Spectral and scattering theory
for the AharonovBohm operators
Abstract
We review the spectral and the scattering theory for the AharonovBohm model on . New formulae for the wave operators and for the scattering operator are presented. The asymptotics at high and at low energy of the scattering operator are computed.
Laboratoire de Mathématiques d’Orsay, CNRS UMR 8628, Université ParisSud XI, Bâtiment 425, 91405 Orsay Cedex, France;
Email:Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, United Kingdom; Email:
On leave from Université de Lyon, Université Lyon I, CNRS UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622 Villeurbanne Cedex, France
1 Introduction
The AharonovBohm (AB) model describing the motion of a charged particle in a magnetic field concentrated at a single point is one of the few systems in mathematical physics for which the spectral and the scattering properties can be completely computed. It has been introduced in [3] and the first rigorous treatment appeared in [23]. A more general class of models involving boundary conditions at the singularity point has then been developed in [2, 10] and further extensions or refinements appeared since these simultaneous works. Being unable to list all these subsequent papers, let us simply mention few of them : [25] in which it is proved that the AB models can be obtained as limits in a suitable sense of systems with less singular magnetic fields, and [24] in which it is shown that the low energy behavior of the scattering amplitude for two dimensional magnetic Schrödinger operators is similar to the scattering amplitude of the AB models. Concerning the extensions we mention the papers [12] which considers the AB operators with an additional uniform magnetic field and [18] which studies the AB operators on the hyperbolic plane.
The aim of the present paper is to provide the spectral and the scattering analysis of the AB operators on for all possible values of the parameters (boundary conditions). The work is motivated by the recent result of one of the authors [22] showing that the AB wave operators can be rewritten in terms of explicit functions of the generator of dilations and of the Laplacian. However, this astonishing result was partially obscured by some too complicated expressions for the scattering operator borrowed from [2] and by a certain function presented only in terms of its Fourier transform. For those reasons, we have decided to start again the analysis from scratch using the modern operatortheoretical machinery. For example, our computations do not involve an explicit parametrization of which leads in [2] or in [10] to some unnecessary complications. Simultaneously, we recast this analysis in the uptodate theory of selfadjoint extensions [9] and derive rigorously the expressions for the wave operators and the scattering operator from the stationary approach of scattering theory as presented in [27].
So let us now describe the content of this review paper. In Section 2 we introduce the operator which corresponds to a Schrödinger operator in with a type magnetic field at the origin. The index corresponds to the total flux of the magnetic field, and on a natural domain this operator has deficiency indices . The description of this natural domain is recalled and some of its properties are exhibited.
Section 3 is devoted to the description of all selfadjoint extensions of the operator . More precisely, a boundary triple for the operator is constructed in Proposition 3. It essentially consists in the definition of two linear maps from the domain of the adjoint of to which have some specific properties with respect to , as recalled at the beginning of this section. Once these maps are exhibited, all selfadjoint extensions of can be labeled by two matrices and satisfying two simple conditions presented in (7). These selfadjoint extensions are denoted by . The field and the Weyl function corresponding to the boundary triple are then constructed. By taking advantage of some general results related to the boundary triple’s approach, they allow us to explicit the spectral properties of in very simple terms. At the end of the section we add some comments about the role of the parameters and and discuss some of their properties.
The short Section 4 contains formulae on the Fourier transform and on the dilation group that are going to be used subsequently. Section 5 is the main section on scattering theory. It contains the time dependent approach as well as the stationary approach of the scattering theory for the AB models. Some calculations involving Bessel functions or hypergeometric functions look rather tricky but they are necessary for a rigorous derivation of the stationary expressions. Fortunately, the final expressions are much more easily understandable. For example, it is proved in Proposition 10 that the channel wave operators for the original AB operator are equal to very explicit functions of the generator of dilation. These functions are continuous on and take values in the set of complex number of modulus . Theorem 11 contains a similar explicit description of the wave operators for the general operator .
In Section 6 we study the scattering operator and in particular its asymptotics at small and large energies. These properties highly depend on the parameters and but also on the flux of the singular magnetic field. All the various possibilities are explicitly analysed. The statement looks rather messy, but this simply reflects the richness of the model.
The parametrization of the selfadjoint extensions of with the pair is highly non unique. For convenience, we introduce in the last section a onetoone parametrization of all selfadjoint extensions and explicit some of the previous results in this framework. For further investigations in the structure of the set of all selfadjoint extensions, this unique parametrization has many advantages.
Finally, let us mention that this paper is essentially selfcontained. Furthermore, despite the rather long and rich history of the AharonovBohm model most of the our results are new or exhibited in the present form for the first time.
Remark 1.
After the completion of this paper, the authors were informed about the closely related work [7]. In this paper, the differential expression on is considered and a holomorphic family of extensions for is studied. Formulae for the wave operators similar to our formula (14) were independently obtained by its authors.
2 General setting
Let denote the Hilbert space with its scalar product and its norm . For any , we set by
corresponding formally to the magnetic field ( is the Dirac delta function), and consider the operator
Here denotes the set of smooth functions on with compact support. The closure of this operator in , which is denoted by the same symbol, is symmetric and has deficiency indices [2, 10]. For further investigation we need some more information on this closure.
So let us first decompose the Hilbert space with respect to polar coordinates: For any , let be the complex function defined by . Then, by taking the completeness of the family in into account, one has the canonical isomorphism
(1) 
where and denotes the one dimensional space spanned by . For shortness, we write for , and often consider it as a subspace of . Clearly, the Hilbert space is isomorphic to , for any
In this representation the operator is equal to [10, Sec. 2]
(2) 
with
and with a domain which depends on . It clearly follows from this representation that replacing by , , corresponds to a unitary transformation of . In particular, the case is equivalent to the magnetic fieldfree case , i.e. the Laplacian and its zerorange perturbations, see [4, Chapt. 1.5]. Hence throughout the paper we restrict our attention to the values .
So, for and , the domain is given by
For , let denote the Hankel function of the first kind and of order , and for let stand for the Wronskian
One then has
where and . It is known that the operator for are selfadjoint on the mentioned domain, while and have deficiency indices . This explains the deficiency indices for the operator .
The problem of the description of all selfadjoint extensions of the operator can be approached by two methods. On the one hand, there exists the classical description of von Neumann based on unitary operators between the deficiency subspaces. On the other hand, there exists the theory of boundary triples which has been widely developed for the last twenty years [9, 11]. Since our construction is based only on the latter approach, we shall recall it briefly in the sequel.
Before stating a simple result on for let us set some conventions. For a complex number , the branch of the square root is fixed by the condition . In other words, for with and one has . On the other hand, for we always take the principal branch of the power by taking the principal branch of the argument . This means that for with and we have . Let us also recall the asymptotic behavior of as in and for :
(3) 
Proposition 2.
For any with , the following asymptotic behavior holds:
Proof.
Let us set , and recall that implies and that the Hankel function satisfies . By taking this and the asymptotics (3) into account, the condition implies that
(4) 
and that
(5) 
Multiplying both terms of (5) by and subtracting it from (4) one obtains that
(6) 
On the other hand, considering (5) as a linear differential equation for : , and using the variation of constant one gets for some :
Now Eq. (6) implies that , and by using l’Hôpital’s rule, one finally obtains:
3 Boundary conditions and spectral theory
In this section, we explicitly construct a boundary triple for the operator and we briefly exhibit some spectral results in that setting. Clearly, our construction is very closed to the one in [10], but this paper does not contain any reference to the boundary triple machinery. Our aim is thus to recast the construction in an uptodate theory. The following presentation is strictly adapted to our setting, and as a general rule we omit to write the dependence on on each of the objects. We refer to [9] for more information on boundary triples.
Let be the densely defined closed and symmetric operator in previously introduced. The adjoint of is denoted by and is defined on the domain
Let , be two linear maps from to . The triple is called a boundary triple for if the following two conditions are satisfied:


the map is surjective.
It is proved in the reference mentioned above that such a boundary triple exists, and that all selfadjoint extensions of can be described in this framework. More precisely, let be matrices, and let us denote by the restriction of on the domain
Then, the operator is selfadjoint if and only if the matrices and satisfy the following conditions:
(i) is selfadjoint, (ii) .  (7) 
Moreover, any selfadjoint extension of in is equal to one of the operator .
We shall now construct explicitly a boundary triple for the operator . For that purpose, let us consider and choose with . It is easily proved that the following two functions and define an orthonormal basis in , namely in polar coordinates:
where is the normalization such that . In particular, by making use of the equality
valid for , one has
Let us also introduce the averaging operator with respect to the polar angle acting on any and for almost every by
Following [10, Sec. 3] we can then define the following four linear functionals on suitable :
For example, by taking the asymptotic behavior (3) into account one obtains
(8)  
with
(9) 
The main result of this section is:
Proposition 3.
The triple , with defined on by
is a boundary triple for .
Proof.
We use the schema from [8, Lem. 5]. For any let us define the sesquilinear forms
and
We are going to show that these expressions are well defined and that .
i) Clearly, is well defined. For , let us first recall that and are well defined on the elements of and . We shall now prove that for , which shows that is also well defined on . In view of the decomposition (2) it is sufficient to consider functions of the form for any and with . Obviously, for such a function with one has for almost every , and thus . For the equalities follow directly from Proposition 2. . It has already been proved above that the four maps
ii) Now, since for all , the only non trivial contributions to the sesquilinear form come from . On the other hand one also has for . Thus, we are reduced in proving the equalities
for any and .
Observe first that for and arbitrary one has
since . Now, for one has , and hence . For one easily calculate with that
and then .
We now consider and . One has
and again . It then follows that .
So it only remains to show that . For that purpose, observe first that
On the other hand, one has
with . By inserting (9) into this expression, one obtains (with and )
Finally, by taking the equality
into account, one obtains
which implies .
iii) The surjectivity of the map follows from the equalities (8). ∎
Let us now construct the Weyl function corresponding to the above boundary triple. The presentation is again adapted to our setting, and we refer to [9] for general definitions.
As already mentioned, all selfadjoint extensions of can be characterized by the matrices and satisfying two simple conditions, and these extensions are denoted by . In the special case , then is equal to the original AharonovBohm operator . Recall that this operator corresponds to the Friedrichs extension of and that its spectrum is equal to . This operator is going to play a special role in the sequel.
Let us consider and . It is proved in [9] that there exists a unique with . This solution is explicitly given by the formula: with
The Weyl function is then defined by the relation . In view of the previous calculations one has
In particular, one observes that for one has .
In terms of the Weyl function and of the field the Krein resolvent formula has the simple form:
(10) 
for . The following result is also derived within this formalism, see [5] for i), [11, Thm. 5] and the matrix reformulation [13, Thm. 3] for ii). In the statement, the equality has already been taken into account.
Lemma 4.

The value is an eigenvalue of if and only if , and in that case one has

The number of negative eigenvalues of coincides with the number of negative eigenvalues of the matrix .
We stress that the number of eigenvalues does not depend on , but only on the choice of and .
Let us now add some comments about the role of the parameters and and discuss some of their properties. Two pairs of matrices and satisfying (7) define the same boundary condition (i.e. the same selfadjoint extension) if and only if there exists some invertible matrix such that and [20, Prop. 3]. In particular, if satisfies (7) and if , then the pair defines the same boundary condition (and is selfadjoint). Hence there is an arbitrariness in the choice of these parameters. This can avoided in several ways.
First, one can establish a bijection between all boundary conditions and the set of the unitary matrices by setting
(11) 
see a detailed discussion in [14]. We shall comment more on this in the last section.
Another possibility is as follows (cf. [21] for details): There is a bijection between the set of all boundary conditions and the set of triples , where , is an identification map (identification of as a linear subspace of ) and is a selfadjoint operator in . For example, given such a triple the corresponding boundary condition is obtained by setting
with respect to the decomposition . On the other hand, for a pair satisfying (7), one can set with , is the identification map of with and . In this framework, one can check by a direct calculation that for any such that is invertible, one has
(12) 
where is the adjoint of , i.e. the composition of the orthogonal projection onto together with the identification of with .
Let us finally note that the conditions (7) imply some specific properties related to commutativity and adjointness. We shall need in particular:
Lemma 5.
Let satisfies (7) and with . Then

The matrices and are invertible,

The equality holds.
Proof.
i) By contraposition, let us assume that . Passing to the adjoint, one also has , i.e. there exists such that . By taking the scalar product with one obtains that . The righthand side is real due to (i) in (7). But since , the equality is possible if and only if . It then follows that , which contradicts (ii) in (7). The invertibility of can be proved similarly.
ii) If , then the matrix is selfadjoint and it follows that
If , then the equality is trivially satisfied. Finally, if but one has . Furthermore, let us define by and let be its adjoint map. Then, by the above construction there exists such that . It is also easily observed that is just the multiplication by some with , and hence . Similarly one has . Taking the adjoint of the first expression leads directly to the expected equality. ∎
4 Fourier transform and the dilation group
Before starting with the scattering theory, we recall some properties of the Fourier transform and of the dilation group in relation with the decomposition (1). Let be the usual Fourier transform, explicitly given on any and by
where denotes the convergence in the mean. Its inverse is denoted by . Since the Fourier transform maps the subspace of onto itself, we naturally set by the relation for any . More explicitly, the application is the unitary map from to given on any and almost every by
where denotes the Bessel function of the first kind and of order . The inverse Fourier transform is given by the same formula, with replaced by .
Now, let us recall that the unitary dilation group is defined on any and by
Its selfadjoint generator is formally given by , where is the position operator and is its conjugate operator. All these operators are essentially selfadjoint on the Schwartz space on .
An important property of the operator is that it leaves each subspace invariant. For simplicity, we shall keep the same notation for the restriction of to each subspace . So, for any , let be an essentially bounded function on . Assume furthermore that the family is bounded. Then the operator defined on by is a bounded operator in .
Let us finally recall a general formula about the Mellin transform.
Lemma 6.
Let be an essentially bounded function on such that its inverse Fourier transform is a distribution on . Then, for any one has
where the r.h.s. has to be understood in the sense of distributions.
Proof.
The proof is a simple application for of the general formulae developed in [15, p. 439]. Let us however mention that the convention of this reference on the minus sign for the operator in its spectral representation has not been adopted. ∎
As already mentioned leaves invariant. More precisely, if for some , then with