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Open graph built out of a periodic linear chain with a unit cell made of two bonds with scattering matrices (68) at each vertex. The figure shows a chain with N = 2 unit cells.
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We study the classical limit of quantum mechanics on graphs by introducing a Wigner function for graphs. The classical dynamics is compared to the quantum dynamics obtained from the propagator. In particular, we consider extended open graphs whose classical dynamics generate a diffusion process. The transport properties of the classical system are...
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... One advantage of the scattering approach is that eigenvalue conditions can be written in terms of a secular equation involving the determinant of a unitary matrix of finite dimension N, where N typically equals twice the number of edges on the graph. Similarly, the scattering matrix of an open quantum graph can be given in terms of a closed form expression involving finite dimensional matrices of size N [16,17]. ...
... In 2001, Barra and Gaspard [17] used the scattering approach to express the Green's function of a quantum graph as a sum over trajectories in the spirit of semiclassical quantum mechanics. At the time, it was not yet clear within the physics community what scattering matrices are connected to matching conditions related to a well-defined self-adjoint Schrödinger operator on the metric graph. ...
... At the time, it was not yet clear within the physics community what scattering matrices are connected to matching conditions related to a well-defined self-adjoint Schrödinger operator on the metric graph. We generalize and simplify the approach [17] by using a simple three step procedure that leads to the Green's function for general self-adjoint matching conditions for closed and open graphs with a finite number of edges. This directly provides a number of closed form expressions that, to the best of our knowledge, have not been given before (though implied in [17], see also [18], where closed form expressions are given for a few simple examples). ...
In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard [Barra F and Gaspard P 2001, Phys. Rev. E 65, 016205] and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate.
The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived.
The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.
... [44-47] for a discussion and further references. Particular worth mentioning is the tail ρ(Γ) ∝ 1/Γ 3/2 [46,48,49] originally discovered numerically in a system with quasi-1D dynamics [48] and qualitatively explained as a characteristic feature reflecting classical diffusion in disordered samples of size L ξ taking place beyond strictly 1D chains. To the best of our knowledge, no controllable ab initio analytic derivation of such a behaviour was reported in the literature. ...
... [44][45][46][47] for a discussion and further references. Particular worth mentioning is the tail ρ(Γ) ∝ 1/Γ 3/2 [46,48,49] originally discovered numerically in a system with quasi-1D dynamics [48] and qualitatively explained as a characteristic feature reflecting classical diffusion in disordered samples of size L ξ taking place beyond strictly 1D chains. To the best of our knowledge, no controllable ab initio analytic derivation of such a behaviour was reported in the literature. ...
We develop a general non-perturbative characterisation of universal features of the density $\rho(\Gamma)$ of $S$-matrix poles (resonances) $E_n-i\Gamma_n$ describing waves incident and reflected from a disordered medium via a single $M$-channel waveguide/lead. Explicit expressions for $\rho(\Gamma)$ are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels ($M\sim 1$) the most salient features are tails $\rho(\Gamma)\sim 1/\Gamma$ for narrow resonances reflecting exponential localization and $\rho(\Gamma)\sim 1/\Gamma^2$ for broad resonances reflecting states located in the vicinity of the attached wire. For multimode wires with $M\gg 1$, intermediate asymptotics $\rho(\Gamma)\sim 1/\Gamma^{3/2}$ is shown to emerge reflecting diffusive nature of decay into wide enough contacts.
... To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2] (see also [3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs whenh (the reduced Planck constant) goes to zero. ...
... while, to prove the statement for W ND − we have to study the limit t → −∞ of (U N −t U D t + F * c F s )ϕ 2 L 2 (R+) = 2 π ∞ 0 dx ∞ 0 dk e ikx+ik 2 |t| (F s ϕ)(k) 2 . ...
... Additionally, for any ϕ ∈ C ∞ 0 (R + ), F c ϕ belongs to C ∞ (R + ), it decays at infinity faster than any polynomial in 1/k, and |(F c ϕ)(k)| ≤ 2 √ 2π Starting from the identity e −iηy−iη 2 = i y+2η d dη e −iηy−iη 2 , by integration by parts, we obtain F(y, t) = F 1 (y, t) + F 2 (y, t) , with F 1 (y, t) := ∞ 0 dη e −iηy−iη 2 2i (y + 2η) 2 (F s ϕ)(η/t 1/2 ) and F 2 (y, t) := 1 t 1/2 ∞ 0 dη e −iηy−iη 2 1 i(y + 2η) F c ((·)ϕ) (η/t 1/2 ) . ...
We consider the dynamics of a quantum particle of mass m on a n -edges star-graph with Hamiltonian $$H_K=-(2m)^{-1}\hbar ^2 \Delta $$ H K = - ( 2 m ) - 1 ħ 2 Δ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kreĭn’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $$(H_K,H_{D}^{\oplus })$$ ( H K , H D ⊕ ) , where $$H_{D}^{\oplus }$$ H D ⊕ is the Hamiltonian with Dirichlet conditions in the vertex.
... Coming back to the existence of a resonance gap, i.e., to the question of whether resonances lie in a strip of the form Im z ∈ [−K, −K ] for some K > 0, the analogy with semiclassical Schrödinger operators may lead us to look for criteria that are less related to arithmetic of the edge lengths, and more to some classical dynamics. In the physics literature, this was proposed in [4], where the authors introduce some classical dynamics on a quantum graph, introduce a topological pressure 1 , and claim that, if this pressure is negative, there should be a resonance gap. ...
In this paper, we consider a sequence of open quantum graphs, with uniformly bounded data, and we are interested in the asymptotic distribution of their scattering resonances. Supposing that the number of leads in our quantum graphs is small compared to the total number of edges, we show that most resonances are close to the real axis. More precisely, the asymptotic distribution of resonances of our open quantum graphs is the same as the asymptotic distribution of the square-root of the eigenvalues of the closed quantum graphs obtained by removing all the leads.
... To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2] (see also [3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs whenh (the reduced Planck constant) goes to zero. ...
... while, to prove the statement for W ND − we have to study the limit t → −∞ of (U N −t U D t + F * c F s )ϕ 2 L 2 (R+) = 2 π ∞ 0 dx ∞ 0 dk e ikx+ik 2 |t| (F s ϕ)(k) 2 . ...
... Additionally, for any ϕ ∈ C ∞ 0 (R + ), F c ϕ belongs to C ∞ (R + ), it decays at infinity faster than any polynomial in 1/k, and |(F c ϕ)(k)| ≤ 2 √ 2π Starting from the identity e −iηy−iη 2 = i y+2η d dη e −iηy−iη 2 , by integration by parts, we obtain F(y, t) = F 1 (y, t) + F 2 (y, t) , with F 1 (y, t) := ∞ 0 dη e −iηy−iη 2 2i (y + 2η) 2 (F s ϕ)(η/t 1/2 ) and F 2 (y, t) := 1 t 1/2 ∞ 0 dη e −iηy−iη 2 1 i(y + 2η) F c ((·)ϕ) (η/t 1/2 ) . ...
We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\u{\i}n's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{\oplus})$, where $H_{D}^{\oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex.
... To understand fundamental aspects of quantum mechanics, graphs are idealized exactly soluble models to address, e.g., band spectrum properties of lattices [30,31], the relation between periodic-orbit theory and Anderson localization [32], general scattering [33], chaotic and diffusive scattering [34][35][36], and quantum chaos [37]. In particular, quantum graphs relevance in grasping distinct features of quantum chaotic dynamics have been demonstrated in two pioneer papers [38,39]. ...
... We examine a schematic way to regroup the multi-scattering contributions (essentially a factorization method [134,[153][154][155]), leading to a final closed analytic expression for G. This particular procedure to construct the exact G is very useful to interpret many results concerning quantum graphs, like interference in transport processes [35,156,157]. With the help of illustrative examples, we elaborate on how to extract from G the graphs quantum properties. ...
... Our analysis generalizes the quantum version of the Kirchhoff's rules discussed in Ref. [134], because we construct the scattering matrix in a systematic and very general way, and also generalizes the results of Ref. [35], where was studied only open quantum graphs and was not showed how to classify and sum up all the classical trajectories. We illustrate our method by some concrete examples like binary tree, cube and Sierpiński-like quantum graphs, obtaining the transmission (as well the reflection) amplitudes as functions of wave number of incident plane waves, by using k-dependent boundary conditions. ...
Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green’s function (G) framework. This approach is particularly interesting because G can be written as a sum over classical-like paths, where local quantum effects are taken into account through the scattering matrix elements (basically, transmission and reflection amplitudes) defined on each one of the graph vertices. Hence, the exact G has the functional form of a generalized semiclassical formula, which through different calculation techniques (addressed in detail here) always can be cast into a closed analytic expression. It allows to solve exactly arbitrary large (although finite) graphs in a recursive and fast way. Using the Green’s function method, we survey many properties of open and closed quantum graphs as scattering solutions for the former and eigenspectrum and eigenstates for the latter, also considering quasi-bound states. Concrete examples, like cube, binary trees and Sierpiński-like topologies are presented. Along the work, possible distinct applications using the Green’s function methods for quantum graphs are outlined.
... The only attempts for treatment of Green function on metric graphs are those in Refs. [12][13][14]. In [12] the Green function is constructed for the regular Shturm-Liouville problem on the interval via different approaches. ...
... In [12] the Green function is constructed for the regular Shturm-Liouville problem on the interval via different approaches. The Green function for the Schrödinger operator on the open star graph is studied by the multiscattering expansion in [13]. Schmidt, Cheng and da Luz [14] presented a recursive prescription to calculate the exact Green function for general quantum graphs via a scattering approach. ...
We treat the problem of the Green function for quantum graphs by focusing on such topologies as star and tree graphs. The exact Green function for the Schrödinger equation on primary star graphs is derived in the form of 3 × 3− matrix using the vertex boundary conditions providing continuity and current conservation. Extension of the approach for the derivation for the Green function on tree graph is presented. Possible practical applications of the obtained results are discussed.
... Quantum walks represent quantum evolution in continuous spaces, discrete lattices and graphs by the motion of a quantum walker under the action of unitary operators, derived from the specific Hamiltonians [2,3]. Quantum walk theory has been developing rapidly and has become a powerful model for quantum system evolution with direct connections to Feynman propagators and quantum cellular automata [4][5][6][7][8][9]. Among others, quantum walks have been applied to solve decision problems in terms of quantum walks on decision trees, to model breakdown of electric field driven systems, to study nanotubules and to develop new quantum algorithms [10][11][12][13]. ...
Both discrete and continuous quantum walks on graphs are universal for quantum computation. We define and use discrete quantum walks on the graphene honeycomb lattice to investigate the possibility of using graphene armchair and zigzag nanoribbons to implement quantum gates. The probability distribution of the quantum walker location represents the particle (electron) density distribution on the graphene lattice. We use a universal set of quantum gates as coins that drive the quantum walk and show that different quantum gates result in distinguishable particle distributions on the graphene lattice.
... Notice that the summands in the second identity in (175) can be interpreted as contributions of free particle kernels at a fixed time t corresponding to the four types of paths p(x 0 , x) (see section 6) joining x 0 and x (see e.g. [52,58]). For this reason, we define (cf. ...
The Berry-Keating operator H_{BK}:= -i\hbar \big(x\frac{d}{dx}+\frac{1}{2}\big) (Berry and Keating 1999 SIAM Rev. 41 236) governing the Schrödinger dynamics is discussed in the Hilbert space L^2({\mathbb R}_\gt,dx) and on compact quantum graphs. It has been proved that the spectrum of HBK defined on L^2({\mathbb R}_\gt,dx) is purely continuous and thus this quantization of HBK cannot yield the hypothetical Hilbert-Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of HBK acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of HBK. In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the 'squared' Berry-Keating operator H_{BK}^2:= -x^2\frac{d^2}{dx^2}-2x\frac{d}{dx}-\frac{1}{4} which is a special case of the Black-Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for H2BK on compact quantum graphs. While the spectra of both HBK and H2BK on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither HBK nor H2BK can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail.
... On a Kirchhoff quantum graph the semiclassical approximation again reduces to the (exact) method of images, which is, nevertheless, quite intricate to execute. Solutions for the heat and resolvent equations have been known for some time [27] [19] [4] [18]. The corresponding construction for the cylinder kernel has been extensively investigated by Wilson et al. [32] [12] [13] [14] [6]. ...
... On a Kirchhoff quantum graph the semiclassical approximation again reduces to the (exact) method of images, which is, nevertheless, quite intricate to execute. Solutions for the heat and resolvent equations have been known for some time [27,19,4,18]. The corresponding construction for the cylinder kernel has been extensively investigated by Wilson et al. [32, 12, 13, 14, 6]. ...
Vacuum energy and other spectral functions of Laplace-type differential operators have been studied approximately by classical-path constructions and more fundamentally by boundary integral equations. As the first step in a program of elucidating the connections between these approaches and improving the resulting calculations, I show here how the known solutions for Kirchhoff quantum graphs emerge in a boundary-integral formulation. Comment: 18 pages
