, several aspects of it has not yet been understood. The experimental observations has also accentuated many theoretical problems related…
The validity of the Luttinger sum rule is considered for finite systems of interacting electrons, where the Fermi volume is determined by location of zeroes of Green's function. It is shown that the sum rule in the paramagnetic state is…
We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type boundary…
We show that the spectrum of the Schrodinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel…
The study of the scattering data for a star-shape network of LC-transmission lines is transformed into the scattering analysis of a Schr\"odinger operator on the same graph. The boundary conditions coming from the Kirchhoff rules ensure the…