arXiv subjects

Mats-Erik Pistol

Publications and source records attributed to Mats-Erik Pistol.

9 recordsLinked to original sources

arXiv2025

Generating isospectral but not isomorphic quantum graphs

Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions although we do a slight excursion into graphs with Dirichlet and $δ$-type boundary condititons towards the end of the paper. The spectrum of $\mathbf{L}$ does not determine the graph uniquely, that is, there exist non-isomorphic graphs with the same spectra. There are few known examples of pairs of non-isomorphic but isospectral quantum graphs. In this paper we start to correctify this situation by finding hundreds of isospectral sets, using computer algebra. We have found all sets of isospectral but non-isomorphic equilateral connected quantum graphs with at most nine vertices. This includes thirteen isospectral triplets and one isospectral set of four. One of the isospectral triplets involves a loop where we could prove isospectrality. We also present several different combinatorial methods to generate arbitrarily large sets of isospectral graphs, including infinite graphs in different dimensions. As part of this we have found a method to determine if two vertices have the same Titchmarsh-Weyl $M$-function. We give combinatorial methods to generate sets of graphs with arbitrarily large number of vertices with the same $M$-function. We also find several sets of graphs that are isospectral under more general, permutation invariant, boundary conditions. This necessitates a study of eigenvalue zero where we prove several results. We discuss the possibilities that our program is incorrect, present our tests and open source it for inspection at http://github.com/meapistol/Spectra-of-graphs

math.SP↗

arXiv2025

Families of isospectral and isoscattering quantum graphs

A concept of germ graphs and the M-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the M-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our novel approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.

quant-ph↗

arXiv2024

Cospectral trees indistinguishable by scattering

Let v_1 and v_2 be two distinct vertices of a tree T_0. Let ϕ_N^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 rooted at v_i with Neumann conditions at the root and let ϕ_D^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 with Dirichlet conditions at the root. We prove that if attaching any tree to T_0 at the vertices v_1 and v_2 leads to cospectral trees and d(v_1)=d(v_2) then ϕ_N(λ)^{(1)}\equiv ϕ_N(λ)^{(2)} and ϕ_D(λ)^{(1)}\equiv ϕ_D(λ)^{(1)} (which means that the scattering is the same at v_1 and v_2).

math-ph↗

arXiv2022

Spectrally grown graphs

Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A problem studying such quantum graphs is that the spectrum is timeconsuming to compute by hand and the inverse problem of finding a quantum graph having a specified spectrum is difficult. We solve the forward problem, to find the eigenvalues, using a previously developed computer program. We obtain all eigenvalues analytically for not too big graphs that have rationally dependent edges. We solve the inverse problem using "spectrally grown graphs". The spectrally grown graphs are evolved from a starting (parent) graph such that the child graphs have eigenvalues are close to some criterion. Our experiments show that the method works and we can usually find graphs having spectra which are numerically close to a prescribed spectrum. There are naturally exceptions, such as if no graph has the prescribed spectrum. The selection criteria (goals) strongly influence the shape of the evolved graphs. Our experiments allows us to make new conjectures concerning the spectra of quantum graphs. We open-source our software at https://github.com/meapistol/Spectra-of-graphs.

math.SP↗

arXiv2017

Single-nanowire, low-bandgap hot carrier solar cells with tunable open-circuit voltage

Compared to traditional pn-junction photovoltaics, hot carrier solar cells offer potentially higher efficiency by extracting work from the kinetic energy of photogenerated "hot carriers" before they cool to the lattice temperature. Hot carrier solar cells have been demonstrated in high-bandgap ferroelectric insulators and GaAs/AlGaAs heterostructures, but so far not in low-bandgap materials, where the potential efficiency gain is highest. Recently, a high open-circuit voltage was demonstrated in an illuminated wurtzite InAs nanowire with a low bandgap of 0.39 eV, and was interpreted in terms of a photothermoelectric effect. Here, we point out that this device is a hot carrier solar cell and discuss its performance in those terms. In the demonstrated devices, InP heterostructures are used as energy filters in order to thermoelectrically harvest the energy of hot electrons photogenerated in InAs absorber segments. The obtained photovoltage depends on the heterostructure design of the energy filter and is therefore tunable. By using a high-resistance, thermionic barrier an open-circuit voltage is obtained that is in excess of the Shockley-Queisser limit. These results provide generalizable insight into how to realize high voltage hot carrier solar cells in low-bandgap materials, and therefore are a step towards the demonstration of higher efficiency hot carrier solar cells.

physics.app-ph↗

arXiv2015

Atomistic $k.p$ theory

Pseudopotentials, tight-binding models, and $k\cdot p$ theory have stood for many years as the standard techniques for computing electronic states in crystalline solids. Here we present the first new method in decades, which we call atomistic $k\cdot p$ theory. In its usual formulation, $k\cdot p$ theory has the advantage of depending on parameters that are directly related to experimentally measured quantities, however it is insensitive to the locations of individual atoms. We construct an atomistic $k\cdot p$ theory by defining envelope functions on a grid matching the crystal lattice. The model parameters are matrix elements which are obtained from experimental results or {\it ab initio} wave functions in a simple way. This is in contrast to the other atomistic approaches in which parameters are fit to reproduce a desired dispersion and are not expressible in terms of fundamental quantities. This fitting is often very difficult. We illustrate our method by constructing a four-band atomistic model for a diamond/zincblende crystal and show that it is equivalent to the $sp^3$ tight-binding model. We can thus directly derive the parameters in the $sp^3$ tight-binding model from experimental data. We then take the atomistic limit of the widely used eight-band Kane model and compute the band structures for all III-V semiconductors not containing nitrogen or boron using parameters fit to experimental data. Our new approach extends $k\cdot p$ theory to problems in which atomistic precision is required, such as impurities, alloys, polytypes, and interfaces. It also provides a new approach to multiscale modeling by allowing continuum and atomistic $k\cdot p$ models to be combined in the same system.

cond-mat.mtrl-sci↗

arXiv2004

N-representability of two-electron densities and density matrices and the application to the few-body problem

We have found a (dense) basis for the N-representable, two-electron densities, in which all N-representable two-electron densities can be expanded, using positive coefficients. The inverse problem of finding a representative wavefunction, giving a prescribed two-electron density, has also been solved. The two-electron densities are found to lie in a convex set in a vector space. We show that density matrices are more complicated objects than densities, and density matrices do not seem to lie in a convex set. An algorithm to compute the ground-state energy of a few-particle system is proposed, based on the obtained results, where the correlation is treated exactly.

cond-mat.str-el↗

arXiv2000

Boundary conditions in the envelope function approximation as applied to semiconductor heterostructures: the multi-band case

We have found the equations that determine the self-adjoint extensions, and thus the boundary conditions, of the differential operator used in the multi-band k.p-theory, when the coefficients in the Kane-matrix are piecewise constant. Both the one-dimensional and the three-dimensional case have been investigated. A numerical calculation shows that the choice of boundary conditions affects the energy eigenvalues for a quantum well.

cond-mat.mtrl-sci↗

arXiv1997

Anisotropic GaAs island phase grown on flat GaP: spontaneously formed quantum wire array

A dense phase of GaAs wires forms in the early stages of strained growth on GaP,assembling from elongated Stranski-Krastanow islands. The electron diffraction during growth is consistent with long, faceted GaAs islands that are anisotropically deformed without dislocations. The lateral wire period and long shapes are not predicted by published models, though we conclude that the island orientation is picked out by facet energy inequivalencies not present in the analogous system of Ge islands on Si.

cond-mat↗