Acta Polytechnica doi:10.14311/AP.2016.56.0224 Acta Polytechnica 56(3):224–235, 2016 © Czech Technical University in Prague, 2016 available online at http://ojs.cvut.cz/ojs/index.php/ap THE AHARONOV-BOHM HAMILTONIAN WITH TWO VORTICES REVISITED Petra Košťáková, Pavel Šťovíček∗ Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00 Praha, Czech Republic ∗ corresponding author: stovipav@kmlinux.fjfi.cvut.cz Abstract. We consider an invariant quantum Hamiltonian H = −∆LB + V in the L2 space based on a Riemannian manifold M̃ with a discrete symmetry group Γ. To any unitary representation Λ of Γ one can relate another operator on M = M̃/Γ, called HΛ, which formally corresponds to the same differential operator as H but which is determined by quasi-periodic boundary conditions. As originally observed by Schulman in theoretical physics and Sunada in mathematics, one can construct the propagator associated with HΛ provided one knows the propagator associated with H. This approach is reviewed and demonstrated on a quantum model describing a charged particle on the plane with two Aharonov-Bohm vortices. The construction of the propagator is explained in full detail including all substantial intermediate steps. Keywords: Aharonov-Bohm effect; propagator; covering space; Bloch decomposition. 1. Introduction Suppose there is given a Riemannian manifold M̃ with a discrete symmetry group Γ and a Γ-periodic Hamilton operator H on L2(M̃). To any unitary representation Λ of Γ one can relate another operator on M = M̃/Γ, called HΛ, which is determined by quasi-periodic boundary conditions. A formula relating the propagators KΛt (x,x0) and Kt(x,x0) associated with HΛ and H, respectively, has been derived in the framework of the Feynman path integral [18, 19]. An analogous formula is also known for heat kernels [4]. An opposite point of view is taken when one decomposes the operator H into a direct integral with components HΛ where Λ runs over all irreducible unitary representations of Γ [3, 6, 23]. The evolution operator then decomposes correspondingly. This type of decomposition is a substantial step in the Bloch analysis. The both relations, the propagator formula on one hand and the generalized Bloch decomposition on the other hand, are in a sense mutually inverse [11, 12]. In the current paper we wish to demonstrate how this relationship can be effectively used on a concrete example of interest. We consider the formula for propagators in the case of the Aharonov-Bohm effect with two vortices. In this quantum model, M̃ is identified with the universal covering space of the plane with two excluded points and Γ is the fundamental group of the same manifold. This problem has already been treated by one of the authors quite a long time ago in [21]. But the topic is in no way exhausted completely, and this quantum model was intensively discussed in a number of papers, in some cases even very recently. These discussions rely on completely different approaches, however, like asymptotic methods for largely separated vortices, semiclassical analysis and a complex scaling method [1, 2, 9, 25]. One may also mention more complex models comprising, apart of magnetic vortices, also additional potentials or magnetic fields [14, 16], or models with an arbitrary finite number of magnetic vortices or even with countably many vortices arranged in a lattice [15, 17, 22]. On the other hand, the method stemming from the original ideas of Schulman and Sunada turned out to be fruitful also in analysis of other interesting models like Brownian random walk on the twice punctured plane [5, 7]. Here we return to the article [21] which is in its character a brief letter presenting the final formulas without a detailed derivation. But the technique applied therein is of independent interest and can prove useful in other situations as well, as already mentioned above. This is why we focus, in the present paper, primarily on the method itself and aim to explain the approach on a concrete example while indicating all necessary intermediate steps in full detail. Hopefully, the provided analysis may open the way to new applications of the method. The paper is organized as follows. The main ideas and results of the general approach are outlined in Section 2 following papers [11, 12]. Section 3 is the key section of the present paper. In Subsection 3.1, a formula for the propagator on the universal covering space of the twice punctured plane, as originally presented in [21], is briefly recalled. Subsection 3.2 has a preliminary character and provides a summary of some auxiliary useful identities. Subsection 3.3 is fully dedicated to the proof of the propagator formula, as given in (9), (10). More precisely, the goal of this subsection is a verification of equation (17). As a corollary, in Section 4, more details 224 http://dx.doi.org/10.14311/AP.2016.56.0224 http://ojs.cvut.cz/ojs/index.php/ap vol. 56 no. 3/2016 The Aharonov-Bohm Hamiltonian with Two Vortices Revisited are provided, if compared to [21], about a formal application of the Schulman-Sunada formula, as recalled in (5), to the studied example while making use of the knowledge of the propagator on the covering space. 2. A summary of the general approach 2.1. Periodic Hamiltonians Let M̃ be a connected Riemannian manifold with a discrete and at most countable symmetry group Γ. The action of Γ on M̃ is assumed to be smooth, free and proper. Let us recall that under these assumptions any element s ∈ Γ different from the unity has no fixed points on M̃, and for any compact set K ⊂ M̃, the intersection K ∩s ·K is nonempty for at most finitely many elements s ∈ Γ. This also implies that any point y ∈ M̃ has a neighborhood U such that the sets s ·U, s ∈ Γ, are mutually disjoint [13, Corollary 12.10]. Denote by µ̃ the measure on M̃ induced by the Riemannian metric. The quotient M = M̃/Γ is a connected Riemannian manifold with an induced measure µ. This way one gets a principal fiber bundle π : M̃ → M with the structure group Γ. All L2 spaces based on the manifolds M and M̃ are everywhere tacitly understood with the measures µ and µ̃, respectively. In a number of important examples, M̃ is the universal covering space of M and Γ = π1(M) is the fundamental group of M. In particular, this is the case when one is considering the Aharonov-Bohm effect. To a unitary representation Λ of Γ in a separable Hilbert space LΛ one relates the Hilbert space HΛ formed by Λ-equivariant vector-valued functions on M̃. This means that any function ψ ∈ HΛ is measurable with values in LΛ and satisfies ∀s ∈ Γ, ψ(s ·y) = Λ(s)ψ(y) almost everywhere on M̃. Moreover, the norm of ψ induced by the following scalar product is required to be finite. If ψ1,ψ2 ∈ HΛ then the function y 7→ 〈ψ1(y),ψ2(y)〉LΛ defined on M̃ is Γ-invariant and so it projects to a function sψ1,ψ2 defined on M, and the scalar product is defined by 〈ψ1,ψ2〉 = ∫ M sψ1,ψ2 (x) dµ(x). Our discussion focuses on Γ-periodic Hamiltonians on M̃ of the form H = −∆LB + V where ∆LB is the Laplace-Beltrami operator and V is a Γ-invariant semibounded and locally integrable real function on M̃. Clearly, the differential operator −∆LB + V is semibounded on the domain formed by test functions (i.e. smooth and compactly supported functions), and H is defined as its Friedrichs extension. The same choice will also be made in other instances below in the paper. This is to say that in the presented approach we distinguish the Friedrichs extension as the preferred self-adjoint extension of a given semibounded symmetric operator. Here we are referring to the widely used result ensuring the existence of an unambiguously defined and in some sense minimal self-adjoint extension of a semibounded symmetric operator, the so called Friedrichs extension [10, § VI.2]. This choice is encountered very frequently in various applications and it also makes it possible to avoid the discussion of the domain of the self-adjoint operator in question which sometimes may be quite tedious. To the same differential operator, −∆LB + V , one can relate a self-adjoint operator HΛ in the space HΛ for any unitary representation Λ of Γ in LΛ. Let us define ΦΛ : C∞0 (M̃) ⊗ LΛ → HΛ by ∀ϕ ∈ C∞0 (M̃),∀v ∈ LΛ, (ΦΛϕ⊗v)(y) = ∑ s∈Γ ϕ(s ·y)Λ(s−1)v. (1) Since the action of Γ is proper, the vector-valued function ΦΛϕ⊗v is smooth. Moreover, ΦΛϕ⊗v is Λ-equivariant, the norm of ΦΛϕ⊗v in HΛ is finite, and the range of ΦΛ is dense in HΛ. The Laplace-Beltrami operator is well defined on Ran(ΦΛ) and it holds ∆LBΦΛ[ϕ⊗v] = ΦΛ[∆LBϕ⊗v]. One can also verify that the differential operator −∆LB is positive on the domain Ran(ΦΛ) ⊂ HΛ. Since the function V (y) is Γ-invariant, the multiplication operator by V is well defined in the Hilbert space HΛ. The Hamiltonian HΛ is defined as the Friedrichs extension of the differential operator −∆LB + V considered on the domain Ran ΦΛ. The reader is referred to [11, 12] for more details. 225 Petra Košťáková, Pavel Šťovíček Acta Polytechnica 2.2. A generalization of the Bloch decomposition Let Γ̂ be the dual space to Γ (the quotient space of the space of irreducible unitary representations of Γ). In the first step of the generalized Bloch analysis one decomposes H into a direct integral over Γ̂ with components being equal to HΛ. To achieve this goal a well defined harmonic analysis on Γ is necessary. It is known that the harmonic analysis is well established for locally compact groups of type I [20]. So all formulas presented below are well defined provided Γ is a type I group. As shown in [26, Satz 6], a countable discrete group is of type I if and only if it has an Abelian normal subgroup of finite index. This means that there exist multiply connected configuration spaces of interest whose fundamental groups are not of type I. For example, the fundamental group in the case of the Aharonov-Bohm effect with two vortices is the free group with two generators, and it is not of type I. This problem is avoided, however, if M̃ is the maximal Abelian covering of M rather than the universal covering [12, 24]. Let us recall basic properties of the harmonic analysis on discrete type I groups [20]. The Haar measure on Γ is simply the counting measure. Let dm̂ be the Plancherel measure on Γ̂. It is known that if Γ is a countable discrete group of type I then dim LΛ, the dimension of the carrier representation space, is a bounded function of Λ ∈ Γ̂ [26, Korollar I]. Denote by I2(LΛ) ≡ LΛ ⊗ L ∗Λ the Hilbert space formed by Hilbert-Schmidt operators on LΛ (L ∗Λ is the dual space to LΛ). The Fourier transform is defined as a unitary mapping F : L2(Γ) → ∫ ⊕ Γ̂ I2(LΛ) dm̂(Λ). Note that in this situation f ∈ L1(Γ) just means that the values of f on Γ represent a summable sequence. Since every summable sequence is also square summable we have f ∈ L1(Γ) ⊂ L2(Γ), and then F [f](Λ) = ∑ s∈Γ f(s)Λ(s). Conversely, if f is of the form f = g ∗h (the convolution) where g,h ∈ L1(Γ), and f̂ = F [f] then f(s) = ∫ Γ̂ Tr [ Λ(s)∗f̂(Λ) ] dm̂(Λ). Using unitarity of the Fourier transform one finds that m̂(Γ̂) ≤ 1. The following rule is of crucial importance: ∀s ∈ Γ, ∀f ∈ L2(Γ), F [f(s ·g)](Λ) = Λ(s−1)F [f(g)](Λ). Now we are going to construct a unitary map Φ : L2(M̃) → ∫ ⊕ Γ̂ HΛ ⊗ L ∗Λ dm̂(Λ) making it possible to decompose H. Observe that the tensor product HΛ ⊗ L ∗Λ can be naturally identified with the Hilbert space of Λ-equivariant operator-valued functions on M̃ with values in I2(LΛ). For f ∈ L2(M̃) and y ∈ M̃ set ∀s ∈ Γ, fy(s) = f(s−1 ·y). The norm ‖fy‖ in L2(Γ) is a Γ-invariant function of y ∈ M̃ whose projection onto M is square integrable. Hence for almost all x ∈ M and all y ∈ π−1({x}) one has fy ∈ L2(Γ). We define components Φ[f](Λ), Λ ∈ Γ̂, by( Φ[f](Λ) ) (y) = F [fy](Λ) ∈ I2(LΛ). In particular, if f ∈ L1(M̃) ∩L2(M̃) then( Φ[f](Λ) ) (y) = ∑ s∈Γ f(s−1 ·y)Λ(s). Equivalently, referring to (1), one can define Φ in the following way. For ϕ ∈ C∞0 (M̃), v ∈ LΛ and y ∈ M̃ set( Φ[ϕ](Λ) ) (y)v = (ΦΛϕ⊗v)(y). (2) Then Φ introduced in (2) is an isometry and extends unambiguously to a unitary mapping. Finally one can verify the formula ΦHΦ−1 = ∫ ⊕ Γ̂ HΛ ⊗ 1 dm̂(Λ) which represents the sought Bloch decomposition. As a corollary we have ΦU(t)Φ−1 = ∫ ⊕ Γ̂ UΛ(t) ⊗ 1 dm̂(Λ). (3) 226 vol. 56 no. 3/2016 The Aharonov-Bohm Hamiltonian with Two Vortices Revisited 2.3. Propagators associated with periodic Hamiltonians In (3), the evolution operator U(t) is expressed in terms of UΛ(t), Λ ∈ Γ̂. It is possible to invert this relationship and to derive a formula for the propagator associated with HΛ which is expressed in terms of the propagator associated with H. The propagators are regarded as distributions which are introduced as kernels of the corresponding evolution operators. Recall that, by the Schwartz kernel theorem (see, for example, [8, Theorem 5.2.1]), to every B ∈ B(L2(M̃)) there exists one and only one β ∈ D′(M̃ ×M̃) such that ∀ϕ1,ϕ2 ∈ C∞0 (M̃), β(ϕ1 ⊗ϕ2) = 〈ϕ1,Bϕ2〉. One calls β the kernel of B. The kernel theorem can be extended to Hilbert spaces formed by Λ-equivariant vector-valued functions. In this case the kernels are operator-valued distributions. To every B ∈ B(HΛ) there exists one and only one β ∈ D′(M̃ ×M̃) ⊗ B(LΛ) such that ∀ϕ1,ϕ2 ∈ C∞0 (M̃), ∀v1,v2 ∈ LΛ,〈 v1,β(ϕ1 ⊗ϕ2)v2 〉 LΛ = 〈 ΦΛϕ1 ⊗v1,BΦΛϕ2 ⊗v2 〉 . The distribution β is Λ-equivariant: ∀s ∈ Γ, β(s ·y1,y2) = Λ(s)β(y1,y2) and β(y1,s ·y2) = β(y1,y2)Λ(s−1) (4) Denote by Kt ∈ D′(M̃ ×M̃) the kernel of U(t) ∈ B(L2(M̃)), and by KΛt ∈ D′(M̃ ×M̃) ⊗B(LΛ) the kernel of UΛ(t) ∈ B(HΛ). Here and everywhere in this section, t is a real parameter. The kernel KΛt is Λ-equivariant in the sense of (4). First, we can rewrite the Bloch decomposition (3) in terms of kernels. For all ϕ1,ϕ2 ∈ C∞0 (M̃), Kt(ϕ1 ⊗ϕ2) = ∫ Γ̂ Tr [ KΛt (ϕ1 ⊗ϕ2) ] dm̂(Λ), with the integral being convergent. An inverse relation was derived by Schulman in the framework of path integration [18, 19] and reads KΛt (x,y) = ∑ s∈Γ Λ(s)Kt(s−1 ·x,y). (5) It is possible to give (5) the following rigorous interpretation [11, 12]. Suppose that ϕ1,ϕ2 ∈ C∞0 (M̃) are fixed but otherwise arbitrary. Set Ft(s) = Kt ( ϕ1(s−1 ·y1) ⊗ϕ2(y2) ) for s ∈ Γ, Gt(Λ) = KΛt (ϕ1 ⊗ϕ2) ∈ I2(LΛ) for Λ ∈ Γ̂. One can show that Ft ∈ L2(Γ) and Gt is bounded on Γ̂ in the Hilbert-Schmidt norm. Recalling that m̂(Γ̂) ≤ 1 we have ‖Gt(·)‖∈ L1(Γ̂) ∩L2(Γ̂). Then Ft = F−1[Gt] and, consequently, Gt = F [Ft]. (6) Rewriting (6) formally yields (5). 3. The propagator on the universal covering space 3.1. A formula for the propagator The configuration space for the Aharonov-Bohm effect with two vortices is the plane with two excluded points, M = R2 \{a,b}. This is a flat Riemannian manifold and the same is true for the universal covering space M̃. Let π : M̃ → M be the projection. It is convenient to complete the manifold M̃ by a countable set of points A∪B lying on the border of M̃ and projecting onto the excluded points, π(A) = {a} and π(B) = {b}. M̃ looks locally like R2 but differs from the Euclidean space by some global features. First of all, not every two points from M̃ can be connected by a geodesic segment. Fix a point x ∈ M̃. The symbol D(x), as introduced below in (20), stands for the set of points y ∈ M̃ which can be connected with x by a segment. Then D(x) is a sheet of the covering M̃ → M. It can be identified with R2 cut along two half-lines with the limit points a and b, respectively. The border ∂D(x) is formed by four half-lines. The universal covering space M̃ can be imagined as a result of an infinite process of gluing together countably many copies of D(x) with each copy having four neighbors. 227 Petra Košťáková, Pavel Šťovíček Acta Polytechnica The fundamental group of M, called Γ, is known to be the free group with two generators ga and gb. For the generator ga one can choose the homotopy class of a simple positively oriented loop winding once around the point a and leaving the point b in the exterior. Analogously one can choose gb by interchanging the role of a and b. One-dimensional unitary representations Λ of Γ are determined by two numbers α, β, 0 ≤ α,β < 1, so that Λ(ga) = e2πiα, Λ(gb) = e2πiβ. The standard way how to define the Aharonov-Bohm Hamiltonian HAB with two vortices is to choose a vector potential −→ A for which curl −→ A = 0 on M and such that the nonintegrable phase factor [27] for a closed path from the homotopy class ga or gb equals e2πiα or e2πiβ, respectively (assuming that 0 < α,β < 1). HAB then acts as the differential operator (−i∇− −→ A)2 in L2(M). Here again, to be more rigorous, HAB is the Friedrichs extension of the positive operator (−i∇− −→ A)2 defined on test functions on M. For our purposes it would be more convenient to pass to a unitarily equivalent formulation. This is done in two steps. First, the differential operator (−i∇− −→ A)2 is lifted to M̃. Then the unitarily equivalent operator is H̃AB acting as a differential operator (−i∇− −→̃ A)2 in the Hilbert space of Γ-periodic functions on M̃ which are square integrable over a fundamental domain of the Γ action. Once more, H̃AB is rigorously introduced with the aid of the Friedrichs extension. Second, curl −→̃ A = 0 holds again on M̃. But this time M̃ is simply connected and therefore the vector potential can be removed by a globally well defined gauge transformation. This gauge transformation induces a unitary mapping between the Hilbert space of Γ-periodic functions on M̃ and the Hilbert space HΛ of Λ-equivariant functions on M̃, as introduced in Subsection 2.1. The resulting operator which is unitarily equivalent to HAB is nothing but the Hamiltonian HΛ = −∆ acting in HΛ, as it has been introduced in the same subsection. Remember that simultaneously one considers the free Hamiltonian H = −∆ in L2(M̃). H is Γ-periodic. In order to apply (5) and compute the propagator KΛ(t,x,y) associated with HΛ one has to rely on a known formula for the free propagator K(t,x,y) on M̃. Let us recall a formula for K(t,x,y), as presented in [21]. Let ϑ stand for the Heaviside step function. For x,y ∈ M̃ ∪A∪B set χ(x,y) = 1 if the points x, y can be connected by a geodesic segment, and χ(x,y) = 0 otherwise. Given t ∈ R we define Z(t,x,y) = ϑ(t)χ(x,y) 1 4πit exp ( i 4t dist2(x,y) ) , (7) furthermore, for x1,x3 ∈ M̃ ∪A∪B and x2 ∈A∪B obeying χ(x1,x2) = χ(x2,x3) = 1, and for t1, t2 > 0, we set V ( x3,x2,x1 t2, t1 ) = 2i (( θ −π + i ln t2r1 t1r2 )−1 − ( θ + π + i ln t2r1 t1r2 )−1) (8) where θ = ∠x1,x2,x3 ∈ R is the oriented angle and r1 = dist(x1,x2), r2 = dist(x2,x3). Note that θ can take any real value. We claim that the free propagator on M̃ equals K(t,x,x0) = ∑ γ∈C (x,x0) Kγ(t,x,x0), (9) where C (x,x0) stands for the set of all piecewise geodesic curves γ : x0 → C1 → ··· → Cn → x with the inner vortices Cj, 1 ≤ j ≤ n, belonging to the set of extreme points A∪B. This means that it should hold χ(x0,C1) = χ(C1,C2) = · · · = χ(Cn,x) = 1. Let us denote by |γ| = n the length of the sequence (C1,C2, . . . ,Cn). In particular, if |γ| = 0 then γ designates the geodesic segment x0 → x. To simplify notation we set everywhere where convenient C0 = x0 and Cn+1 = x. With this convention, the terms in (9) equal Kγ(t,x,x0) = ∫ Rn+1 dtn · · ·dt0 δ(tn + · · · + t0 − t) n−1∏ j=0 V ( Cj+2,Cj+1,Cj tj+1, tj ) n∏ j=0 Z(tj,Cj+1,Cj). (10) In particular, if |γ| = 0 then Kγ(t,x,x0) = Z(t,x,x0), and if |γ| = 1 then γ designates a path composed of two geodesic segments x0 → C → x, with C ∈A∪B, and Kγ(t,x,x0) = ϑ(t) ∫ t 0 V ( x,C,x0 t−s,s ) Z(t−s,x,C)Z(s,C,x0) ds. In what follows we aim to provide a detailed verification of formulas (9), (10). 228 vol. 56 no. 3/2016 The Aharonov-Bohm Hamiltonian with Two Vortices Revisited 3.2. Auxiliary relations In R2, it holds true that ( ∂ ∂x + i ∂ ∂y ) 1 x + iy = 2πδ(x)δ(y) and ∆ 1 x + iy = 2π ( δ(y)δ′(x) − iδ(x)δ′(y) ) . It follows that ( ∂2 ∂r2 + 1 r ∂ ∂r + 1 r2 ∂2 ∂θ2 )( θ + i ln t r )−1 = 2πt r2 ( δ(t−r)δ′(θ) − irδ′(t−r)δ(θ) ) (11) holds on the domain t > 0, r > 0, θ ∈ R. On the same domain, ( r ∂ ∂r + t ∂ ∂t )( θ + i ln t r )−1 = 0. (12) Combining (11) and (12) one finds that ( i ∂ ∂t + ∂2 ∂r2 + 1 r ∂ ∂r + 1 r2 ∂2 ∂θ2 )( θ + i ln t r )−1 1 t exp ( i r2 4t ) = 2π r2 exp ( i r2 4t )( δ(t−r)δ′(θ) − irδ′(t−r)δ(θ) ) . (13) Equipped with (13) one can prove the identity ( i ∂ ∂t + ∂2 ∂r2 + 1 r ∂ ∂r + 1 r2 ∂2 ∂θ2 )∫ t 0 ( θ + i ln (t−s)r0 sr )−1 1 t−s exp ( i r2 4(t−s) ) f(s) ds = 2πr0 r2(r + r0) exp ( i (r + r0)r 4t )[ f ( tr0 r + r0 ) δ′(θ) − i r r + r0 (( 1 + i r0(r + r0) 4t ) f ( tr0 r + r0 ) + tr0 r + r0 f′ ( tr0 r + r0 )) δ(θ) ] , (14) which is true in the sense of distributions for any r0 > 0 and f ∈ C1([0, +∞[), again on the domain t > 0, r > 0, θ ∈ R. Note that 1 ε exp ( i r2 4ε ) → 0 as ε → 0+ in D′(]0, +∞[). In particular, letting f(s) = (1/s) exp(ir 20 /(4s)) one derives the identity ( i ∂ ∂t + ∂2 ∂r2 + 1 r ∂ ∂r + 1 r2 ∂2 ∂θ2 )∫ t 0 ( θ + i ln (t−s)r0 sr )−1 1 (t−s)s exp ( i ( r2 4(t−s) + r 20 4s )) ds = 2π tr2 exp ( i (r + r0)2 4t ) δ′(θ). (15) Let us further recall a basic fact concerning the generalized Laplacian. If G ⊂ M̃ is an open set with a piecewise smooth boundary, χG is the characteristic function of G, −→n is the normalized outer normal vector field on ∂G and η is a smooth function on M̃ then, in the sense of distributions, ∆(ηχG) = (∆η)χG − ∂η ∂−→n δ∂G − ∂ ∂−→n (ηδ∂G). (16) The distribution δ∂G is a single layer supported on the curve ∂G and fulfilling ∀ϕ ∈ C∞0 (M̃), δ∂G(ϕ) = ∫ ∂G ϕd`. The double layer ∂/∂−→n (ηδ∂G) is defined by ∀ϕ ∈ C∞0 (M̃), ( ∂ ∂−→n (ηδ∂G) ) (ϕ) = − ∫ ∂G ∂ϕ ∂−→n d`. 229 Petra Košťáková, Pavel Šťovíček Acta Polytechnica 3.3. Verification of the propagator formula We have to show that, for x0 ∈ M̃ fixed, the propagator K(t,x,x0) defined in (9), (10) verifies the condition( i ∂ ∂t + ∆ ) K(t,x,x0) = iδ(t)δ(x,x0) on R×M̃. (17) This is equivalent to showing that lim t→0+ K(t,x,x0) = δ(x,x0) (18) and ( i ∂ ∂t + ∆ ) K(t,x,x0) = 0 for t > 0, x ∈ M̃. (19) Equation (18) is obvious. In fact, since the form of Z(t,x,x0) on the sheet {x; χ(x,x0) = 1} is that of the free propagator on R2 we have lim t→0+ Z(t,x,x0) = δ(x,x0). By a similar reasoning, limt→0+ Z(t,x,C) = 0 if C ∈A∪B and x runs over M̃. Hence lim t→0+ Kγ(t,x,x0) = 0 if |γ| ≥ 1. Concerning (19), we first introduce some notation related to the geometry of the universal covering space. Denote by % the distance dist(a,b). Observe that if C1,C2 ∈ A ∪ B then χ(C1,C2) = 1 if and only if dist(C1,C2) = %. If this is the case then necessarily C1 ∈A and C2 ∈B or vice versa. For x ∈ M̃ ∪A∪B set D(x) = {y ∈ M̃; χ(x,y) = 1}. (20) If x ∈ M̃ then D(x) can be identified with the plane cut along two half-lines with the limit points a and b, respectively. The border of D(x) consists of two pairs of half-lines. One pair has a common limit point A ∈A and is denoted ∂D(x; A), the other pair has a common limit point B ∈B and is denoted ∂D(x; B). We have ∂D(x) = ∂D(x; A) ∪∂D(x; B). (21) If C ∈A∪B then D(C) resembles the universal covering space in the one-vortex case. It can be viewed as a union of countably many sheets glued together in a staircase-like way. Each sheet contributes to the border of D(C) by a pair of half-lines with a common limit point C′. Thus the border ∂D(C) is formed by a countable union of pairs of half-lines: ∂D(C) = ⋃ C′∈D, dist(C,C′)=% ∂D(C; C′), (22) where D = B if C ∈A and D = A if C ∈B. Let us first examine the case |γ| = 0. One has ( i ∂ ∂t + ∆ ) Z(t,x,x0) = 0 for t > 0 and x ∈ D(x0). Observe also that ∂ ∂−→n Z(t,x,x0) = 0 for x ∈ ∂D(x0). This is so since, in polar coordinates centered at x0, Z(t,x,x0) does not depend on the angle variable. Let us also note that Z(t,x,x0) can be continued smoothly in the variable x over the borderline of the domain D(x0). Thus, in virtue of (16), for t > 0 and x ∈ M̃, ( i ∂ ∂t + ∆ ) Z(t,x,x0) = − ∂ ∂−→n ( Z(t,x,x0)δ∂D(x0) ) . (23) Remark. In (23) as well as everywhere in this section we use the following convention. The value of a density (which is in this case Z(t,x,x0)) on the border ∂D(x0) is understood as the limit value taken from the interior of the domain D(x0). 230 vol. 56 no. 3/2016 The Aharonov-Bohm Hamiltonian with Two Vortices Revisited Next we discuss the case |γ| = 1. Then γ designates a piecewise geodesic curve x0 → C → x, with C ∈A∪B. Denote by γ′ the geodesic segment x0 → x provided x ∈ D(x0). We have Kγ(t,x,x0) = 1 8π2i χ(x,C)χ(C,x0) × ∫ t 0 (( θ −π + i ln (t−s)r0 sr )−1 − ( θ + π + i ln (t−s)r0 sr )−1) 1 (t−s)s exp ( i ( r2 4(t−s) + r20 4s )) ds (24) where r = dist(x,C), r0 = dist(C,x0) and θ = ∠x0,C,x. Application of the differential operator (i∂t + ∆) to the RHS of (24) in the sense of distributions results in several singular terms supported on one-dimensional submanifolds. First, due to the discontinuity of the characteristic function χ(x,C), the application of ∆ leads to two terms supported on the boundary ∂D(C) (see (16)). Second, as it follows from (15), the singularity of the integrand for the values θ = ±π and r0/s = r/(t−s) produces terms supported on the submanifold determined by θ = ±π, and this set is nothing but a part of the boundary of the domain D(x0), namely ∂D(x0; C). Notice that for θ = ±π it holds r + r0 = dist(x,x0) and ∂/∂−→n = ±r−1∂/∂θ. Moreover, in polar coordinates centered at C, δ∂D(x0;C) = 1 r ( δ(θ −π) + δ(θ + π) ) . Thus the latter contribution takes the form 1 4πitr2 ∂ ∂θ ( exp ( i 4t dist(x,x0)2 )( δ(θ −π) − δ(θ + π) )) = ∂ ∂−→n ( Kγ′(t,x,x0)δ∂D(x0,C) ) , where Kγ′(t,x,x0) = Z(t,x,x0). In summary, we obtain ( i ∂ ∂t + ∆ ) Kγ(t,x,x0) = − ( ∂ ∂−→n Kγ(t,x,x0) ) δ∂D(C) − ∂ ∂−→n ( Kγ(t,x,x0)δ∂D(C) ) + ∂ ∂−→n ( Kγ′(t,x,x0)δ∂D(x0,C) ) . (25) Finally, let us consider the case |γ| ≥ 2. Thus γ is a piecewise geodesic curve x0 → C1 → ··· → Cn → x, n ≥ 2. Denote by γ′ the truncated geodesic curve x0 → C1 → ··· → Cn−1 → x provided x ∈ D(Cn−1). Recalling (7), (8), one can express Kγ(t,x,x0) = ∫ Rn dtn−1 . . . dt0V ( x,Cn,Cn−1 t− τ,tn−1 ) Z(t− τ,x,Cn)Fγ(t0, . . . , tn−1,x0) = 1 2π χ(x,Cn) ∫ Rn−1 dtn−2 . . . dt0 ∫ t−τ′ 0 dtn−1 (( θ −π + i ln (t− τ′ − tn−1)% tn−1r )−1 − ( θ + π + i ln (t− τ′ − tn−1)% tn−1r )−1) 1 t− τ′ − tn−1 exp ( i r2 4(t− τ′ − tn−1) ) Fγ(t0, . . . , tn−1,x0), (26) where τ = t0 + · · · + tn−2 + tn−1, τ′ = t0 + · · · + tn−2, r = dist(Cn,x), θ = ∠Cn−1,Cn,x, and Fγ(t0, . . . , tn−1,x0) = n−2∏ j=0 V ( Cj+2,Cj+1,Cj tj+1, tj )n−1∏ j=0 Ztj (Cj+1,Cj). Application of (i∂t + ∆) to the RHS of (26) in the sense of distributions again produces several singular terms. As a consequence of the discontinuity of the characteristic function χ(x,Cn) a single and a double layer supported on the boundary ∂D(Cn) occur (see (16)). The singularity of the integrand for the values θ = ±π and %/tn−1 = r/(t− τ′ − tn−1) produces terms supported on the part of the boundary of the domain D(Cn−1), namely on ∂D(Cn−1; Cn). This time one can apply identity (14). In order to treat the resulting terms the following equations are useful. Suppose that θ = ±π and so x ∈ ∂D(Cn−1; Cn). Set r′ = r + % = dist(Cn−1,x), θ′ = ∠Cn−2,Cn−1,x. If %/tn−1 = r/(t− τ′ − tn−1) then tn−1 = %(t− τ′) r′ and t− τ′ − tn−1 r = t− τ′ r′ . 231 Petra Košťáková, Pavel Šťovíček Acta Polytechnica Moreover, % r′ exp ( ir2 4(t− τ) ) Z(tn−1,Cn,Cn−1) = Z(t− τ′,x,Cn−1) and V ( Cn,Cn−1,Cn−2 %s2/r ′,s1 ) = V ( x,Cn−1,Cn−2 s2,s1 ) . Observe also that ∂ ∂s ( exp ( ir2 4(t− τ′ −s) ) exp (i%2 4s ))∣∣∣∣ s=%(t−τ′)/r′ = 0, exp ( ir2 4(t− τ′ −s) ) exp (i%2 4s )∣∣∣∣ s=%(t−τ′)/r′ = exp ( ir′ 2 4(t− τ′) ) , and for θ = π, ∂ ∂s V ( Cn,Cn−1,Cn−2 s,tn−2 )∣∣∣∣ s=%(t−τ′)/r′ = ir′ %(t− τ′) ∂ ∂θ′ V ( x,Cn−1,Cn−2 t− τ′, tn−2 ) . A similar relation holds for θ = −π. After a bit tedious but quite straightforward manipulations one arrives at the final identity ( i ∂ ∂t + ∆ ) Kγ(t,x,x0) = − ( ∂ ∂−→n Kγ(t,x,x0) ) δ∂D(Cn) − ∂ ∂−→n ( Kγ(t,x,x0)δ∂D(Cn) ) + ( ∂ ∂−→n Kγ′(t,x,x0) ) δ∂D(Cn−1;Cn) + ∂ ∂−→n ( Kγ′(t,x,x0)δ∂D(Cn−1;Cn) ) . (27) Now we can show (19) when taking into account (23), (25) and (27). It is true that ( i ∂ ∂t + ∆ ) K(t,x,x0) = ∑ |γ|≥2 [ − ( ∂ ∂−→n Kγ(t,x,x0) ) δ∂D(Cn) − ∂ ∂−→n ( Kγ(t,x,x0)δ∂D(Cn) ) + ( ∂ ∂−→n Kγ′(t,x,x0) ) δ∂D(Cn−1;Cn) + ∂ ∂−→n ( Kγ′(t,x,x0)δ∂D(Cn−1;Cn) )] + ∑ |γ|=1 [ − ( ∂ ∂−→n Kγ(t,x,x0) ) δ∂D(C) − ∂ ∂−→n ( Kγ(t,x,x0)δ∂D(C) ) + ∂ ∂−→n ( Z(t,x,x0)δ∂D(x0;C) )] − ∂ ∂−→n ( Z(t,x,x0)δ∂D(x0) ) = 0, where we have used (21) and (22). 4. Conclusion. The propagator for two Aharonov-Bohm vortices In conclusion we present a formula for the propagator of a charged particle on the plane pierced by two Aharonov-Bohm magnetic fluxes. Without loss of generality we can suppose that the vortices are located at the points a = (0, 0) and b = (%, 0). In order to express the propagator for the Aharonov-Bohm Hamiltonian HAB we again pass to a unitarily equivalent formulation. Let us cut the plane along two half-lines, La = ]−∞, 0[ ×{0} and Lb = ]%, +∞[ ×{0}. Let (ra,θa) be polar coordinates centered at the point a and (rb,θb) be polar coordinates centered at the point b. The angle variables are chosen so that the values θa = ±π correspond to the two sides of the cut La, and similarly for θb and Lb. Then an explicit and commonly used choice of the Aharonov-Bohm vector potential reads −→ A = α∇θa + β∇θb. Denote by U a unitary operator in L2(R2,d2x) acting as the multiplication operator Uψ = ei(αθa+βθb)ψ, 232 vol. 56 no. 3/2016 The Aharonov-Bohm Hamiltonian with Two Vortices Revisited and let H′Λ = U −1HABU. Then H′Λ acts as −∆ in L 2(R2,d2x), and its domain is determined by the boundary conditions along the cut La ∪Lb: ψ(ra,θa = π) = e2πiαψ(ra,θa = −π), ∂θaψ(ra,θa = π) = e 2πiα∂θaψ(ra,θa = −π), ψ(rb,θb = π) = e2πiβψ(rb,θb = −π), ∂θbψ(rb,θb = π) = e 2πiβ∂θbψ(rb,θb = −π). In addition, one imposes the regular boundary condition at the vortices, namely ψ(a) = ψ(b) = 0. We wish to find a formula for the propagator KAB(t,x,x0) associated with the Hamiltonian HAB. Note that KAB(t,x,x0) = expi(αθa(x)+βθb(x)) K′ Λ(t,x,x0)e−i(αθa(x0)+βθb(x0)) where K′Λ(t,x,x0) is the propagator associated with H′Λ. Let us denote D = R2 \ (La ∪Lb). Then one can embed D ⊂ M̃ as a fundamental domain. K′Λ(t,x,x0) is simply obtained as the restriction to D of the propagator KΛ(t,x,x0) associated with the Hamiltonian HΛ. On the other hand, to construct KΛ(t,x,x0) one can apply formula (5) and the knowledge of the free propagator on M̃, see (9), (10). Thus we get KΛ(t,x,x0) = ∑ g∈Γ ∑ γ∈C (g·x,x0) Λ(g−1)Kγ(t,g ·x,x0). (28) Fix t > 0 and x0,x ∈ D. One can classify piecewise geodesic paths in M̃, γ : x0 → C1 →···→ Cn → g ·x, (29) with Cj ∈ A∪B and g ∈ Γ, according to their projections to M. Let γ be a finite alternating sequence of points a and b, i.e. γ = (c1, . . . ,cn), cj ∈ {a,b} and cj 6= cj+1. The empty sequence γ = () is admissible. Relate to γ a piecewise geodesic path in M, namely x0 → c1 → ··· → cn → x. Suppose that this path is covered by a path γ in M̃, as given in (29). Then Cj ∈ A iff cj = a and Cj ∈ B iff cj = b. Denote the angles ∠x0,c1,c2 = θ0 and ∠cn−1,cn,x = θ. Then the angles in the path γ in (29) take the values ∠x0,C1,C2 = θ0 + 2πk1, ∠Cn−1,Cn,g · x = θ + 2πkn and ∠Cj,Cj+1,Cj+2 = 2πkj+1 for 1 ≤ j ≤ n − 2 (if n ≥ 3), where k1, . . . ,kn are integers. Any values k1, . . . ,kn ∈ Z are possible. In that case the representation Λ applied to the group element g occurring in (29) takes the value Λ(g) = exp ( 2πi(k1σ1 + · · · + knσn) ) where σj ∈{α,β} and σj = α if cj = a, and σj = β if cj = b. Using the equation∑ k∈Z e−2πiαk ( 1 θ + 2πk −π + is − 1 θ + 2πk + π + is ) = −2 sin(πα) e−α(s−iθ) 1 + e−s+iθ which is valid for 0 < α < 1, |θ| < π, one can carry out a partial summation in (28) over the integers k1, . . . ,kn. This way the double sum in (28) reduces to a sum over finite alternating sequences γ. Here is the resulting formula for K′Λ(t,x,x0). We set ζa = 1 or ζa = e2πiα or ζa = e−2πiα depending on whether the segment x0x does not intersect La, or x0x intersects La and x0 lies in the lower half-plane, or x0x intersects La and x0 lies in the upper half-plane. Analogously, ζb = 1 or ζb = e2πiβ or ζb = e−2πiβ depending on whether the segment x0x does not intersect Lb, or x0x intersects Lb and x0 lies in the upper half-plane, or x0x intersects Lb and x0 lies in the lower half-plane. Furthermore, let us write ζa = eiαηa, ζb = eiβηb, where ηa,ηb ∈{0, 2π,−2π}. Then one has K′Λ(t,x,x0) = ζaζb 1 4πit exp ( i |x−x0|2 4t ) − ∑ c∈{a,b} ζc sin(πσ) 4π2i ∫ ∞ 0 dt1 t1 ∫ ∞ 0 dt0 t0 δ(t1 + t0 − t) exp ( i ( r 2c 4t1 + r 20c 4t0 ))exp[−σ(sc − i(θc −θ0c −ηc)] 1 + exp(−sc + iθc − iθ0c) + 1 4πi ∑ γ,n≥2 (−1)n ∫ ∞ 0 dtn tn . . . ∫ ∞ 0 dt0 t0 δ(tn + · · · + t0 − t) exp ( i 4 (r2 tn + %2 tn−1 + · · · + %2 t1 + r20 t0 )) Sγ(s,θ,θ0), 233 Petra Košťáková, Pavel Šťovíček Acta Polytechnica where Sγ(s,θ,θ0) = sin(πσn) π exp[−σn(sn − iθ)] 1 + exp(−sn + iθ) sin(πσn−1) π exp(−σn−1sn−1) 1 + exp(−sn−1) ×···× sin(πσ1) π exp[−σ1(s1 − iθ0)] 1 + exp(−s1 + iθ0) , and sa = ln t1r0a t0ra , sb = ln t1r0b t0rb , sj = ln tjrj−1 tj−1rj for 1 ≤ j ≤ n. 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