On a quantum Hamiltonian in a unitary magnetic field with axisymmetric potential
We study a magnetic Schrödinger Hamiltonian, with axisymmetric potential in any dimension. The associated magnetic field is unitary and non constant. The problem reduces to a 1D family of singular Sturm-Liouville operators on the half-line indexed by a quantum number. We study the associated band functions. They have finite limits that are the Landau levels. These limits play the role of thresholds in the spectrum of the Hamiltonian. We provide an asymptotic expansion of the band functions at infinity. Each Landau level concerns an infinity of band functions and each energy level is intersected by an infinity of band functions. We show that among the band functions that intersect a fixed energy level, the derivative can be arbitrary small. We apply this result to prove that even if they are localized in energy away from the thresholds, quantum states possess a bulk component. A similar result is also true in classical mechanics.
- 1 Reduction to one-dimensional Hamiltonians
- 2 Basics about the eigenpairs of the fiber operator
- 3 Asymptotic behaviour of the band functions
- 4 Velocity operator
The motion of a spinless quantum particle in is described by the spectral properties of the associated Hamiltonian. When the particle moves in a magnetic field, it is the magnetic Laplacian acting on , where is a magnetic potential.
One of the simplest example of magnetic field is the constant one. In the case , this model has been studied from the beginning of quantum mechanics [Lan77] and also more recently for the general case [Hel96, Dim01].
The variations of a non constant field can induce transport properties for the particle. In this context, we focus on magnetic fields that are translationally invariant along one direction. For such fields, the Hamiltonian has a band structure and transport properties in the direction of invariance are linked to the study of band functions (also called dispersion curves) that are the eigenvalues of the fibered operators. Moreover, the propagation of the particle in this direction is determined by the derivatives of these band functions that play the role of group velocities [Yaf08, Exn99].
In the case , one of the studied models of this class is the Iwatsuka model [Iwa85, Man97]. For , similar models are the planar translationally invariant magnetic fields [Yaf08, Rai08]. Let denote the cylindrical coordinates of . The potential takes the form , where is the intensity of the potential. The associated magnetic field is therefore given by
Thus this field is planar and its norm is . Moreover the associated field lines are circles contained in planes with center on the invariant axis (see Figure 1).
In view of the form of the magnetic field (1), two specific cases are relevant. The first model consists of a magnetic field generated by an infinite rectilinear wire bearing a constant current [Yaf03, Bru15]. If we assume that the wire coincides with the axis, then the Biot & Savard law states that the generated magnetic fields writes as the field (1) for the intensity . Here all the band functions are decreasing from to . Hence the spectrum of is . The band functions tend exponentially to and it provides a reaction of the ground state energy of under an electric perturbation [Bru15]. Moreover the particle has a preferable direction of propagation along the axis [Yaf03].
It is also natural to consider the case of a unitary magnetic field. For the field (1), it corresponds to the intensity . In this case the band functions tend to finite limits that are the Landau levels [Yaf08, Proposition 3.6]. Therefore the bottom of the spectrum of is positive. An approximated value has been calculated and used to compare the energy on a wedge in a magnetic model and the one coming from the regular part of the wedge [Pop12, Pop15].
In this article we continue to study this magnetic field in the case and we generalize the framework to any dimension . In particular we will show that the derivatives of the band functions possess an new type of behavior.
Spectral decomposition of the Hamiltonian and description of the model
For every , we set and we define the magnetic potential by
We define the Hamiltonian as the following operator, self-adjoint in :
In order to define the magnetic field we consider, we identify this potential with the -differential form . We define the magnetic field as . We calculate , . Therefore is unitary since [Hel96, Section 1].
After a partial Fourier transform in the variable, is unitarily equivalent to the direct integral in of the family of operators , self-adjoint in and defined by
Moreover for any frequency , reduces to the orthogonal sum over (called the magnetic quantum numbers) of operators self-adjoint in and defined by
The spectrum of each is discrete. Let , be the increasing sequence of its eigenvalues. The are the band functions (also called dispersion curves).
We say that an operator is fibered [Ree78d, Section XIII.16] if it can be written as
with a -finite measure space. An important class of fibered operators is the one of analytically fibered operators introduced in [Ger98a]. In this framework, is a real analytic manifold and thresholds can be defined [Ger98a, Section 3]. They form a discrete set and their definition remains to stratification. Moreover away from them, some spectral results are rather standard. For example a limiting absorption principle as well as propagation estimates hold [Ger98a, Theorem 3.3] and it is tied to Mourre estimates. For a fibered operator , we define the energy-momentum set as
One of the conditions for the operator to be analytically fibered in this sense is that the projection defined as is proper. Finally, notice that if is a 1-dimensional manifold, then these thresholds correspond to the critical points of the band functions and can be referred to as attained thresholds [Geu97, Hel01, Soc01, Bri09].
Other examples of fibered magnetic models can be found in the literature, in dimension 2 [Iwa85], on the half-plane [Bru14] or in dimension 3 [Yaf08]. In these models, the considered Hamiltonian is also fibered along and the band functions tend to finite limits. The sets of frequencies associated with the energy levels concentrated in the neighborhood of these limits are unbounded. Hence the previous projection, , is not proper. So these magnetic models are not contained in the class of analytically fibered operators that we described above. Nevertheless thresholds can still be defined as the limits of the band functions.
The model described in this article remains in this case. Indeed it is already known that the band functions tend to the Landau levels [Yaf08, Proposition 3.6]. Our first goal is to precise the convergence of the band functions to these levels. To that aim we provide an asymptotic expansion for as (see Theorem 3.1). The method used to prove this theorem is inspired by the method of quasi-modes [Dim99] that has already been used in the proof of similar result [Bru15, His16].
For the previous magnetic models, some studies of classical spectral problems already exists [Man97, Deb99, His15, His16, Pop16]. Our model contains one additional challenge. Actually for the Iwatsuka model and for half-plane model, each limit corresponds to a finite number of band functions. In this article, each threshold is the limit of all the band functions for . Therefore any interval of energy is intersected by an infinity of band functions (see equation (3.15)) and the set of frequencies associated with (even if is away from the Landau levels) is unbounded (see proposition 3.2). Furthermore we will prove in Theorem 3.2 that even if is away from the Landau levels, the group velocity tends to at any energy as . Therefore it is not clear at first sight that the Mourre estimates used in the case of the analytically fibered operators still hold. The proof of Theorem 3.2 use a convenient formula for the derivative (see Proposition 2.2) which links it to the normalized eigenfunctions of operators as well as an exponential decay of these eigenfunctions that is uniform with respect to and relies on Agmon estimates.
These properties have consequences on the transport properties associated with the magnetic field that we consider. Indeed these propreties are determined from the behavior of the velocity operator (see Definition (4.3)) that is linked to the multiplication operator by the family of the derivatives of the band functions (see formula (4.4)). In the previous models [Iwa85, Bru14], the velocity operator is bounded from below by a positive constant in any energy interval , away from the thresholds. Hence every quantum state localized in energy away from them carries a non trivial current [Man97, Deb99] and is called edge state. Such an estimate does not hold anymore if the infimum of the derivatives of the band functions on is . This situation occurs in the previous cases when there is a threshold in [His16] and according to Theorem 3.2, everywhere in the model of our work. In this sense the definition of “thresholds” as the Landau levels seems not to be relevant. Indeed for the Iwatsuka model and for the Half-plane model, only the quantum state that are localized in energy around the thresholds possess a bulk component [Man97, His16]. In the case of the model considered here, we show in Theorem 4.1 that any quantum state, even if it is localized in energy away from the Landau levels can be decomposed into one state that carries an arbitrary small current and one that carries a non trivial one.
In classical mechanics, such a magnetic fields also induces transport properties. Indeed a charged particle follows the Newton law . This equation can be integrated [Yaf03, Section 4] and we ploted the classical trajectories (Figure 2) in the case . We can observe that the particle propagate in the direction and one can show that it has an effective velocity in this direction: there is a constant such that [Yaf03, Theorem 4.2]. Furthermore, denote by the total energy of the particle and by its areal velocity that are constants fixed by the initial conditions. These quantities satisfy the estimate . Moreover, for , with , one can find initial conditions such that is the energy of the particle and its areal velocity. Therefore one can find initial conditions such that is arbitrarily small, namely such that the particle propagate arbitrarily slowly along the axis.
In Sections 1, the Hamiltonian is reduced to a family of 1D singular Sturm-Liouville operators. The band functions are introduced and described in Section 2. Section 3 presents the results concerning the asymptotic behaviors of these band functions. In Subsection 3.1, we prove Theorem 3.1 that provides an asymptotic expansion of as gets large. Subsection 3.2 presents the asymptotic study of the derivative. In particular, Theorem 3.2 provides the asymptotic behavior of as and as is fixed far from the Landau level . In Section 4, we analyze the current carried by quantum states that are localized in energy far from the thresholds.
1 Reduction to one-dimensional Hamiltonians
In this section we define precisely the operators that we consider and we explain how is reduced to 1 dimensional operators.
This form, initially defined on , is semi-bounded from below. Thus it admits a Friedrichs extension: . Let be the quadratic form defined by
This form, initially defined on and then closed in , is the quadratic form associated with the operator (4). Denotes by the Fourier-transform with respect to , which is defined by
The forms and are related through the relation
Therefore the operator is decomposed as follows:
We now reduce the problem to a -dimensional one using both the cylindrical symmetry and the following Laplace-Beltrami formula:
Recall that is essentially self-adjoint on and that its spectrum is discrete. Its eigenvalues are , . Denote by the corresponding eigenspaces. Remember that has finite dimension: . The spaces are invariant by . In addition, the restrictions of the operator on these spaces are identified with the operators
These operators act on . They are associated with the bilinear forms
Denote by the angular Fourier transform. The operator is decomposed as:
Finally, it is more convenient to consider operators acting on the Hilbert space . To proceed we use the isometry defined by . We define as
and the functions as
So where is defined by
This operator acts on with domain . It is associated with the quadratic form
2 Basics about the eigenpairs of the fiber operator
In this section we prove that the dispersion curves are analytic functions, we calculate their derivative and we investigate the behavior of the eigenfunctions at .
2.1 Behavior of the eigenfunctions at
First we investigate the behavior of the functions of at , namely:
Lemma 2.1 ()
Let , and .
The bilinear form associated with is given by relation (1.1). For every and every , we have . Notice that . We integrate by part the first term of the form which yields:
We apply this formula to an arbitrary function and to functions that satisfy for any
We deduce that
Therefore integrating this condition, we deduce that
Thus remembering that , we conclude that relation (2.1) holds. ∎
Note that . So if then awing to a Sobolev embedding, . Hence . Thus if , then is bounded as . Combine it with the fact that and it provides the embedding (2.2).
Notice that as . Therefore the operator has compact resolvent. So for every and for every the spectrum of is an increasing sequence of positive eigenvalues , . We conclude this subsection by proving the following proposition.
Proposition 2.1 (Behavior of the eigenfunctions at )
Let , and . The eigenvalue is non-degenerate. Let be the normalized eigenfunction associated with it. There exists an analytic function such that and such that in a neighborhood of ,
First, consider the differential equation
We look for solutions that admit a series expansion in a neighborhood of . By the Frobenius method, if a solution is given by where is an analytic function such that , then satisfies the indicial equation
This equation has as solutions. Thus the equation (2.4) admits a solution of the form with an analytic function such that . In order to have a basis of solutions for equation (2.4) we look for a solution of the form . By straightforward calculations we find that as , so
if , then ,
in the other cases, .
Finally, we deduce from Lemma 2.1 that in both cases . Hence . This concludes the proof since is an eigenvalue of . ∎
Remark 2.1: We deduce from this proposition that the embedding (2.2) is optimal.
2.2 Derivative of the band functions
Here we give a formula for the derivative of the band functions.
Proposition 2.2 ()
Let, for , . The derivative is given by:
In the case , this proposition has already been proved [Yaf08, Theorem 4.3]. The way to prove it in the general case is the same as in this particular case so we refer to this proof for more details. We still present the main ideas of the proof.
The Feynman-Hellmann formula [Ism88] yields that
We apply integrations by parts to get the result. We use the super-exponential decay of eigenfunctions for handling the non-integral terms corresponding to [Shn57, Olv97] and Proposition 2.1 for handling the non-integral term at . In the particular case , the result of Proposition 2.1 is not sharp enough. In order to improve it, we inject the identity (2.3) into the following eigenvalue equation:
Therefore we obtain that as and we use it for handling non-integral term at . ∎
2.3 Global behavior of the band functions
The min-max principle implies that
Indeed first note that if , then . Therefore
On the other hand, we define for the operator , self-adjoint on ,
This operator has compact resolvent, therefore its spectrum is discrete. Let be the increasing sequence of its eigenvalues. Note that . Hence, for any , . Thus,
From Proposition 2.2 we deduce that if , then for every is negative on . Therefore in this case the band functions are decreasing. So these functions admit finite limits at . In the case the min-max principle yields that these limits are the Landau levels [Yaf08, Proposition 3.6], namely
This proof is still valid if and Subsection 3.1 provides an asymptotic expansion of when tends to . In the case then . Therefore we will deduce from Theorem 3.1 (see remark 3.1) that for every admits local minima (the question of the number of minima stays open). In the other cases, according to Proposition 2.2, for every is decreasing from to .
We use a finite difference method to compute numerical approximations of the band function with , and . We compute for on the interval with an artificial Dirichlet boundary condition at .
On Figure 3, we have ploted the numerical approximation of for , and . According to the theory, decrease from to . We also ploted this level. Note that different band function may intersect for different values of .
Figure 4 presents a zoom on the first level: for and .
3 Asymptotic behaviour of the band functions
In this section we provide an asymptotic expansion for the band functions and their derivative. First we provide an asymptotic expansion for as with and fixed. In a second time we estimate the behavior of as is fixed and as and tend to and are related to eachother by the condition where is a constant.
3.1 Near thresholds: high frequency
In this subsection we study the behaviour of the spectrum of near the thresholds. Namely we describe the behaviour of when and are fixed and . More precisely, this subsection is devoted to the proof of the following theorem.
Theorem 3.1 (Asymptotic expansion of the band functions)
Let and . There is a sequence of real numbers such that
To prove this theorem we consider the operators defined by relation (1.4) and we apply the method of the harmonic approximation [Hel88, Dim99] to derive an asymptotic expansion of its eigenvalues.
Remark 3.1: In the case , that is , Theorem 3.1 states that , as . In this case, the operator is with Dirichlet boundary condition at . This operator has already been studied and we know [His16, Theorem 1.4] [Ivrii, Section 15.A] that there are some constant such that
So we focus on the proof in the particular case .
Remark 3.2: We compute that and . Therefore for , Theorem 3.1 yields
In the case and , . Therefore for every , tend to from below. Hence the have local minima.
Canonical transformation and asymptotic expansion of the operator
For we apply the change of variable . It shows that is unitarily equivalent to the following operator acting on :
A Taylor expansion of the potential for large provides
Estimation on the remainder term will be written later (see equation (3.8)). We define a sequence of formal operators by
For every , we set
with the convention . We set . For every , the operator can be formally decomposed into:
First we look for quasi-modes for the formal operator acting on . This formal procedure provides functions defined on and we use a suitable cut-off function in to derive quasi-modes for .
Calculation of the quasi-modes
We look for quasi-eigenpairs of of the form
where the functions are mutually orthogonal in . Note that the functions may depend on . We are led to solve the system
We solve it by induction:
Note that is the quantum harmonic oscillator. Hence we choose for a couple for where is a Landau level, and is the corresponding normalised Hermite function with the convention that . So from now on we set for a certain , fixed. All the quantities considered in what follows may depend on the choice of . We simplify the notations with ommiting this index.
We assume that there exists such that for every , and have been constructed.
The scalar product of the second equation of the system (3.3) with provides the value of :
So is known, therefore the Fredholm alternative provides a unique value for such that for every .
The quasi-modes can be computed using the Hermite functions. The Hermite functions satisfy the following results
Combining them with the system (3.3) we infer that for every , there exist polynomial functions such that
Evaluation of the quasi-mode
Previously we have obtained quasi-eigenpairs for . The functions are defined on . We now use a suitable cut-off function to get quasi-modes for .
Let such that
For , we define the cut-off function on by
Note that this function is supported in and is equal to on . Let, for , be defined by
Since , can be used as a quasi-mode for .
Lemma 3.1 (Control of the quasi-mode)
Let . Recalling that , and are fixed, there is a constant such that
First, observe that
We proceed to control the right hand side term by term:
We use the definition of to compute the first term:
Thus we deduce that
Note that may depend on .
Finally notice that . Moreover, and are supported in . Therefore we deduce from formula (3.4) that
Proof of Theorem 3.1
We deduce from the spectral theorem and from Lemma 3.1 that