Hidden superconformal symmetry of spinless Aharonov-Bohm system

A hidden supersymmetry is revealed in the spinless Aharonov-Bohm problem. The intrinsic supersymmetric structure is shown to be intimately related with the scale symmetry. As a result, a bosonized superconformal symmetry is identified in the system. Different self-adjoint extensions of the Aharonov-Bohm problem are studied in the light of this superconformal structure and interacting anyons. Scattering problem of the original Aharonov-Bohm model is discussed in the context of the revealed supersymmetry.


Introduction
The Aharonov-Bohm (AB) effect was discovered theoretically fifty years ago [1,2]. Since that time it found various experimental confirmations [3], and has been transformed into one of the most studied problems in planar physics [4,5,6,7,8,9,10]; for a nice review we refer the reader to [11]. The AB effect underlies the dynamical realization of anyons [12,13,14], which currently are supposed to play the key role in fractional Hall effect [15]. It appears in the analysis of cosmic strings [16,17], and planar gravity [18,19]. This effect plays also important role in the physics of graphene and nanotubes [20,21,22].
In their original work [1], Aharonov and Bohm pointed out the importance of the vector potential in quantum theory. Unlike in classical mechanics, it has direct impact on the quantum dynamics even when the electromagnetic field vanishes everywhere in the regions accessible for a charged particle. Such a situation is realized when the magnetic flux penetrating perpendicularly the plane is contained in finite regions bounded by impenetrable barrier. As a limit case we can consider the model given by the vector potential which corresponds to a singular flux that punctures the plane in the origin x = y = 0. In comparison with the free particle on the punctured plane, the physics is changed via a nontrivial phase that wavefunction acquires when is moving around the point where the flux dwells. This is the core of the AB effect.
In this work, we are going to testify this model on the presence of a hidden supersymmetry [23]. We will show that the Hamiltonian of a spinless charged particle moving in presence of the vector potential (1.1) 1 , 3) x = r cos ϕ, y = r sin ϕ, possesses a rich algebraic structure of both exact (not dependent on time explicitly) and dynamical (time dependent) integrals of motion, that close for a superconformal superalgebra.
The key ingredients of a supersymmetric structure are supercharges Q a , Hamiltonian H, and a grading operator Γ. The grading operator separates the set of relevant operators into families of bosonic and fermionic observables in accordance with whether they commute or anticommute with it. Supercharges are supposed to be fermionic while Hamiltonian is the bosonic operator, {Γ, Q a } = [H, Γ] = 0 , Γ 2 = 1 .
(1. 4) We speak about hidden supersymmetry when the operators Q a and Γ can be found despite the lack of fermionic (spin) degrees of freedom in a system. The hidden supersymmetric structure has been observed in various physically interesting one-dimensional models, including the Dirac delta function potential problem, the reflectionless Pöschl-Teller system [24], and periodic finite-gap quantum systems [25,26]. It was also observed in the bound state Aharonov-Bohm effect [24], that corresponds to a particle confined to a circle. In those systems, the hidden supersymmetry reflects their peculiar spectral and scattering properties. We will seek for the operators Γ, Q 1 and Q 2 that would satisfy (1.4) and (1.5) These relations correspond to Lie superalgebra of quantum mechanical N = 2 supersymmetry 2 . The supercharges Q 1 and Q 2 can be nonlocal in general, as they correspond to the square roots of the spinless differential operator H α .
The Hamiltonian H α does not determine the dynamics of the particle uniquely until its actual domain of definition is fixed. The ambiguity in the proper definition of the system is intimately related to the self-adjoint extensions of the Hamiltonian. Physically, this corresponds to different possibilities to realize the condition of impenetrability of the region x = y = 0. The task of self-adjoint extensions has been analyzed extensively in the literature. The case of a single magnetic vortex has been studied as a limit case of an impenetrable tube of finite radius with internal magnetic field [29]. It was also analyzed directly with making use of the von Neumann theory of self-adjoint extensions [30,17,31].
Having in mind our objective, we cannot use these results directly as they do not contain any information on the existence of the supersymmetric structure described by (1.4), (1.5). Our approach will be different: we will identify first grading operator Γ, and construct operators Q 1 and Q 2 that will satisfy (1.4) and (1.5) formally. Then we will find their self-adjoint extensions. Hamiltonian, defined as the square of supercharges, will be self-adjoint by construction [32]. The obtained results will be compared with the known ones. As we will see, the self-adjoint extension with regular wave functions at the origin will be unitarily equivalent to the free particle system for integer values of the magnetic flux, meanwhile it will match exactly with the model discussed by Aharonov and Bohm for non-integer values of α. We also find two other self-adjoint extensions of H α , which for non-integer values of α possess hidden supersymmetry and correspond to supersymmetric two-anyon systems with contact interaction.
The work is organized as follows. In the next Section, we construct a formal supercharge that satisfies the required properties. We then specify its self-adjoint extensions, and discuss the existence of N = 2 supersymmetry in the system. Finding the eigenfunctions of the associated Hamiltonian, we show that the obtained system coincides with the original model discussed by Aharonov and Bohm. We analyze the action of supercharges on the wave functions to clarify whether we have exact or spontaneously broken supersymmetry. In Section 3, we consider other two self-adjoint extensions of H α , which posses hidden supersymmetry. A particular attention is given to the case of semi-integer flux in Section 4, where an su(2) family of grading operators exists. In Section 5, we discuss conformal symmetry of the systems and confirm their scale invariance. Sequently, we extend the algebraic structure of the hidden supersymmetry by conformal symmetry. In Section 6, we provide an alternative interpretation of the model in terms of anyons. The last section is devoted to a brief summary and discussion of the results, with emphasis on their physical aspects. Particularly, we discuss the scattering problem in the original Aharonov-Bohm model in the light of the hidden supersymmetry and related translation symmetry breaking. We list also there some open problems to be interesting for a future research. Appendices include details on self-adjoint extensions of the supercharges considered in Sections 2 and 3, and explicit formulas for the domains of the operators discussed in Section 5.
2 Hidden N=2 supersymmetry in spinless AB system In general, a formal Hamiltonian operator H α (1.2) admits a four-parametric U (2) family of self-adjoint extensions, which specify physically different configurations, distinct in their spectral and scattering properties [31,33]. The spectrum depends strongly on the actual choice of the domain of definition of H α ; besides a continuous part of non-negative energy scattering states, it may contain up to two bound states of negative energy. As we stated above, our goal is to examine the model for the presence of the hidden supersymmetry (1.4), (1.5) generated by self-adjoint supercharges. This excludes immediately those selfadjoint extensions of (1.2) in which bound states are present, since negative energy levels would imply purely imaginary eigenvalues for the supercharges.
The general solution of the partial-wave stationary Schrödinger equation for non-negative energy E = This gives rise to a unique fixing of the self-adjoint extension of the operator H α that corresponds to the Aharonov-Bohm system, which we denote by H AB α . The aim of the present section is to reveal a hidden supersymmetry in the Aharonov-Bohm system. We proceed as follows: first of all, we identify the Z 2 -grading operator of the bosonized supersymmetry. Then we define a formal supercharge operator, find its self-adjoint extension, and obtain the second odd generator of the N = 2 supersymmetry. After that we show that the square of the found supercharges coincides with the Hamiltonian H AB α of the Aharonov-Bohm system.
Consider a nonlocal operator of rotation in π, which is presented in terms of the total angular momentum J = −i∂ ϕ + α as It is a unitary, Hermitian involutive operator, R 2 = 1, which commutes with Hamiltonian (1.2), and can be identified as the grading operator Γ. Consider a formal nonlocal differential operator (2.5) This operator and operator iRQ α satisfy formally relations On the other hand, we have The commutator [P x , P y ] is just the two-dimensional Dirac delta function. Unlike the one-dimensional case, such a term is not uniquely defined in the planar quantum systems [34,35]. As it was discussed in [36], the self-adjoint extension of Hamiltonian H α has to be specified to define consistently the operator. When we specify the actual domain of the self-adjoint extension of H α , the Dirac delta function term is redundant in the potential since its manifestation can be understood in asymptotic behavior of the wave functions near the origin. 3 In our current case, it will suffice to fix the self-adjoint extension of Q α since the square of self-adjoint operator is self-adjoint as well. Before we step to the analysis of the self-adjoint extension of Q α , let us note that the actual choice of the signs in definition of R(α) in (2.5) is crucial. Alternative choice of the sign for the same flux value case leads to a different self-adjoint extension of H α , and will be discussed in the next section. As we will see later in this section, the exception is the case of integer flux values. For α ∈ Z, both choices R(α) = R and R(α) = −R lead to the same result.
Operator Q α (2.5) defined on the smooth functions with compact support is symmetric. Hence, the machinery of von Neumann theory can be applied to find its self-adjoint extensions. It can be checked that Q α is essentially self-adjoint for any α ∈ R. Indeed, the equations (Q α ) † f (r, ϕ) = ±if (r, ϕ) do not have square integrable in R 2 solutions. The deficiency index is equal to (0, 0), and the operator Q α does have a unique self-adjoint extension, its closure, which we denote as Q AB α . Its domain of definition D(Q AB α ) is given by Eqs. (A.10) and (A.11) in Appendix A.
To play the role of the supercharge, the operator Q AB α has to anticommute with the grading operator. The operator R is essentially self-adjoint on D(Q AB α ) and leaves this space invariant. Hence, the anticommutation relation {Q AB α , R} = 0 is well defined on D(Q AB α ). This allows us to construct immediately the second self-adjoint supercharge iRQ AB α , defined on D(Q AB α ) as well. The square of the supercharges gives the self-adjoint Hamiltonian H c α that is defined as Let us show now that the system described by H c α coincides with the model proposed by Aharonov and Bohm. To do this, we shall find eigenfunctions of H c α . To simplify the forthcoming analysis, let us comment on relation between the systems H c α and H c α+n with magnetic flux values different in integer number n ∈ Z. A simple formal operator equality suggests that the unitary transformation U n = e inϕ is associated with the change of the magnetic flux of the system. It is indeed the case. First, we have U −1 n RU n = (−1) n R, and for α / ∈ Z there holds The case of α ∈ Z has, however, a peculiarity, and deserves a separate comment. For α = n, there exists a system with hidden supersymmetry represented by self-adjoint operators H c n and Q AB n defined on corresponding domains. We can use the transformation U 1 to construct another system with the same flux, described by H c n = U −1 1 H c n−1 U 1 and U −1 1 Q AB n−1 U 1 . For the transformed supercharge domain, there holds a relation , which means that the independent integrals of motion Q AB n , R and U −1 1 Q AB n−1 U 1 = P x −iRP y coexist in the same domain D(Q AB n ). Their linear combinations (including their multiplications by R) lead to another set of integrals of motion, given by R, P x , P y and their multiples by R. But P x and P y are the generators of translation in the plane, and, hence, the system described by H c n has a translational symmetry. As we will see, H c 0 corresponds to a free particle in the plane, which, of course, possesses translational invariance. Then, the revealed translational symmetry of H c n can be understood as a consequence of the unitary equivalence of H c 0 and H c n .
Note that the Hamiltonian H c 0 is invariant, in addition, under spatial reflections. As there is no preferential direction in the plane, we can consider two reflections R x g(x, y)R x = g(−x, y) , R y g(x, y)R y = g(x, −y) , (2.11) which satisfy the relations In the polar coordinates their action is given by They commute with the operator R, therefore they should be treated as nonlocal even integrals of motion within the supersymmetric structure. Despite their involutive nature, neither of these two operators can be identified as the grading operator since they do not anticommute with the supercharge (2.5) [they do not commute with (2.5) either]. As we shall see in Section 4, the twisted analogs of the operators (2.11) emerge nontrivially in the systems with half-integer flux. We conclude that the change of the sign of R(α) in definition (2.5) for integer flux value case reduces to a unitary transformation, and that this sign ambiguity gives rise to the translational invariance of H c n . At the same time, we can see that the complete knowledge of the system for α ∈ [−1, 0) (or for α ∈ [0, 1)) provides a detailed description for any other value of the magnetic flux as well. We will employ this fact in the forthcoming analysis of the spectral properties and supersymmetric structure of the system.
Let us fix the flux to be α ∈ [−1, 0). In the polar coordinates, the supercharge Q AB where are the projectors on the subspaces of even (Π + ) and odd (Π − ) partial waves. It preserves subspaces H l , where L e i(2l−1)ϕ , e i2lϕ is a linear space spanned by the indicated vectors. Then, the eigenvalue problem can be solved separately in each H l . The equation is rewritten with help of (2.14) in the form The general solution of (2.18) for nonzero eigenvalues λ are linear combinations of the Bessel functions of the first, J ν (|λ|r), and second, Y ν (|λ|r), kinds. The first is regular while the other one is singular at the origin, but both are not normalizable. To keep their interpretation in terms of scattering states, we require the wavefunctions not to have too strong divergence at infinity 4 and to respect the behavior near the origin, prescribed by the domain of definition. Since the singular solution violates the first requirement due to its divergence at r = 0, it has to be discarded. Then the acceptable solutions of (2.17) for λ = 0 are The solutions of the equations (2.18) for λ = 0 with admissible behavior at the origin are Φ 0,l ∼ r 2l+α−1 e i(2l−1)ϕ for 2l + α ≥ 1 , r −2l−α e i2lϕ for 2l + α ≤ 0 . (2.20) We pass now to the analysis of the eigenfunctions of H c α . Hamiltonian commutes with the generator of rotations since D(H c α ) is invariant with respect to the action of J. Hence, one can find their common eigenfunctions Ψ |λ|,j , H c α Ψ |λ|,j = λ 2 Ψ |λ|,j , JΨ |λ|,l = (l + α)Ψ |λ|,l . (2.21) They can be composed from the eigenvectors of Q AB α corresponding to different signs of λ, The zero-energy eigenstates of H c α are Ψ 0,l ∼ r |l+α| e ilϕ . (2.23) Note that the wave functions (2.22) vanish at the origin except the special case of integer flux such that 2l + α = β ∈ {0, 1}. In this case, J |β−2l−α| (|λ|r) = J 0 (|λ|r) → 1 for r → 0, that is in agreement with the results on the self-adjoint extension of the free particle in the punctured plane [38]. The exclusion of the origin is of no importance here since the considered functions are regular at this point. In fact, the considered self-adjoint extension H c 0 of H α with α = 0 is in correspondence with the system of the free particle, since its domain of definition is spanned by the same complete basis of partial waves J |m| (kr)e imϕ .
We can compare the system represented by H c α with the original setting of Aharonov and Bohm in the similar vein. The behavior of the wave functions near the origin is prescribed in the same way in both systems. This leads to the same complete basis of partial waves given by (2.19), (2.20). Hence, H c α and H AB α represent the same self-adjoint extension of H α . Thus, the system described by H c α coincides with that discussed originally by Aharonov and Bohm, This means that the Aharonov-Bohm model possesses the hidden N = 2 supersymmetry generated by the supercharges Q AB α and iRQ AB α , in which the role of the grading operator is played by the operator R. This result is valid for any value of the magnetic flux. Now, let us discuss the nature of the revealed supersymmetry, and the action of the supercharges. The spectrum of the operator H AB α consists of continuous part only, which covers nonnegative real numbers. Any value of energy E is infinitely degenerate since there is an inifinite set of linearly independent generalized wave functions (2.22) corresponding to the given energy E = λ 2 . Let us discuss the action of the supercharges Q AB α and iRQ AB α . The second supercharge interchanges the eigenfunctions of Q AB α with different sign of λ = 0, i.e. there holds Consequently, with the direct use of this relation and (2.22), we can write where Q a is Q AB α or iRQ AB α . The spectrum of H AB α includes infinitely degenerate zero energy level. We restrict our consideration to the subspace H l where all the energy levels are doubly degenerate. This subspace is invariant under the action of the supercharges. Taking into account Eq. (2.20) for the zero modes of Q AB α , we conclude that there exists just a single state in H l annihilated by Q AB α . Fixing l ≥ 0, we can write explicit form of the involved functions, (2.16). We illustrate the action of the supercharges in these subspaces for l = 0, 1, 2. The zero energy states (2.23) are represented by the circles, the black circles correspond to the zero modes of Q a . The arrows between the same energy levels in each H AB l correspond to relations (2.26) for E > 0 and to the relations (2.27) for E = 0.
This resembles the Jordan blocks structure, which can appear in diagonalization of a finite-dimensional matrix. It does not contradict the self-adjointness of Q AB α , the supercharge can be diagonalized with making use of its eigenstates (2.19) and (2.20). Hence, the supercharges annihilate just one half of the zero-energy states. The rest of these states is transformed into the kernel of the supercharges (see Fig.1).
This picture can be compared with the cases of unbroken and broken supersymmetry in non-periodic one-dimensional systems. There, particularly, the unbroken supersymmetry is related to the existence of a singlet bound state of zero energy, annihilated by supercharge. The second, nonphysical solution corresponding to zero energy is transformed to a physical one by a supercharge. In the present case, the continuous nature of the spectrum together with the infinite degeneracy of the energy levels prevents us from a similar classification of the revealed hidden supersymmetry. On the other hand, there is some similarity of the revealed hidden supersymmetric structure with that appearing in one-dimensional finitegap periodic quantum systems, cf. [26].
In conclusion of this section, let us make a few comments on the structure of the revealed supersymmetry, which further on will provide an alternative interpretation of the system in terms of anyons. In (2.15) we introduced projectors Π ± on the subspaces of even and odd orbital angular momentum. This allows us to separate the domain D(H AB α ) into two subsets Π ± D(H AB α ), each of which consists of eigenvectors of R with fixed eigenvalue +1 or −1. We can employ the matrix representation of the projectors, The Hamiltonian H AB α as well as other operators can be rewritten in the matrix form, where H AB α,± = Π ± H AB α . The supercharge Q AB α is antidiagonal operator and its explicit form for α ∈ [−1, 0) can be deduced from (2.14) and (2.28). In this framework, the wave function ψ from the domain of H AB α is just a column vector, whose upper element is composed of even partial waves, ψ + = Π + ψ = l∈Z g + l (r)e i2lϕ , while the lower component consists of odd partial waves, ψ − = Π − ψ = l∈Z g − l (r)e i(2l−1)ϕ . Such a representation reveals an obvious similarity of the hidden supersymmetry of the spinless Aharonov-Bohm system with supersymmetry of a usual form, associated with introduction of the spin degrees of freedom [39,40].

Exotic models
The choice of the signs we made in definition of R(α) in (2.5), and the observed ambiguity for α ∈ Z case, led us to the revealing of the hidden N = 2 supersymmetry in the original Aharonov-Bohm system. In this section we investigate the consequences of the alternative choice of the signs in (2.5) for non-integer flux values.
So, let us consider the operator The formal relations (2.6) and (2.7) imposed on the supercharge remain intact, up to the sign of the commutator term [P x , P y ] in the square ofQ α . This suggests that the difference, if any, could appear in self-adjoint extensions of the supercharge operator (3.1). The transformation U 1 changes the flux of the system in one unit. It maintains the self-adjointness of the operators, i.e. when an . This means that when we find all the admissible self-adjoint extensions ofQ α for α ∈ (−1, 0)| mod 2 , we can get all the self-adjoint extensions of the operator for α ∈ (0, 1)| mod 2 just by application of this transformation. The inverse is also true with changing of the transformation U 1 for Without loss of generality, we restrict our analysis to α ∈ (0, 1)| mod 2 . As the operatorQ α for α ∈ (0, 1)| mod 2 coincides formally with the operator Q α for α ∈ (−1, 0)| mod 2 , we can use directly Eq. (2.14) to express the operator in polar coordinates, just keeping in mind the different range of α. The operatorQ α preserves the subspaces (2.16), and is symmetric on . The domains of its conjugate and its closure are presented in Appendix A.
We have to solve the deficiency equationsQ † α ψ = ±iψ to reveal the bases of the deficiency subspaces. The relation (2.14) together with (2.16) simplify this task since the problem can be inspected for each subspace H l separately. The deficiency indexes are vanishing again in all the subspaces H l except the subspace H l0 given by the integer l 0 such that 2l 0 + α ∈ (0, 1). In contrary to (2.5), the deficiency indexes ofQ α are (1, 1), so that there exists a U (1) family of self-adjoint extensionsQ γ α ofQ α . The detailed derivation of the result is rather technical (see Appendix A), and we present the final form of the domain of the self-adjoint operatorQ γ α : where Q α is the closure ofQ α . Expanding the function f (r, ϕ) ∈ D(Q α ) in partial waves f (r, ϕ) = l f l (r)e ilϕ , we find that the radial parts f l (r) have to have the following asymptotic behavior near the origin: . The functions Φ ± (r, ϕ) form the basis of deficiency subspaces,Q † α Φ ± (r, ϕ) = ±iΦ ± (r, ϕ), and can be written in terms of McDonald functions The operator R is essentially self-adjoint on D Q γ α , but the requirement {Q γ α , R} = 0 is consistent if and only if the operator R leaves D(Q γ α ) invariant. Using (3.2) and the fact that The requirement on the invariance of D(Q γ α ) holds true for two values of parameter γ only, Hence, the N = 2 supersymmetric structure is admissible just for these values of the parameter γ. If not stated otherwise, we will restrict γ ∈ {0, π} from now on. In this case, the domains ofQ γ α and iRQ γ α coincide. It is worth to mention that both D(Q 0 α ) and D(Q π α ) acquire particularly simple form, which manifests their invariance with respect to rotations generated by J.
The structure of N = 2 supersymmetry is completed by the following definition of the self-adjoint Hamiltonian H γ α : Hence, taking the different definition (3.1) of the supercharge, we reveal two distinct self-adjoint extensions H γ α of the formal Hamiltonian operator H α , which, like the Aharonov-Bohm system considered in the previous section, are characterized by the hidden N = 2 supersymmetry. As the domains of Hamiltonians are invariant with respect to J, the systems have rotational symmetry as well.
In the next step we shall analyze the spectrum of H γ α and find the associated wavefunctions. Since H γ α is the square of the self-adjoint operatorQ γ α , and D(H γ α ) is a subset of D(Q γ α ), we conclude that in correspondence with the hidden supersymmetric structure, the spectrum is non-negative.

Half-integer flux and twisted reflections
The operator R commutes formally with H α for any value of the magnetic flux α. In contrary, the reflection operators R x and R y defined in (2.11) are exclusive integrals of motion of the free particle. When the magnetic flux is switched on, they provoke a change of the sign of the magnetic flux in H α , i.e.
We can define the "twisted" reflection operatorsR x = e iαπ e −2iαϕ R x andR y = e −2iαϕ R x , for which For a general value of α, however, they do not preserve the space of 2π-periodic functions.R x andR y are defined consistently for α = m or α = m + 1 2 only, where m ∈ Z. For α = m, these operators are related to non-twisted reflections (2.11) by the unitary transformation U m = e imϕ , and they commute, therefore, with R. The situation is essentially different for half-integer values of α. For α = m + 1 2 we get and where for the sake of convenience we included in definition ofR x an additional numerical factor (−1) m+1 . The operators satisfy also The zero modes of H γ α as well as H AB α can be understood as a low energy limit of the properly normalized scattering states.
Relations (4.3), (4.4) mean that the twisted reflection operators (4.2) together with R satisfy exactly the same set of algebraic relations as the three Pauli matrices, i.e. up to the numerical factor 1 2 they are generators of the spinorial representation of su (2). In this section, we discuss the role of the triplet of reflection operators for the hidden supersymmetry of the systems with half-integer flux 6 .
Without loss of generality, set α = 1/2. The operators R,R x = −ie −iϕ R x andR y = e −iϕ R y are symmetric on D(Q AB 1/2 ), or D(Q γ 1/2 ), γ ∈ [0, 2π). In addition, neitherR x f = ±if norR y f = ±if have nontrivial solutions. The operators are essentially self-adjoint both on D(Q AB 1/2 ) and D(Q γ 1/2 ). As R 2 x =R 2 y = 1, they are unitary as well. The described properties of the triplet of reflection operators allows us to introduce a three-parametric family of SU (2)-transformations Therefore, the set of integrals of motion of H AB 1/2 consisting of Q AB 1/2 , iRQ AB 1/2 , R and J has to be extended by the operatorsR x andR y .
The question is then how they could be incorporated into the superalgebraic structure of the system. Keeping R as the grading operator, the new integrals of motion are of the fermionic nature. Hence, the anticommutators with the supercharges Q AB 1/2 and iRQ AB 1/2 should be computed, as well as their commutators with J. Both twisted reflections, however, anticommute with angular momentum generator J. As a consequence, the repeated commutators with J give and analogous relations forR y . Subsequent anticommutators of the odd integralsR x J n andR y J k , n, k = 0, 1 . . ., produce the integrals of the form RJ n . In the same way, the anticommutators of the twisted reflections with the supercharges Q AB 1/2 and iRQ AB 1/2 , and corresponding repeated (anti)commutation relations reproduce the basic integrals multiplied by (H AB 1/2 ) n . We see that the inclusion of the twisted reflection operators into the superalgebraic structure leads to its nonlinear deformation characterized by appearance of the multiplicative factors (J) n and (H AB 1/2 ) k in (anti)commutation relations, cf. [24,26,27,28].
Since the integralsR x andR y satisfy the relations (4.1), any of them can also be taken as the Z 2grading operator instead of R. Such a possibility for alternative choice of the grading operator resembles the tri-supersymmetric structure studied in [26]. The difference is that here the three involutive integrals mutually anticommute, while in the tri-supersymmetric structure analogous integrals mutually commute 7 . If, for instance,R x is identified as the grading operator, the operators Q AB 1/2 and iR x Q AB 1/2 will be nontrivial supercharges, and the angular momentum J has also to be treated as an odd generator. The anticommutator of the supercharge iR x Q AB 1/2 with R generates then iR y Q AB 1/2 , that has to be treated as an even integral. Further computing shows that with R x taken as the grading operator, we have, again, a nonlinear superalgebraic structure.
The picture is completely different in the case of the systems described by H γ 1/2 (γ ∈ {0, π}). The operatorsR x andR y are no longer symmetries of the system as the domain D(H γ α ) is not invariant under their action. The unitary transformations (4.5) can be used to map the system H γ 1/2 (γ ∈ {0, π}) to another, equivalent one, with the same supersymmetric structure. Let us discuss a few particular examples, where the transformed grading operator acquires particularly simple form. We are interested in the mappings which would interchange the operators R,R x andR y in the role of the grading operator, where the explicit form of the SU (2)-transformations is (4.9) Let us note that the transformations (4.8) together with (2.4) suggest that the twisted reflectionsR x and R y can be written formally in the following way : Inspect now how these transformations change the other constituents of the supersymmetry, Hamiltonian and supercharges. Formally, Hamiltonian H 1/2 commutes with any of R,R x orR y so that it is invariant with respect to the SU (2)-transformations (4.5). The formal operatorQ 1/2 is transformed as Let us suppose that we take the self-adjoint extension H 0 1/2 (with the superchargeQ 0 1/2 ) as the initial system. The transformations (4.5) are unitary, and applied toQ 0 α produce self-adjoint operators, defined on U k D(Q 0 α ) for k ∈ {0, 1, 2, 3, 4}. The transformed supercharges in the upper line of (4.11) coincide formally withQ 1/2 . As we found in the previous section, there exists one-parameter family of the selfadjoint extensions of this operator, labeled by γ. Hence, the systems produced by U 0 , U 1 and U 3 should fit into this classification scheme. This is indeed the case: we can write where the value of the index γ coherently reflects the domain of definition, given by (3.2). The remaining systems with the supercharges of the lower line in (4.11) do not belong to the family of self-adjoint operators Q γ 1/2 as neither of the supercharges coincides formally withQ 1/2 . The explicit form of the domains of definition of the new supercharges for k ∈ {1, 2, 3, 4} can be written in the following compact form : Hence, for the semi-integer values of the magnetic flux α we have a three parametric family of the systems with hidden supersymmetry, associated with the formal supercharge operator (4.14) These systems fit into the general scheme of the self-adjoint extensions of the Aharonov-Bohm model discussed in [31], where the self-adjoint extensions of H α with broken rotational symmetry were observed. Despite the rotational symmetry is broken in our present case as well (see (4.13)), domains of definition are invariant with respect to the operatorJ(β x , β y , β) = UJU † , i.e. the systems associated with (4.14) are unitarily equivalent to the systems with rotational symmetry.

Superconformal symmetry
Jackiw showed that like a charge-monopole system [44], the original Aharonov-Bohm model is characterized by a dynamical conformal so(2, 1) symmetry [45]. We revealed the hidden N = 2 supersymmetry not only in the Aharonov-Bohm system characterized by a regular behavior of the wave functions at the origin, but also in exotic models corresponding to some special cases of the U (2) family of self-adjoint extensions of the formal Hamiltonian operator (1.2). On the other hand, if we look at the U (2) family of the self-adjoint extensions requiring the scale symmetry, this also excludes immediately those cases which are characterized by the presence of the bound states. Such a similarity with restrictions imposed by the requirement of the presence of the hidden supersymmetry, certainly, is worth a more in-depth look. In this section, we study the question of compatibility of the revealed hidden supersymmetric structure with the dynamical conformal symmetry. Besides the Hamiltonian of the system, which we denote here by H, the dynamical conformal symmetry [46] is generated by the operators D and K that depend explicitly on time. They satisfy equation d dt C = ∂ t C − i[H, C] = 0, C = D, K, and their explicit form is given by The operator D generates dilatations, while K is the generator of the special conformal transformations. The conformal algebra so(2, 1) is established by the formal commutation relations The domain D c where the commutators are well defined has to be specified. It has to be located in the intersection of the domains of all the involved operators. Also, the action of each of the operators H, K, and D has to keep the wave function in the domains of the two remaining operators. For t = 0, the explicit form of D and K in polar coordinates is This set is dense in L 2 (R 2 ) as it contains smooth functions with compact support (C ∞ 0 (R 2 )). The generalized eigenvectors (scattering states) of H do not have compact support, and are not square integrable. However, they can serve to construct the wave packets which are normalizable, and represent physical states. These square integrable functions inherit some of the properties of the scattering states; they do not belong to C ∞ 0 (R 2 ), and have a specific behavior of partial waves near the origin, dictated by D(H). We can ask whether they are present in D c . The necessary condition is that the operators K and D do not alter asymptotic behavior of the partial waves near the origin.
The domain of definition of either H AB α or H γ α is rotationally invariant. The partial waves near the origin may not be more divergent than a fixed power of r, prescribed by the domain of definition. Keeping in mind explicit form (5.3), we see that neither K nor D violate this restriction on asymptotic behavior of partial waves. Hence, the domain D c includes the physically interesting states 8 composed of the scattering states. This conclusion is not evident for other self-adjoint extentions H γ α when a general value of γ is considered. Let us just note that the invariance with respect to D is broken in general. The scale invariance is recovered for γ = 0 or γ = π when α is treated as a free parameter. For fixed value of the magnetic flux α = 1/2 mod 1, the scale symmetry appears in the whole family of self-adjoint extensions H γ 1/2 for any value of γ, see Appendix B. Therefore, the hidden supersymmetry of the systems represented by H AB α and H γ α comes hand in hand with conformal symmetry and the scale invariance in particular. Below we show that the both structures are compatible in the Lie algebraic sense, and give rise to the superconformal osp(2|2) symmetry. The operators K and D commute with R, their domains are invariant with respect to the action of R, and they can be treated as bosonic generators in the framework of the extended superalgebra. The relevant commutation and anticommutation relations have to be computed to verify that the superalgebra is closed. The computation does not depend on the actual choice of the self-adjoint extension, so that we adopt the notation H for H AB α or H γ α , and, respectively, Q 1 = P x + iεRP y for Q AB α orQ γ α , and Q 2 = −iεRQ 1 , where ε = +1 or −1 in dependence on the value of the flux α, see Eqs. (2.5) and (3.1). The self-adjoint generator of dilatations is denoted below by D. To close the superalgebra, two additional integrals of motion (explicitly dependent on time) have to be involved. In the commutator of K and Q j , there appear new integrals of motion The mixed anticommutator of Q j and S k brings a new conserved quantity, {Q 1 , S 2 } = 2F , Completing the remaining relations dictated by the superalgebra, we end up with Instead of the even generators J and R, in addition to the linear combination (5.6) we define the operator which commutes with all the other even and odd generators of superalgebra, playing the role of its central charge. The introduced operators S j , F and Z are essentially self-adjoint on their natural domains of definition, see Appendix B. Note that from relation J = F + 2Z it follows that Q i and S i are vector operators. Likewise in the case of the conformal symmetry, the actual domain of definition D sc has to be specified to make the relations (5.2) and (5.7) consistent. It has to be an intersection of the domains of the involved operators (just let us remind that D(Q 1 ) = D(Q 2 ) and D(S 1 ) = D(S 2 )), and the action of any of them has to keep the function in the intersection of the domains of the remaining operators.
The same analysis applies as in the case of D c . The domain D sc is dense in L 2 (R 2 ) as it contains the set of smooth functions with compact support. We require that neither of the operators violates asymptotics of the functions near the origin -they should maintain or increase the power of the leading term in the asymptotic expansion. This requirement is met by all the new operators S j , F and Z. Hence, the domain D sc can support physically interesting states represented particularly by wave packets. We conclude that the three self-adjoint extensions H AB α , H 0 α and H π α possess the scale invariance as a consequence of their conformal symmetry. The conformal and hidden supersymmetric structures of these systems are compatible, and lead to the superconformal symmetry. The resulting algebraic structure corresponds to the superalgebra osp(1|2) × o(2), which was observed earlier in various physical models [47,48,49,50], including a spin-1/2 particle in the presence of a magnetic vortex [50] 9 . For spin-1/2 particle systems possessing the superconformal symmetry, the role of the grading operator is played by the matrix σ 3 . We revealed here the same superalgebraic structure in the system without fermionic degrees of freedom.

Hidden supersymmetry and anyons
In early eighties, Wilczek proposed a dynamical mechanism for realization of anyons that is based on the Aharonov-Bohm effect [13]. Here we show that the anyon picture provides a rather natural interpretation for the hidden supersymmetric structure described in the previous sections.
Consider a two-anyon, planar system described by the formal Hamiltonian operator where p I = −i∂/∂ x I , r is a relative coordinate, r = x 1 − x 2 , and we set the masses of the constituents m 1 = m 2 = 4. The constituent point particles are 'statistically charged', and each carries a 'magnetic' vortex described by the statistical vector potential, In the center of mass reference frame, Hamiltonian (6.1) takes the form (1.2).
The two-anyon system may be composed from statistically interacting identical bosons, or fermions. This means that when the statistical interaction is switched off (α = 0), the constituent particles should obey either Bose, or Fermi statistics. The identity of the particles and their statistics are taken into account by requiring that under the exchange, r → − r ⇒ ϕ → ϕ + π, the wave function obeys the relation ψ α=0 (r, ϕ+ π) = ηψ α=0 (r, ϕ) with η = +1 for boson, or η = −1 for fermion constituents. This requirement is maintained when the statistical interaction is switched on. Therefore, we have Requiring the Hamiltonian (6.1) to be self-adjoint operator, its domain of definition has to be specified. The nontrivial behavior of the wave functions (6.3) near the origin, dictated by the particular choice of the self-adjoint extension, is interpreted then as a contact (zero-range) interaction of the anyons [51].
The anyon framework provides an interesting interpretation for the systems we studied in Sections 2 and 3. As follows from the discussion at the end of Section 2, Hamiltonians H AB α , H 0 α and H π α can be described by diagonal two-by-two matrices in representation where the grading operator R is given by the Pauli matrix σ 3 , see (2.29). The upper and lower components of the states in this representation correspond to π-periodic and π-antiperiodic in ϕ parts of the wave functions. Due to the described correspondence between the Hamiltonian (6.1) of the two-anyon system and the Hamiltonian (1.2), we conclude that the diagonal components Π ± H AB α and Π ± H γ α of the studied spinless systems can be understood as the Hamiltonians that describe the relative motion of the two-anyon systems. The upper component represents the system based on bosons (as it acts on π-periodic functions) while the lower-diagonal operators rule the dynamics of the system based on fermions. The self-adjoint Hamiltonians Π ± H γ α imply additional contact interaction of the anyons. Therefore, the hidden superconformal symmetry that we revealed in spinless Aharonov-Bohm system corresponds to an explicit center-of-mass supersymmetric structure of the system composed from the two two-anyon systems based on bosons and on fermions.

Discussion and outlook
Let us summarize and discuss the main results by stressing the physical aspects that are behind the revealed hidden supersymmetric structure.
For integer values of the flux, α = n, n ∈ Z, the Aharonov-Bohm system is unitary equivalent to a planar free particle system (α = 0). The latter possesses the rotation and translation symmetries generated by the angular momentum operator J, and by mutually commuting momenta operators P x and P y . In correspondence with this, Hamiltonian operator (1.2) can be factorized as H n = (P x + iǫP y )(P x − iǫP y ), (7.1) or, alternatively, can be presented as a perfect square, where the parameter ǫ can take any of two values, +1 or −1, and R is a nonlocal operator of rotation for angle π. For α = n, the formal Aharonov-Bohm Hamiltonian (1.2) can also be factorized in the form (7.1), or (7.2). However, in the case of non-integer flux, the operators P x and P y are not physical, and the translation invariance is broken, see below. Thus, for α = n, (7.1) is a purely formal factorization.
In contrast with (7.1), representation (7.2) can be well defined. A nontrivial property associated with factorization (7.2) is that for a given flux α = n, two different choices for the value of the parameter ǫ correspond to physically distinct systems. For α ∈ (−1, 0) mod 2, ǫ = +1, and α ∈ (0, 1) mod 2, ǫ = −1, factorization (7.2) corresponds to the original system H AB α investigated by Aharonov and Bohm [1,2], which is characterized by a regular at the origin behavior of the Hamiltonian eigenfunctions. Alternative choice of the values of the parameter ǫ in (7.2) gives rise to two different, exotic models given by selfadjoint Hamiltonians H γ α with γ = 0, π, which are characterized by a singular behavior at the origin of their eigenfunctions in one specific partial wave correlated with the value of the flux, see Eqs. (3.14)-(3.17). For half-integer values α = n + 1/2, both exotic systems with γ = 0 and γ = π are unitary equivalent, and like the Aharonov-Bohm model H AB α , they possess additional nonlocal integrals of motion in the form of the twisted reflection operatorsR x andR y . These nonlocal integrals together with R satisfy the same algebraic relations as the three Pauli matrices, i.e. generate a spinorial representation of su(2) realized on the states of the corresponding system.
Identifying the nonlocal operator R as the Z 2 -grading operator, we interpret the self-adjoint operator appearing in factorization (7.2) as the supercharge Q 1 , another self-adjoint supercharge is Q 2 = iRQ 1 . Therefore, for non-integer flux values, the translation symmetry of the Aharonov-Bohm system H AB n is broken, and corresponding mutually commuting generators P x and P y are substituted by nonlocal, mutually anti-commuting, odd operators Q 1 = P x + iǫRP y and Q 2 = −ǫP y + iRP x 10 . By taking into account the dynamical conformal symmetry, the revealed hidden supersymmetric structure of the spinless Aharonov-Bohm system is extended to the superconformal osp(2|2) symmetry. By this superconformal symmetry, one can relate not only the states with the same value of the angular momentum and different values of the energy, see Ref. [45], but also the states with different energy values and different in one angular momentum in correspondence with Fig. 1 [53].
We have shown that the hidden superconformal symmetry of the spinless Aharonov-Bohm system is in one-to-one correspondence with explicit center-of mass supersymmetric structure of the system composed from the two two-anyon subsystems, the composites of one of which before switching on statistical interaction (α = 0) satisfy boson statistics, while another subsystem is formed by two identical fermion particles. The exotic models given by the Hamiltonians H γ α , γ = 0, π, with nontrivial behavior of the wave functions near the origin correspond in this interpretation to the case of anyons with a contact (zero-range) interaction.
The non-physical nature of the operators P x and P y can be revealed immediately if to apply them to the Hamiltonian eigenfunction (7.3). The action of the operator P x + iP y produces a state, in which the l = 1 partial wave is multiplied by the function J |α|−1 (kr), that has a not permitted, singular behavior at the origin. Analogously, the state (P x − iP y )ψ contains a partial wave with l = 0 multiplied by the singular at the origin function J −|α| (kr). The supercharge (2.14) can be written in the form Q AB α = Π + (P x + iP y ) + Π − (P x − iP y ). Its projectors on the subspaces with even and odd l, Π ± = 1 2 (1 ± R), just annul the singularities produced by nonphysical operators P x ± iP y in corresponding partial waves. One can show that in the case of the exotic systems H γ α , γ = 0, π, considered in Section 3, the picture is similar: the operators P x ± iP y acting on the states of the domain of the Hamiltonian H γ α , in contrast with the action of the supercharges, produce the states that do not belong to the domain. This explains the mechanism of translation symmetry breaking, and its substitution for the hidden supersymmetry, as well as a purely formal character of factorization (7.1). Note also here that in the case α = n, the action of the operators P x ± iP y on the energy eigenstates (7.3) does not produce singularities, and operators P x and P y commute on the domain of the Hamiltonian H AB n . This corresponds to a unitary equivalence of the model H AB n to a free planar particle system discussed in Section 2.
Partial wave analysis applied to the wave function (7.3) gives the scattered wave with asymptotic behavior for large r, see [1,54], ψ sc → r −1/2 e ikr f (ϕ) , where the phase shifts are given by With taking into account (7.4) and (7.5), we get the relation e 2iδ 2l = e 2iδ 2l−1 . (7.10) This relation between the phase shifts reflects coherently with the picture presented on Fig. 1 a hidden supersymmetry in the scattering problem of the spinless Aharonov-Bohm model in the case α ∈ (−1, 0) mod 2. In the case α ∈ (0, 1) mod 2, index 2l − 1 on the right hand side of relation (7.10) is changed for 2l + 1 in correspondence with Fig. 2.
Finally, we note that the original Aharonov-Bohm calculation of the scattering amplitude [1], mathematically more justified in comparison with partial wave analysis, see [54], was based on separation of the wave function (7.3) into three functions, ψ = ψ 1 + ψ 2 + ψ 3 . In the case α ∈ (−1, 0), this corresponds to separation of wave function ψ in partial wave with l = 0 (ψ 3 ), and in the infinite sums with l > 0 (ψ 1 ) and l < 0 (ψ 2 ) [1,54]. For the function ψ 1 the equivalent integral representation was found in [1], that allowed the authors to find its asymptotic expansion, and then to calculate the scattering amplitude. The function ψ 1 is nothing else as the second series in (7.6). This means that the original method used in [1] is coherent with the hidden supersymmetric structure revealed in the present paper.
In conclusion, let us discuss some open problems to be interesting for further investigation. The stationary Schrödinger equation of the Aharonov-Bohm model is separable in polar coordinates. Its radial equation corresponds to stationary Schrödinger equation of Calogero model. When we specify the self-adjoint extension of the formal Hamiltonian operator H α , the self-adjoint extension of the radial part of H α is fixed as well. In other words, fixing the value of the angular momentum, the (rotationally invariant) self-adjoint extension of H α fixes the self-adjoint extensions of Calogero model [55]. In [56], Gitman et al. discussed recently the dilatation symmetry of the self-adjoint extensions of this one-dimensional system. They concluded that there are only few self-adjoint extensions of the Calogero model which possess scale invariance. We described three Aharonov-Bohm type systems, represented by H AB α and H γ α , γ = 0, π. These systems proved to be scale invariant. It is quite intriguing question, whether these two distinct symmetries, scale invariance and hidden supersymmetry, are interrelated somehow. We suppose that this is indeed the case. Verification of this hypothesis could provide a deeper insight into the physical system and its symmetries as well.
Recently, the hidden supersymmetry of the reflectionless Pöschl-Teller system was explained in [57] in the context of non-relativistic AdS/CFT correspondence [58,59]. The rather natural question is then whether some AdS/CFT holography interpretation exists for the hidden superconformal symmetry observed here.
The Aharonov-Bohm type systems described formally by H α , can have up to two bound states. The systems with negative energies were disqualified in our framework from the very beginning by requirement of the presence of a self-adjoint supercharge. This is in correlation with spontaneous breakdown of their scale invariance. However, such systems could fit into the framework of the nonlinear supersymmetry. Analysis of this possibility requires a separate consideration. We analyzed the spinless particle case. It would be interesting to consider the systems with spin degrees of freedom as well [53]. The spin one-half system would be governed by the Pauli Hamiltonian, whose diagonal components would differ formally just in the sign of the magnetic field, cf. [60,50]. This suggests that the actual self-adjoint extensions of the upper-and the lower-diagonal elements of the matrix Hamiltonian could differ in some way. The standard supersymmetry should be present then in addition to the hidden supersymmetry, at least in some particular cases. The presence of both, explicit and hidden, supersymmetries should give rise to the structure of tri-supersymmetry [26,61,53].
As we observed in Section 4, in the case of half-integer flux values there exists a three parameter family of unitary transformations (4.5), generated by R,R x andR y . These transformations do not change formal Hamiltonian H α , but interchange its self-adjoint extensions. Hence, there exists a three parametric family of self-adjoint extensions of H α which allow the existence of the hidden supersymmetry, see (4.14). We discussed few particular cases in (4.12), where the systems associated withQ γ 1/2 for γ ∈ {0, π/2, π, 3π/2} were interrelated by these unitary mappings. The family of all the self-adjoint extensions of H α is four parametric [31]. So it seems that a great part of the self-adjoint extensions of H α possesses hidden supersymmetry for semi-integer values of α. It would be interesting to clarify this point.
We investigated the question of the presence of the hidden supersymmetry in spinless quantum mechanical Aharonov-Bohm type systems. The intriguing open question is whether such a symmetry may be present in related field systems. The simplest system for such a generalization could be a non-relativistic (2+1)-dimensional model of a boson field minimally coupled to a Chern-Simons field [62,35,63]. If the hidden bosonized supersymmetry of the nature discussed here is present in such a field system, then its supersymmetrically extended (by inclusion of a fermion field) version [48] would be described by a more reach than the osp(2|2) superconformal structure [49], related to the tri-supersymmetry [26,61].
Since functions ψ(r, ϕ) ∈ D(Q) identically vanish in some neighborhood of the origin, the right hand side of Eq. (A.1) vanishes for any function φ(r, ϕ) such thatQφ(r, ϕ) ∈ L 2 R 2 . Therefore, the adjoint operator is densely defined in where AC R 2 \ {0} is a set of absolutely continuos functions in punctured plane [30].
Since the set e imϕ , m ∈ Z is a complete orthogonal system in L 2 (S 1 ), we can write where φ m (r) ∈ AC (R + \ {0}) ∩ L 2 (R + ; r dr). Then, the conditionQφ(r, ϕ) ∈ L 2 R 2 for α / ∈ Z reduces to |φ 2l (r)| = For α = β − 2l 0 ∈ Z, β ∈ {0, 1}, the partial waves φ j are subject to the following restrictions The closure Q. The minimal closed extension of Q is called the closure of this operator, which is defined as Q := Q † † . According to the previous discussion on the definition of the adjoint operator and Eq. (A.1), it follows that its domain is the set of functions f (r, ϕ) for whichQf (r, ϕ) ∈ L 2 R 2 and (see  7)). This means that the domains of definition Q † and Q are identical and the operator Q is essentially self-adjoint 11 in this case. Having in mind the note in the begining of the Appendix, we conclude that the operator Q α defined in (2.5) has unique self-adjoint extension Q AB This means that the restriction of Q † to the subspace H l0 has a larger domain than the restriction of Q to this subspace. Since these domains do not coincide, Q is not essentially self-adjoint. Let us remind that for these values of α, Q corresponds toQ α , see (3.1). Deficiency subspaces. We shall find solutions φ = φ 2l e 2liϕ + φ 2l−1 e (2l−1)iϕ of Q † φ = ±iµφ for φ ∈ D(Q † ). We can use directly the equation (2.18) for λ = ±iµ. It reduces to This differential equation has solutions of the form φ 2l (r) = C 1 K |2l+α| (µr) + C 2 I |2l+α| (µr), where I ν and K ν are the modified Bessel functions of the first and second (or, Macdonald function) kinds, respectively. The modified Bessel function of the first kind (I ν ) has to be discarded as it diverges for f → +∞, C 2 = 0 for all l. The function K ν decreases exponentially in infinity. For r → +0, it reads (A.14) We require the eigenvectors of Q † to lie in D(Q † ) and to be square integrable in particular. This requirement is met only for 0 < 2l + α < 1, i.e. for l = l 0 . Then there is one (and only one) eigenvector of Q † corresponding to each of the eigenvalues λ = ±iµ, given by Φ ± = e i2l0ϕ K 2l0+α (µr) ± e i(2l0−1)ϕ K 1−(2l0+α) (µr) .

Appendix B
Let us take γ as a free parameter. Hamiltonian H γ α is self-adjoint as it is a square of self-adjoint superchargẽ Q γ α . We can define the domain D c , see (5.4), for the current extension H γ α . It is dense in L 2 (R 2 ) as it contains infinitely smooth functions with compact support as well.
However, D c cannot accomodate the wave packets (normalizable combinations of scattering states) for general value of γ. Let us demonstrate this in the following way: we restrict α ∈ [0, 1). Let Φ 0 (r, ϕ) = φ 0 (r)+φ −1 e −iϕ be a function lying in the intersection of H 0 , D(Q γ α ) and D. It has the asymptotic behavior at the origin prescribed by (3.9). Acting with D we get Dφ 0 (r) ∼ A(1 − α)(1 + e iγ ) Γ(α) 2 1−α r −α (1 + O(r 2 )) , (B.1) We require that the resulting function does not leave the domain of definition of Q γ α . It is a necessary condition to keep the wave packets composed of scattering states from H 0 within D c . Considering α as a free parameter, this requirement can be satisfied just for γ = 0 or γ = π. There exists another possibility as well: when α = 1/2, DΦ 0 satisfies (3.9) for any value of γ. This is in agreement with our observation of Section 4, where the broader family of the self-adjoint extensions with hidden superysymmetry generator UQ 0 1/2 U −1 was revealed in the case of the half-integer flux. In particular, we discussed the systems associated withQ