Quantum graph resonances by cut-off technique
We demonstrate how resonances in a quantum graph consisting of a compact core and semi-infinite leads can be identified from the eigenvalue behavior of the cut-off system.
quant-ph↗
arXiv subjects
Jiří Lipovský
Publications and source records attributed to Jiří Lipovský.
The spectral determinant for second order elliptic operators on the real line
We derive an expression for the spectral determinant of a second-order elliptic differential operator $\mathcal{T}$ defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation $\mathcal{T} u=0$. Examples of application of the resulting formula include the explicit calculation of the determinant of harmonic and anharmonic oscillators with an added bounded potential with compact support.
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The role of the branch cut of the logarithm in the definition of the spectral determinant for non-selfadjoint operators
The spectral determinant is usually defined using the spectral zeta function that is meromorphically continued to zero. In this definition, the complex logarithms of the eigenvalues appear. Hence the notion of the spectral determinant depends on the way how one chooses the branch cut in the definition of the logarithm. We give results for the non-self-adjoint operators that state when the determinant can and cannot be defined and how its value differs depending on the choice of the branch cut.
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Magnetic square lattice with vertex coupling of a preferred orientation
We analyze a square lattice graph in a magnetic field assuming that the vertex coupling is of a particular type violating the time reversal invariance. Calculating the spectrum numerically for rational values of the flux per plaquette we show how the two effects compete; at the high energies it is the magnetic field which dominates restoring asymptotically the familiar Hofstadter's butterfly pattern.
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Spectral transition model with the general contact interaction
Using a technique introduced by Sergei Naboko, we analyze a generalization of the parameter-controlled model of spectral transition, originally proposed by Smilansky and Solomyak, to the situation where the singular interaction responsible for the effect is characterized by the `full' family of four real numbers with the `diagonal' part of the coupling being position-dependent.
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Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks
We investigate experimentally a Fermi golden rule in two-edge and five-edge microwave networks with preserved time reversal invariance. A Fermi golden rule gives rates of decay of states obtained by perturbing embedded eigenvalues of graphs and networks. We show that the embedded eigenvalues are connected with the topological resonances of the analyzed systems and we find the trajectories of the topological resonances in the complex plane.
quant-ph↗
Application of quotient graph theory to three-edge star graphs
We apply the quotient graph theory described by Band, Berkolaiko, Joyner and Liu to particular graphs symmetric with respect to $S_3$ and $C_3$ symmetry groups. We find the quotient graphs for the three-edge star quantum graph with Neumann boundary conditions at the loose ends and three types of coupling conditions at the central vertex (standard, $δ$ and preferred-orientation coupling). These quotient graphs are smaller than the original graph and the direct sum of quotient graph Hamiltonians is unitarily equivalent to the original Hamiltonian.
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The determinant of one-dimensional polyharmonic operators of arbitrary order
We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator $P_{n}=(-1)^{n} (\partial_x)^{2n}$ on $(0,T)$ with Dirichlet boundary conditions and $n$ a positive integer, and show that it satisfies the asymptotics $\log{(\det P_{n})} = -n^2 \log{n} + \left[\frac{7ζ(3)}{2π^2}+ \frac{3}{2}+\log\left(\frac{T}{4}\right)\right] n^2 + {\rm O}(n)$ for large $n$. This is a consequence of sharp upper and lower bounds for $\log{(\det P_{n})}$ valid for all $n$ and which coincide in the terms up to order $n$. These results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour.
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Spectral determinant for the damped wave equation on an interval
We evaluate the spectral determinant for the damped wave equation on an interval of length $T$ with Dirichlet boundary conditions, proving that it does not depend on the damping. This is achieved by analysing the square of the damped wave operator using the general result by Burghelea, Friedlander, and Kappeler on the determinant for a differential operator with matrix coefficients.
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Non-Weyl Microwave Graphs
One of the most important characteristics of a quantum graph is the average density of resonances, $ρ= \frac{\mathcal{L}}π$, where $\mathcal{L}$ denotes the length of the graph. This is a very robust measure. It does not depend on the number of vertices in a graph and holds also for most of the boundary conditions at the vertices. Graphs obeying this characteristic are called Weyl graphs. Using microwave networks which simulate quantum graphs we show that there exist graphs which do not adhere to this characteristic. Such graphs will be called non-Weyl graphs. For standard coupling conditions we demonstrate that the transition from a Weyl graph to a non-Weyl graph occurs if we introduce a balanced vertex. A vertex of a graph is called balanced if the numbers of infinite leads and internal edges meeting at a vertex are the same. Our experimental results confirm the theoretical predictions of [E. B. Davies and A. Pushnitski, Analysis and PDE 4, 729 (2011)] and are in excellent agreement with the numerical calculations yielding the resonances of the networks.
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Smilansky-Solomyak model with a $δ'$-interaction
We investigate a strongly singular version of the model of irreversible dynamics proposed by Smilansky and Solomyak in which the interaction responsible for an abrupt change of the spectrum is of $δ'$ type. We determine the spectrum in both the subcritical and supercritical regimes and discuss its character as well as its asymptotic properties of the discrete spectrum in terms of the coupling constant.
math-ph↗
Equivalence of resolvent and scattering resonances on quantum graphs
We discuss resonances for Schrödinger operators on metric graphs which consists of a finite compact part and a finite number of halflines attached to it; the vertex coupling is assumed to be of the $δ$-type or certain modifications of it. Using exterior complex scaling on the graph we show that the resolvent and scattering resonances coincide in this case.
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