ap-5-10.dvi Acta Polytechnica Vol. 50 No. 5/2010 Self-adjoint Extensions of Schrödinger Operators with δ-magnetic Fields on Riemannian Manifolds T. Mine Abstract We consider the magnetic Schrödinger operator on a Riemannian manifold M. We assume the magnetic field is given by the sum of a regular field and the Dirac δ measures supported on a discrete set Γ in M. We give a complete characterization of the self-adjoint extensions of the minimal operator, in terms of the boundary conditions. The result is an extension of the former results by Dabrowski-Šťovíček and Exner-Šťovíček-Vytřas. Keywords: Spectral theory, functional analysis, self-adjointness, Aharonov-Bohm effect, quantum mechanics, differen- tial geometry, Schrödinger operator. 1 Introduction Let (M, g) be a two-dimensional, oriented, connected complete C∞-Riemannian manifold, where g is the Riemannianmetric on M. Let dμ be themeasure in- duced fromtheRiemannianmetric. Ifwe take a local chart (U, ϕ), ϕ = (x1, x2), the measure dμ is writ- ten as dμ = √ Gdx1dx2 in U, where G = det(gmn), gmn = g(∂m, ∂n), and ∂m = ∂/∂x m. We denote L2(M) = L2(M;dμ). The set of all 1-forms on M is denoted by Λ1(M). In the coordinate neighbor- hood U, A ∈ Λ1(M) is written as A = A1dx 1 + A2dx 2. In general, the coefficients A1, A2 are complex- valued. We say A is real-valued if the coefficients are real-valued. We say A is of the class CkΛ1(M) if the coefficients are of the class Ck(U) for any local chart (U, ϕ). We define the class LqlocΛ 1(M) (1 ≤ q ≤ ∞)1, etc. similarly. The 2-form dA is called the magnetic field. If A ∈ L1locΛ 1(M), dA can be defined at least in the distribution sense. In U, the magnetic field is given by dA =(∂1A2 − ∂2A1)dx1 ∧ dx2. Let Γ = {γk}Kk=1 be a sequence of mutually dis- tinct points in M. The number K may be infinity, and in this case we assume additionally Γ has no ac- cumulation points in M. Let A be a 1-form on M given by the sum of two 1-forms (A) A = A(0) + A(1). The part A(0) corresponds to the δ magnetic fields, that is, we assume the following. (A0) A(0) ∈ C∞Λ1(M \Γ)∩L1locΛ 1(M), real-valued, and dA(0) = K∑ k=1 2παkδγk , (1) where αk ∈ R, and δγ is the Dirac measure concentrated on the point γ. More precisely, (1) means − ∫ M dϕ ∧ A(0) = K∑ k=1 2παkϕ(γk), for any ϕ ∈ C∞0 (M) (since A (0) ∈ L1locΛ 1(M), the left hand side is well-defined). Notice that this equa- tion is independent of the Riemannian metric g. For the regular part A(1) and the scalar potential V , we assume the following: (A1) A(1) ∈ C1Λ1(M), real-valued. (V) V is real-valued, V ∈ L2loc(M), and is bounded in some open neighborhood of γk for every k =1, . . . , K. Using the local coordinate (x1, x2), we define the Schrödinger operator L in each coordinate neighbor- hood by Lu = − 1√ G ∑ m,n=1,2 (∂m + iAm) · (√ Ggmn(∂n + iAn)u ) + V u, 1The measure dμ is omitted, since the class Lq loc Λ1(M;dμ) is independent of the choice of dμ. The coefficient Am is a function on U ⊂ M, however, we denote the pull-back (ϕ−1)∗Am = Am ◦ ϕ−1 on ϕ(U) ⊂ R 2 by the same symbol Am, for simplicity of notations. This convention is frequently used in this paper. 62 Acta Polytechnica Vol. 50 No. 5/2010 where (gmn) is the inverse matrix of (gmn). This definition is independent of the choice of local coor- dinates (see section 2). Define the minimal operator Hmin by Hminu = Lu, D(Hmin)= C∞0 (M \ Γ), where the overlinedenotes the closurewith respect to the graph norm. Define the maximal operator Hmax by Hmax = H ∗ min. Then we can show that Hmaxu = Lu, D(Hmax) = {u ∈ L2(M) | Lu ∈ L2(M)}, where L is a differential operator on D′(M \ Γ). We assume (SB) The operator Hmin is bounded from below. In the case M is the flat Euclidean plane, it is well-known that the operator Hmin is not essentially self-adjoint and the structure of the self-adjoint ex- tensionsof Hmin canbedeterminedvia the celebrated Krein-VonNeumann theoryof self-adjoint extensions (see e.g. Reed-Simon [13]). In the textbook by Al- beverio et al. [3], the case A(0) = A(1) = 0 and V = 0 (but Γ = ∅) is exhaustively studied. Adami- Teta [1] and Dabrowski-Šťovíček [7] study the case K = 1, α1 ∈ Z, A(1) = 0, and V = 0. Exner- Šťovíček-Vytřas [8] study the case K = 1, α1 ∈ Z, dA(1) = Bdx1 ∧ dx2 for some non-zero constant B (the constant magnetic field), and V =0. Moreover, Lisovyy [11] studies the case M is thePoincarédisk, g is thePoincarémetric, V =0and dA = Bωg+2παδ0, where B is a non-zero constant and ωg is the surface form induced from the Poincarémetric g. In all the results above, they first determine the deficiency subspaces Ker(Hmax ∓ i) and apply the Krein-VonNeumann theory. This method cannot be applied in the case K ≥ 2 and αk ∈ Z, however, this case (and A(1) is the constantfield, V =0)on theflat Euclideanplane is studiedby the author [12], and the structure of the self-adjoint extensions is determined. Our main purpose in this paper is to generalize the result in [12] on general complete Riemannian mani- folds and for more general A and V . Our first result is about the deficiency indices n±(Hmin)=dimKer(Hmax ∓ i). Theorem 1.1 Assume (A), (A0), (A1), (V), and (SB). Then, both deficiency indices n±(Hmin) are equal to 2K1 + K2, where K1 =#{αk | αk ∈ Z}, K2 =#{αk | αk ∈ Z}. Note that Bulla-Gesztesy [4] obtain a similar result in the case A = 0 and V has singularities, and Iwai- Yabu [9] also obtain a similar result on the two- dimensional torus. Next, we shall give a complete characterizationof the self-adjoint extensions of Hmin. To this purpose, we introduce some nice coordinates around singular- ities and some auxiliary functions. For simplicity, we assume K =#Γ is finite for a while. For k = 1, . . . , K, let (Uk, φk), φk = (x 1, x2), be a local chart around γk such that Uk is simply con- nected, φk(γk)= 0, V is bounded in Uk, and {Uk}Kk=1 are disjoint. Let (r, θ) be the radial coordinate in Uk defined by x1 + ix2 = reiθ, r ≥ 0, 0 ≤ θ < 2π. We assume gmn(0,0) = δmn, (2) ∂j gmn(0,0) = 0 (m, n, j =1,2), where δmn is the Kronecker delta. Condition (2) is satisfied, for example, if we take the normal coordinate2 as (x1, x2). Let βk be the fractional part of αk, that is, αk = [αk]+ βk, [αk] ∈ Z and 0 ≤ βk < 1. Put3 Ã(0) = βkr −2(−x2dx1 + x1dx2), Ã(1) = A(1) − A(1)(0). It is well-known that dÃ(0) = 2πβkδ0 (see e.g. Aharonov-Bohm [2, 1] or [7]). Define a phase func- tion ψk ∈ C∞(Uk \ {0}) by ψk(x) = exp 1 i ( A (1) 1 (0)x 1 + A (1) 2 (0)x 2 + (3)∫ x x0 ( A(0) − Ã(0) )) , where A(1) = A (1) 1 dx 1 + A (1) 2 dx 2, x0 is some point in Uk \ {0}, and the path of the line integral ∫ x x0 lies in Uk \ {0}. Notice that the value of the line integral is independent of the choice of paths mod- ulo 2πZ, by the Stokes theorem and the assumption d(A(0) − Ã(0))=2π[αk]δ0 in Uk. Then we have A = à + iψ−1k dψk, à = à (0) + Ã(1) (4) and L = ψkL̃ψ−1k (5) in Uk \ {0}, where L̃ is the operator L corresponding to the vector potential à and the scalar potential V . Let K1, K2 be the numbers in Theorem 1.1. In the sequel, we rearrange the index k so that 0 < βk < 1 for 1 ≤ k ≤ K1. As we prove later, the 2The coordinate defined by the local inverse map of the exponential map from the tangent space at γk to M. 3More precisely, the 1-form A(1) − A(1)(0) is defined as (A(1)1 (x 1, x2) − A(1)1 (0,0))dx 1 +(A (1) 2 (x 1, x2) − A(1)2 (0,0))dx 2, in the coordinate neighborhood Uk. 63 Acta Polytechnica Vol. 50 No. 5/2010 asymptotics of u ∈ D(Hmax) in Uk as r → 0 is given by u = ⎧⎪⎪⎨ ⎪⎪⎩ ψk(c k 1r βk−1e−iθ + ck2r −βk+ ck4r 1−βk e−iθ + ck5r βk)+ ξ (1 ≤ k ≤ K1), ψk(c k 3 logr + c k 6)+ ξ (K1 +1 ≤ k ≤ K), where ck1, . . . , c k 6 are constants and ξ is a regular func- tion in the sense ξ ∈ D(Hmin). Define Φj(u)= { t(c1j , . . . , c K1 j ) ∈ C K1 (j =1,2,4,5), t(cK1+1j , . . . , c K j ) ∈ C K2 (j =3,6), Φ(u)= t(tΦ1(u) · · · tΦ6(u)) ∈ C4K1+2K2. Define a (2K1 + K2) × (2K1 + K2)-diagonal matrix D by D = diag(1 − β1, . . . ,1 − βK1, β1, . . . , βK1, (6) −1/2, . . . , −1/2). Now our theorem is stated as follows. Theorem 1.2 Assume (A), (A0), (A1), (V), (SB) and K < ∞. Let Φ(u), D given above. (i) Let X= ( X1 X2 ) , where X1, X2 are (2K1 + K2)× (2K1 + K2) matrices satisfying rankX =2K1 + K2, X ∗ 1DX2 = X ∗ 2DX1. (7) Then, the operator HX defined by HX u = Lu, D(HX) = {u ∈ D(Hmax) | Φ(u) ∈ RanX} is a self-adjoint extension of Hmin. (ii) For any self-adjoint extension H of Hmin, there exists some matrix X satisfying (7) and H = HX. Wecan consider the case K = ∞, but some technical assumptions are necessary. We shall argue this case in section 5. Thus we can characterize the self-adjoint exten- sions in terms of the boundary conditions. We can easilyprove that theFriedrichsextensioncorresponds to the case X1 = O, X2 = Id. In the case M = R 2 and K =1, similar results are obtained in [7] and [8], and our theorem is a generalization of their results. As stated in their paper, the choice of matrices X is of course not unique: there are infinitely many ma- trices X giving same RanX. The difficulty in the proof is that we cannot de- termine the deficiency subspaces explicitly. To over- come this difficulty, we describe the condition of the self-adjointness only using the quotient subspace D(Hmax)/D(Hmin). This quotient subspace is essen- tially the same object as the sum of deficiency sub- spaces, butmuch easily tractable than the deficiency subspaces themselves. This idea is also used in [4] or [12]. We note that recently self-adjoint extensions of the Schrödinger operators on R2 with δ magnetic fields are studied from the viewpoint of the hidden supersymmetric structure; see Correa et al. [5, 6]. The rest of the paper is organized as follows. In section 2, we review basic notations and facts from the differential geometry and the theory of self- adjoint extensions. In section 3, we shall prove the structure of the self-adjoint extensions depends only on the singular part of the vector potentials. In sec- tion 4, we shall prove the main theorems. In sec- tion 5, we shall consider the case K = ∞ and give a complete characterization of the self-adjoint exten- sions, under some homogeneity conditions. 2 Basic facts 2.1 Formulas in differential geometry We quote some formulas used in Shubin [14] for the convenience of the readers. Take a local chart (U, ϕ), ϕ = (x1, x2), around p ∈ M. Put gmn = g(∂m, ∂n), and let (gmn) be the inverse matrix of (gmn). For α, β ∈ Λ1p(M) (the cotangent space at p), we define the scalar product 〈α, β〉 = ∑ m,n=1,2 gmnαmβn, where α = α1dx 1 + α2dx 2 and β = β1dx 1 + β2dx 2. Put |α|2 = 〈α, α〉, where α = α1dx1 + α2dx2. For a 1-form ω = ω1dx 2 + ω2dx 2, we define a function d∗ω by d∗ω = − 1√ G ∑ m,n=1,2 ∂m (√ Ggmnωn ) . This definition is independent of the choice of local coordinates. Actually, operator d∗ is characterized by the following relation:∫ M 〈du, ω〉dμ = ∫ M ud∗ωdμ for any u ∈ C∞0 (M) and ω ∈ C ∞ 0 Λ 1(M). Let A be a1-formsatisfyingour assumptions. For a function f, we define a 1-form dAf by dAf = df + if A, where d is the exterior derivative, and i = √ −1. For a 1-form ω, we define d∗Aω = d ∗ω − iA∗ω, A∗ω = 〈A, ω〉. 64 Acta Polytechnica Vol. 50 No. 5/2010 Then we obtain a representation of our Schrödinger operator L independent of local coordinates: L = d∗AdA + V. For operator d∗A, the following Leibniz formulas hold: for an appropriate function f and 1-form ω, we have d∗A(f ω) = f d ∗ω − 〈df, ω〉 − if 〈A, ω〉 = f d∗Aω − 〈df, ω〉 = f d ∗ω − 〈dAf, ω〉, (8) d∗AdA(f g) = f d ∗ AdAg − 2〈df, dAg〉 + gd ∗df. (9) Proposition 2.1 Let U, U ′ be open subsets of M \Γ such that U is a compact subset of U ′, and V is bounded in U ′. Then, there exists a constant C > 0 such that∫ U |dAf |2dμ ≤ C ∫ U′ (|f |2 + |Lf |2)dμ (10) for f ∈ D(Hmax). Proof. According to [14, (5.3)],4 we have (L(φf), φf) = �(φLf, φf)+ ∫ M |dφ|2|f |2dμ for f ∈ D(Hmax) and φ ∈ C∞0 (M \ Γ). Take φ ∈ C∞0 (U ′) such that φ =1 on U. Then the conclu- sion follows from the above equality,∫ suppφ |dA(φf)|2 =(L(φf), φf) − (V φf, φf) and assumption V is bounded. � 2.2 Theory of self-adjoint extensions We quote some notation from the textbook [13]. Let H be a separable Hilbert space and denote its inner product by (·, ·), and norm by ‖ · ‖. All the linear operators in this subsection are on the Hilbert space H. For a linear operator X, D(X) denotes the do- main of definition of X, X the closure of X, X∗ the adjoint operator of X. For a linear operator X, the graph inner product of X is defined by (x, y)X =(Xx, Xy)+(x, y) for x, y ∈ D(X), and the graph norm by ‖x‖X = (x, x) 1/2 X . We introduce some equivalent for the sum of the deficiency subspaces, which is also introduced in [4] or [12]. Let X be a closed, densely defined symmet- ric operator. Let D = D(X∗)/D(X), where the right hand side denotes the quotient space. The space D is a Hilbert space equipped with the norm ‖[x]‖2D = min y∈[x] ‖y‖2X∗ = ‖Qx‖ 2 X∗ , where x ∈ D(X∗), [x] = x+D(X) denotes the equiv- alence class of x in the quotient space D(X∗)/D(X), and Q denotes the orthogonal projection onto the orthogonal complement of D(X) in D(X∗). For u, v ∈ D, define [u, v]D = (X ∗x, y) − (x, X∗y), u = [x], v = [y], x, y ∈ D(X∗). The value [u, v]D is independent of the choice of the representatives x, y. Let P be the canonical projec- tion from D(X∗) to D. For a closed subspace V of D, we define a closed linear operator XV by D(XV )= {x ∈ D(X∗) | P x ∈ V }, XV x = X∗x. We also define V [⊥] = {u ∈ D | [u, v]D =0 for any v ∈ V }. Then the following proposition immediately follows from the definition of the self-adjointness. Proposition 2.2 1. For a closed subspace V of D, the operator XV is a self-adjoint extension of X if and only if V [⊥] = V. (11) 2. For any self-adjoint extension X̃ of X, there ex- ists a closed subspace V of D such that XV = X̃. In terms of the above notations, the Krein-Von Neumann theory can be rephrased as follows. Proposition 2.3 Let N± = Ker(X∗ ∓ i) the defi- ciency subspaces of X, n± = dimN± the deficiency indices of X. Then, the following holds. (i) The projection operator P gives a Hilbert space isomorphism from the direct sum N+ ⊕ N− to D. In particular, dimD = n+ + n−. (ii) There exists a one-to-one correspondence be- tween the closed subspaces V of D satisfying (11) and the unitary operators U from H+ to H−, given by V = P(1+ U)H+. This proposition says the space D can play the same role as the sum of deficiency subspaces in the the- ory of self-adjoint extensions. Particularly when N± is difficult to determine explicitly (as in our case), the space D is more tractable, since the element of this space has ambiguity by D(X). Actually, in the next section we shall see that the structure of D for our Schrödinger operator Hmin and the form [·, ·]D is determined only from the singular part A(0) of the vector potential. 4Since the function φ avoids the singularities, the proof of [14, (5.3)] is also available in our case. 65 Acta Polytechnica Vol. 50 No. 5/2010 3 Reduction 3.1 Division to the local potential Let (Uk, φk), φk = (x 1, x2), the local coordinate in- troduced in section 1. Let à be the 1-form given by (4). Take a positive number �k so small that the closed disc {r ≤ 2�k} is contained in Uk. Let ηk ∈ C∞0 (U) such that 0 ≤ ηk ≤ 1, ηk =1 for r ≤ �k, ηk =0 for r ≥ 2�k. Define functions ĝmn, Âm and V̂ on R2 by ĝmn = ηkgmn +(1 − ηk)δmn, Âm = à (0) m + ηkà (1) m , V̂ = ηkV. Define a differential operator Lk on R2 by Lk = − 1√ Ĝ ∑ m,n=1,2 ( ∂ ∂xm + iÂm ) · √ Ĝĝmn ( ∂ ∂xn + iÂn ) + V̂ , where Ĝ = det(ĝmn), and (ĝ mn) is the inverse ma- trix of (ĝmn). Define a linear operator Lk,min on L2(R2;dμk), dμk = √ Ĝdx1dx2, by Lk,minu = Lku, D(Lk,min)= C∞0 (R 2 \ {0}). Let Lk,max = L ∗ k,min. Then Lk,maxu = Lku, D(Lk,max)= {u ∈ L2(R2;dμk) | Lku ∈ L2(R2;dμk)}, where Lk is regarded as a differential operator on D′(R2 \ 0). Let D = D(Hmax)/D(Hmin), Dk = D(Lk,max)/D(Lk,min). Let χk ∈ C∞0 (M) such that 0 ≤ χk ≤ 1, χk = 0 for r ≥ �k and χk = 1 for r ≤ �k/2. Define a map Tk from D to Dk by Tk[f] = [ψ −1 k χkf], where the function ψk is given by (3). Define a map T from D to the direct sum K⊕ k=1 Dk by T [f] = k⊕ k=1 Tk[f]. We also define a map S from K⊕ k=1 Dk to D by5 S K⊕ k=1 [fk] = [ K∑ k=1 ψkχkfk ] . In the sequel, we sometimes write [f, g]D = [[f], [g]]D etc. for simplicity of notations. Lemma 3.1 1. Assume K < ∞. Then, the maps S, T defined above are well-defined andmutually inverse. Moreover, we have [f, g]D = k∑ k=1 [Tk[f], Tk[g]]Dk (12) for any [f], [g] ∈ D. 2. Assume K = ∞. Then themap S is well-defined and injective. Proof. (i) We divide the proof into three steps. Step 1. The map D(Hmax) + f �→ ψ−1k χkf ∈ D(Lk,max) is well-defined and continuous. Proof. Clearly ψ−1k χkf ∈ L 2(R2;dμk), so it suffices to show that Lk(ψ−1k χkf) ∈ L 2(R2;dμk). By (5) and the Leibniz rule (9), we have Lk(ψ−1k χkf)= ψ −1 k L(χkf)= ψ−1k (χkLf − 2〈dχk, dAf 〉 +(d ∗dχk)f) . The first term and the third in the parenthesis of the right hand side are in L2(R2;dμk) and continuous with respect to ‖ · ‖Hmax. Moreover,we can prove the second term is also in L2 and continuouswith respect to ‖ · ‖Hmax by using (10). � Step 2. Let f ∈ D(Hmin). Then, we have ψ−1k χkf ∈ D(Lk,min). Proof. By definition, there exists a sequence {fn}∞n=1 ⊂ C ∞ 0 (M \Γ) such that fn → f in D(Hmin). Then, ψ−1k χkfn ∈ C ∞ 0 (R 2 \ {0}) and ψ−1k χkfn → ψ−1k χkf in D(Lk,max), by Step 1. Since D(Lk,min) is a closed subspace of D(Lk,max), we have the conclu- sion. � Step 1 and 2 imply the map T is well-defined. We can similarly prove that the map S is also well- defined. Step 3. The operator ST is the identity map on D. Proof. By definition, we have (I − ST)[f]= [ψf], ψ =1 − K∑ k=1 χ2k. So it suffices to prove that g = ψf ∈ D(Hmin). 5When K = ∞, we define the map S for the elements of ∞⊕ k=1 Dk having only finite nonzero components [fk]. So there is no difficulty in the definition of S. 66 Acta Polytechnica Vol. 50 No. 5/2010 Let (r, θ) be the radial coordinate in Uk and put Bk,� = {x ∈ Uk | r < �}. Then we have suppψ ⊂ M \ K⋃ k=1 Bk,�k/2. For c > 0, let ξc ∈ C ∞(M) such that 0 ≤ ξc ≤ 1, ξc =1 in M \ K⋃ k=1 B�k/c, ξc =0 in K⋃ k=1 Bk,�k/(2c). Let L0, H0,min and H0,max be the operators corresponding to the potentials ξ4A and ξ4V . These potentials have no singularities, so we have H0,min = H0,max by [14]. Since Lg = L0g ∈ L2, we have g ∈ D(H0,max) = D(H0,min). Thus we can take a sequence {gn} such that gn → g in ‖ · ‖H0,min. Then ξ2gn ∈ C∞0 (M \ Γ) and ξ2gn → ξ2g = g in ‖ · ‖Hmin. Thus we have g ∈ D(Hmin). � Wecanprove T S = I similarly. Then (12) follows from (5) and the equality [f, g]D = K∑ k=1 [χkf, χkg]D (notice that f − ∑ k χkf ∈ D(Hmin) can be proved as in Step 3). (2)Let K = ∞. For anypositive integer n, wecan define T(n) from D to n⊕ k=1 Dk, and S(n) from n⊕ k=1 Dk to D similarly, andprove T(n)S(n) = Id. This implies the map S is well-defined and injective. � 3.2 Analysis of operators on R2 We shall analyze the operator Lk (or Lk,min, Lk,max) defined in the previous subsection. For simplicity of notation, we omitˆand˜in the definition of Lk in the sequel. Then our assumptions are the following: 1. Lk = d∗AdA + V on R 2 \ {0}, A = A(0) + A(1), 2. Lk,min and Lk,max are operators on L 2(R2;dμk), dμk = √ Gdx1dx2, 3. A(0) = βkr −2(−x2dx1 + x1dx2), 0 ≤ βk < 1, 4. A(1) ∈ C10Λ 1({r < 2�k}), real-valued, A(1)(0) = 0, 5. V is bounded, real-valued, 6. gmn(0) = δmn, ∂j gmn(0) = 0, and gmn = δmn for r ≥ 2�k. We shall show that gmn, A (1) and V havenothing todowith the structureof the self-adjoint extensions. To this purpose, define a differential operator Mk on R 2 by Mk = − ∑ n=1,2 ( ∂ ∂xn + iAn )2 . Define a linear operator Mk,min on L 2(R2;dx1dx2) by D(Mk,min) = C∞0 (R 2 \ {0}), Mk,minu = Mku for u ∈ D(Mk,min). Put Mk,max = M ∗ k,min, and Ek = D(Mk,max)/ D(Mk,min). We also define M (0) k , M (0) k,min, M (0) k,max, and E(0)k , by replacing An by A (0) n in the above defi- nition. The operator M(0)k is already studied in [1] and [7]. Here we quote their results and calculate the form [·, ·]E(0) k . Proposition 3.2 Let χ ∈ C∞0 (R 2) such that χ = 1 in some neighborhood of 0. 1. Assume 0 < βk < 1. Put f1k = χe −iθrβk−1, f2k = χr −βk , f4k = χe −iθr1−βk , f5k = χr βk . Then, the deficiency indices n±(M (0) k,min) = 2, dimE(0)k =4 and the vectors {[f n k ]}n=1,2,4,5 form a basis of E(0)k . Moreover, for m, n ∈ {1,2,4,5} with m ≤ n,6 we have [f mk , f n k ]E(0) k = ⎧⎪⎪⎨ ⎪⎪⎩ 4π(βk − 1) for (m, n)= (1,4), −4πβk for (m, n)= (2,5), 0 otherwise. 2. Assume βk =0. Put f3k = χ logr, f 6 k = χ. Then, the deficiency indices n±(M (0) k,min) = 1, dimE(0)k = 2, {[f j k]}j=3,6 form a basis of E (0) k , and [f3k , f 6 k]E(0) k =2π, [f3k , f 3 k]E(0) k = [f6k , f 6 k]E(0) k =0. Proof. (i) The first statement follows from the re- sult in [7] or [1]. For the calculation of [u, v]E(0) k , we use some notation in vector analysis. We use the gradient vector ∇ = t(∂1, ∂2), and identify a 1-form A with the component vector t(A1, A2). The dot · denotes the Euclidean inner product. Then we have [u, v]E(0) k = lim �→0 ∫ r≥� ( −v(∇ + iA(0)) · (∇ + iA(0))u + u(∇ + iA(0)) · (∇ + iA(0))v ) dx1dx2 = 6Notice that [f nk , f m k ]E(0) k = −[f m k , f n k ] E(0) k by definition. 67 Acta Polytechnica Vol. 50 No. 5/2010 lim �→0 ∫ r=� ( vn · (∇ + iA(0))u− un · (∇ + iA(0))v ) rdθ = lim �→0 ∫ r=� ( v∂ru − u∂rv ) rdθ, (13) where n =(cosθ,sinθ), and the line integral is taken counterclockwise. We used the Green formula and the fact n · A(0) = 0. Then we can easily prove the second statement by using (13). (ii) The first part of the statement follows from the results in [3]. The second statement can be jus- tified by using (13). � Next, we prove that the regular part A(1) does not affect the structure of Ek and the corresponding form. Proposition 3.3 All the statements of Proposition 3.2 hold even if we replace M (0) k,min by Mk,min, and E(0)k by Ek. Before the proof, we prepare a perturbative lemma, which is an immediate corollary of [10, Theo- rem IV.5.22]. Lemma 3.4 Let H be a separable Hilbert space and ‖ · ‖ its norm. Let X, Y be densely defined symmetric operators on H. Assume D(X) ⊂ D(Y ) and there exist positive constants C, δ with 0 < δ < 1 and ‖Y u‖ ≤ δ‖Xu‖ + C‖u‖ for every u ∈ D(X). Then, we have D(X + Y ) = D(X) and n±(X + Y ) = n±(X), where the overline denotes the operator closure. Proof of Proposition 3.3 We prove only state- ment (i). Statement (ii) can be proved similarly. By the Leibniz formula (8), we have for u ∈ C∞0 (R 2 \ {0}) (Mk − M (0) k )u = i(d ∗A(1))u − (14) 2i〈A(1), dA(0)u〉 + |A (1)|2u. We denote ‖u‖2 = ∫ R 2 |u|2dx1dx2 for a function u, and ‖ω‖2 = ∫ R 2 |ω|2dx1dx2 for a 1-form ω (notice that |ω|2 = 〈ω, ω〉). We denote the essential supre- mumnormof |u| and |ω| by ‖u‖∞ and ‖ω‖∞, respec- tively. Then we have by the Schwarz inequality ‖(Mk − M (0) k )u‖ ≤ ‖d∗A(1)‖∞‖u‖+2‖A(1)‖∞‖dA(0)u‖+‖A (1)‖2∞‖u‖ ≤ (‖d∗A(1)‖∞ + ‖A(1)‖2∞)‖u‖ + ‖A(1)‖∞(�‖M (0) k u‖ 2 + �−1‖u‖2) for any � > 0, where we used the inequality ‖dA(0)u‖ =(M (0) k u, u) 1/2 ≤ (�‖M(0)k u‖) 1/2(�−1‖u‖)1/2 ≤ 1 2 (�‖M(0)k u‖ + � −1‖u‖). Take � > 0 sufficiently small and apply Lemma 3.4. Then we have n±(Mk,min) = n±(M (0) k,min) = 2, thus dimEk = 4 by (i) of Proposition 2.3. Moreover we have D(Mk,min)= D(M (0) k,min), so the functions {f j k } (j = 1,2,4,5) do not belong to D(Mk,min). And we can prove Mkf jk ∈ L 2(R2) by using (14) and the fact |A(1)| = O(r) near the origin. Thus {[f jk]} form a basis of Ek. For the form [·, ·]Ek, we can prove the formula [u, v]Ek = lim �→0 ∫ r=� ( v(∂r u) − u(∂rv) − 2i(n · A(1))uv ) rdθ in a similarwayas in (13). Thus the value [f mk , f n k ]Ek is not affected by A(1), since |A(1)| = O(r) and |f mk f n k | is at most O(r − max(2βk,2(1−βk))). � Nextwe shall consider the non-flat case. We shall show that metric g also does not affect the structure of Dk and the corresponding form. Proposition 3.5 All the statements of Proposition 3.2 hold even if we replace M (0) k,min by Lk,min and E (0) k by Dk. Since V is bounded, we can assume V = 0. In the sequel, we use the following notation: L = G−1/2(D + A) · G1/2g−1(D + A), where D is the column vector t(D1, D2), Dj = −i∂j, A is identified with the component vector t(A1, A2), and g−1 is the inverse matrix of g =(gmn). We shall prepare some elliptic a priori estimate. Lemma 3.6 Let m, n ∈ {1,2}. Then, there exist Cm > 0 and Cmn > 0 such that ‖(Dm + Am)u‖ ≤ Cm(�‖Mku‖ + �−1‖u‖), ‖(Dm + Am)(Dn + An)u‖ ≤ Cmn(‖Mku‖+‖u‖)(15) for every u ∈ C∞0 (R 2 \ {0}) and every � > 0, where ‖ · ‖ = ‖ · ‖ L2(R 2 ;dx1dx2) . The difficulty is the singularity of our vector poten- tial A at the origin. We can overcome this difficulty by using some commutator technique. Proof of Lemma 3.6 PutΠj = Dj+Aj (j =1,2). Then, since ‖Πju‖2 = (�1/2Π2j u, � −1/2u) ≤ 1 2 ( �‖Π2j u‖ 2 + �−1‖u‖2 ) for u ∈ C∞0 (R 2), it suffices to prove (15). 68 Acta Polytechnica Vol. 50 No. 5/2010 Define auxiliary operators A = iΠ1 +Π2, A† = −iΠ1 +Π2. Let [X, Y ] = XY − Y X be the commutator of oper- ators X and Y . Then we have [Π1,Π2] = [D1, A2] − [D2, A1] = −i(b +2πβkδ0), where b = ∂1A (1) 2 − ∂2A (1) 1 is the magnetic field cor- responding to A(1). Thus we have [A, A†] = 2i[Π1,Π2] = 2(b +2πβkδ0). Particularly for u ∈ C∞0 (R 2 \ {0}), we have (AA† − A†A)u =2bu. Moreover, we have by definition (AA† + A†A)u =2Mku. These equalities imply AA† = Mk + b, A†A = Mk − b (16) on C∞0 (R 2 \ {0}). SinceΠmΠn canbewritten as a finite linear com- binationof the operatorsof the form XY , where X, Y are A or A†, it suffices to showthat there exists some constant C > 0 such that ‖XY u‖ ≤ C(‖Mku‖ + ‖u‖) (17) for u ∈ C∞0 (R 2\{0}). For (X, Y )= (A, A†),(A†, A), (17) follows from (16), since b is bounded. To esti- mate ‖A2u‖2, we assume A(1) ∈ C∞ for a while. Then, we have by (16) ‖A2u‖2 =(A2u, A2u)= ((A†)2A2u, u)= (A†(AA† − 2b)Au, u)= ‖A†Au‖2 − 2(bAu, Au) ≤ ‖A†Au‖2 +2‖b‖∞‖Au‖2 ≤ ‖A†Au‖2 +2‖b‖∞‖A†Au‖‖u‖. When A(1) ∈ C1, we approximate A(1) by C∞- potentials w.r.t. C1-norm on some neighborhood of suppu, thenweget the above inequalityagain. Then, we have (17) by using (16). The case X = Y = A† can be treated similarly. � Proof of Proposition 3.5 First, by assump- tion (vi), we have g−1 = I + ĝ, max |ĝmn| = O(r2), (18) G = 1+ O(r2), |DG| = O(r), as r → 0. Define a unitary operator U from L2(R2; √ Gdx1dx2) to L2(R2;dx1dx2) by U u = G1/4u. Put L̃k = U LkU −1, L̃k,min = U Lk,minU −1, etc. Then we have for v ∈ C∞0 (R 2 \ {0}) L̃k,minv = G −1/4(D + A) · √ Gg−1(D + A)G−1/4v. Thus we have L̃k,min=G −1/4(D + A) · G1/4g−1(D + A)+ G−1/4(D + A) · √ Gg−1(DG−1/4). (19) The second term of (19) is written as G−1/4 ( D · ( √ Gg−1(DG−1/4)) ) + (20) (DG−1/4) · G1/4g−1(D + A). The first term of (20) is bounded, and the second is infinitesimally small w.r.t. Mk,min, by Lemma 3.6. The first term of (19) is written as (D + A) · g−1(D + A)+ (21) G−1/4(DG1/4) · g−1(D + A). The second term of (21) is also infinitesimally small w.r.t. Mk,min, by Lemma 3.6. The first term of (21) is written as Mk,min +(D + A) · ĝ(D + A). The second term of this expression is written as∑ m,n=1,2 (Dmĝmn)(Dn + An)+ (22) ∑ m,n=1,2 ĝmn(Dm + Am)(Dn + An). The first sum of (22) is infinitesimally small w.r.t. Mk,min. If we take �k sufficiently small, the second sum is Mk,min-boundedwith relative bound less than 1, byLemma3.6. Nowwe can applyLemma3.4, and conclude that D(L̃k,min)= D(Mk,min)= D(M (0) k,min), and n±(Lk,min) = n±(L̃k,min) = n±(Mk,min) = n±(M (0) k,min). Moreover, one can show that multi- plication by G1/4 is a bijective continuous map on D(M (0) k,min). Thus we have D(Lk,min)= U −1D(L̃k,min)= G−1/4D(M (0) k,min)= D(M (0) k,min). And then we can prove Lkf mk ∈ L 2(R2;dμk) by the Leibniz formula and (18), and thus {[f mk ]}m form a basis of Dk. 69 Acta Polytechnica Vol. 50 No. 5/2010 In a similar way as in (13), we have [u, v]Dk = lim �→0 ∫ r=� ( vn · √ Gg−1(∇ + iA)u− −un · √ Gg−1(∇ + iA)v ) rdθ. Since √ Gg−1 = I + O(r2), we can replace √ Gg−1 by I in the calculation of [f mk , f n k ]Dk, and we have [f mk , f n k ]Dk = [f m k , f n k ]Ek. Thus we have the conclu- sion. � 4 Proof of main theorems Proof of Theorem 1.1 Since Hmin is semi- bounded, we have n+(Hmin) = n−(Hmin) = dimD/2. By Lemma 3.1 and Proposition 3.5, we have for K < ∞ dimD = K∑ k=1 dimDk =4K1 +2K2, and for K = ∞ dimD ≥ ∞∑ k=1 dimDk = ∞. Thus we have the conclusion. � Proof of Theorem 1.2 ByLemma 3.1 andPropo- sition 3.5, we have for u, v ∈ D(Hmax) [u, v]D =4πΦ(u) ∗ ( O −D D O ) Φ(v), whereΦ(u)∗ is the row-vector tΦ(u) and D is thema- trix given by (6). Let X = t(X1, X2) be the matrix satisfying (7). Then we have X∗ ( O −D D O ) X = O, which implies V ⊂ V [⊥] for V = RanX. Moreover, if rankX =2K1 + K2, we have dimV [⊥] =4K1+2K2 −dimV =2K1+K2 =dimV. Thus we have (11), and therefore HX is self-adjoint. Conversely, for a given self-adjoint extension H of Hmin, we can construct a (4K1+2K2)×(2K1+K2)- matrix X by arranging the coefficients of an arbi- trary basis of V = P D(H) with respect to the basis {[ψkf jk]}. � 5 Infinite singularities Let us consider the case K = ∞, and extend Theo- rem 1.2. Even in this case, for u ∈ D(Hmax) and for each k, we can define the asymptotic coefficients ckj at γk. However, the sequence Φj(u) is an infinite se- quence. We shall findappropriateassumptionswhich make these infinite sequences square summable. In the sequel, Uk, βk, gmn are those introduced in section 1. However, we may replace ψk defined by (3) more appropriate one satisfying (4), if such one exists. For simplicity, we assume V =0. (U) (i) There exists �0 > 0, independent of k, such that Uk = {r < �0} for every k. (ii) There exist β−, β+ such that 0 < β− ≤ βk ≤ β+ < 1 or βk =0, for every k. (iii) There exists C1 > 0 independent of k such that gmn satisfies (2) and |∂i∂j gmn| ≤ C1 in Uk, for every i, j, m, n =1,2. (iv) There exists C2 > 0 independent of k, and phase functions ψk ∈ C∞(Uk \ {0}) satis- fying |ψk| =1, (4) and |∂j A(1)m | ≤ C2 in Uk, for j, m =1,2. Thus we assume some homogeneity for g, A(0), and A(1). Since the open sets {Uk}∞k=1 are required to be disjoint, assumption (i) says the points of Γ are uniformly separated in some sense. Assumption (ii) seems a little strange, butwe need this assumption if we want to make the boundary value Φ(u) square summable.7 Assumption (iii) binds the curvature of M, and (iv) the intensity of the magnetic field. In [12], the author considers a similar assumption when M is the flat Euclidean plane and dA(1) is a constant magnetic field. In the sequel, we use the notation C ∞ = l2 = {(cj)∞j=1 | ∞∑ j=1 |cj |2 < ∞}, and define its inner product by usual l2-inner prod- uct. Let H = CK1 ⊕ CK1 ⊕ CK2. Proposition 5.1 Assume (A), (A0), (A1), (SB), (U), V =0, and K = ∞. Then, the following linear map 7If we consider another type of characterization, assumption (ii) may be dropped. 70 Acta Polytechnica Vol. 50 No. 5/2010 D + [u] �→ Φ(u) ∈ H ⊕ H, (23) is a well-defined homeomorphism. Moreover, [u, v]D = 4π(Φ(u), D̃Φ(v)), (24) D̃ = ( O −D D O ) , where D is a bounded operator on H defined by (6). Once this proposition is established, our theo- rem can be proved similarly as in the proof of Theo- rem 1.2. So we omit the proof. Theorem 5.2 Assume the same conditions as in Proposition 5.1. Then, the statements of Theorem 1.2 hold with the following changes: X1, X2 are bounded operators on H, and condition (7) is re- placed by the condition RanX =KerX∗D̃, where D̃ is the bounded operator on H ⊕ H defined in Proposition 5.1. We conclude this paper by proving Proposi- tion 5.1. Proof of Proposition 5.1. We divide the proof into two steps. Step 1. The map D + [f] �→ ∞⊕ k=1 Tk[f] ∈ ∞⊕ k=1 Dk is continuous, bijective and its inverse is also contin- uous. Proof. By our assumption (U) and the calculation in section 3, we can prove there exists C > 0 inde- pendent of k such that ‖ψ−1k χkf ‖ 2 Lk,max ≤ C ∫ Uk (|Lf |2 + |f |2)dμk. Summing up these equalities with respect to k, we conclude the map D(Hmax) + f �→ ∞⊕ k=1 ψ−1k χkf ∈ ∞⊕ k=1 D(Lk,max) is continuous. Then the well-definedness of the map (23) can be proved similarly as in section 3. Since D is identified with the closed subspace D(Hmin) ⊥ of D(Hmax) and the projection from D(Lk,max) to Dk is continuous, we conclude the map (23) is continuous. Moreover, we can prove the inverse map is also well-defined and continuous, so we have the conclusion. � Step 2. There exists C > 1 independent of k such that C−1|ck| ≤ ‖[u]‖Dk ≤ C|c k| for every [u] ∈ Dk, where ck = (ck1, c k 2, c k 4, c k 5) for 0 < βk < 1, c k = (ck3, c k 6) for βk = 0, and c k j are asymptotic coefficients of u defined in section 1. Proof. We only consider the case 0 < βk < 1. Con- sider the following formula for ck1 ck1 = 1 4π(1 − βk) [f4k , u]Dk , which can be verified by substituting all the basis functions into u. By choosing the representative u ∈ D(Lk,min)⊥ (so ‖u‖Lk,max = ‖[u]‖Dk) and using the Schwarz inequality, we have |ck1| ≤ 1 2π(1 − βk) ‖f4k ‖Lk,max‖[u]‖Dk . The fraction is bounded uniformly w.r.t. k, by our assumption (ii) of (U). Moreover, we can prove ‖f jk ‖Lk,max is also uniformly bounded, by (U) and the calculations in section 3 (first decompose Lk as in section 3, and estimate all terms). Thus we have |ckj | ≤ C‖[u]‖Dk . for j = 1. The case j = 2,4,5 can be treated simi- larly. Conversely,∑ j=1,2,4,5 ‖ckj [f j k]‖Dk ≤ |c k|( ∑ j=1,2,4,5 ‖f jk ‖ 2 Lk,max )1/2, and the sum in the right hand side is uniformly bounded. Thus the conclusion holds. � By Step 1 and 2, we have proved the map (23) is well-defined and homeomorphism. Equation (24) is confirmed by substituting each f jk as u or v. � Acknowledgement This work was partially supported by Doppler In- stitute for mathematical physics and applied math- ematics, KIT Faculty Research Abroad Fellowship Program, and JSPS grant Wakate 20740093. References [1] Adami, R., Teta, A.: On the Aharonov-Bohm Hamiltonian, Lett. Math. Phys. 43 (1998), 43–54. [2] Aharonov, Y., Bohm, D.: Significance of elec- tromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959), 485–491. 71 Acta Polytechnica Vol. 50 No. 5/2010 [3] Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable models in quantum me- chanics. Texts and Monographs in Physics., Springer-Verlag, New York, 1988. [4] Bulla, W., Gesztesy, F.: Deficiency indices and singular boundary conditions in quantum me- chanics, J. Math. Phys. 26 No. 10 (1985), 2520–2528. [5] Correa, F., Falomir, H., Jakubsky, V., Plyu- shchay, M. S.: Hidden superconformal symme- try of spinless Aharonov-Bohmsystempreprint, URL: http://arxiv.org/abs/0906.4055 [6] Correa, F., Falomir, H., Jakubsky, V., Plyu- shchay, M. S.: Supersymmetries of the spin- 1/2 particle in the field of magnetic vortex, and anyons, preprint, URL: http://arxiv.org/abs/1003.1434 [7] Dabrowski, L., Šťovíček, P.: Aharonov–Bohm effectwith δ-type interaction,J.Math. Phys.39, No. 1 (1998), 47–62. [8] Exner, P., Šťovíček, P., Vytřas, P.: Gener- alized boundary conditions for the Aharonov- Bohmeffect combinedwithahomogeneousmag- netic field, J. Math. Phys. 43, No. 5 (2002), 2151–2167. [9] Iwai, T., Yabu, Y.: Aharonov-Bohm quantum systems on a punctured 2-torus, J. Phys. A: Math. Gen. 39 (2006) 739–777. [10] Kato, T.: Perturbation theory for linear opera- tors. Springer, 1966. [11] Lisovyy, O.: Aharonov-Bohm effect on the Poincaré disk, J. Math. Phys. 48 (2007), no. 5, 052112. [12] Mine, T.: The Aharonov-Bohm solenoids in a constant magnetic field. Ann. Henri Poincaré 6 (2005), no. 1, 125–154. [13] Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self- adjointness, AcademicPress,NewYork-London, 1975. [14] Shubin, M.: Essential Self-adjointness for Semi- bounded Magnetic Schrödinger Operators on Non-compact Manifolds, J. Funct. Anal. 186 (2001), 92–116. Dr. Takuya Mine E-mail: mine@kit.ac.jp Kyoto Institute of Technology Matsugasaki, Sakyo-ku Kyoto 606-8585, Japan 72