Comparison theorems for the position-dependent mass Schroedinger equation

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schroedinger equation are established. (i) If a constant mass $m_0$ and a PDM $m(x)$ are ordered everywhere, that is either $m_0\leq m(x)$ or $m_0\geq m(x)$, then the corresponding eigenvalues of the constant-mass Hamiltonian and of the PDM Hamiltonian with the same potential and the BenDaniel-Duke ambiguity parameters are ordered. (ii) The corresponding eigenvalues of PDM Hamiltonians with the different sets of ambiguity parameters are ordered if $\nabla^2 (1/m(x))$ has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.

However, it is known that the PDM Schrödinger equation suffers from ambiguity in operator ordering, caused by the non-vanishing commutator of the momentum operator and the PDM. The PDM Hamiltonians with different ambiguity parameters have been proposed [17][18][19][20], but none of them can be preferred according to the existing reliability tests [21][22][23]. Therefore the attempts are made to settle the issue by fitting the calculated binding energies to the experimental data [24,25].
For generelizing such findings and obtaining additional information, one needs some tools to compare the energy eigenvalues predicted by the different PDM Hamiltonians. Within the constant-mass framework, a convenient tool is provided by the so-called comparison theorems [26][27][28]. For example, the elementary comparison theorem [26,28] states that if two real potentials are ordered, V (1) ≤ V (2) , then each corresponding pair of eigenvalues is ordered, E (1) ≤ E (2) .
The purpose of this paper is to establish the comparison theorems that confront the energy eigenvalues of the constant-mass and PDM Schrödinger equations, as well as the energy eigenvalues of the PDM problems with different ambiguity parameters.
Our presentation is based on the Hellmann-Feynman theorem [30] and makes use of the ideas developed for the constant-mass case [28,29].
The plan of the paper is as follows. In Section 2 we introduce the PDM Hamiltonians and recall the Hellmann-Feynman theorem. In Section 3 the comparison theorems on the PDM background are formulated and proved. In Section 4 we apply these theorems to two PDM problems of current interest. Finally, our conclusions are summarized in Section 5.
The methods we are going to apply are valid for arbitrary dimension N. We suppose that the Hamiltonian operators have domains D(H) ⊂ L 2 (R N ), they are bounded below, essentially self-adjoint, and have at least one discrete eigenvalue at the bottom of the spectrum.
To derive our main results, we need the Hellmann-Feynman theorem [30]. This theorem states that if the Hamiltonian of a system is H(a), where a is a parameter, and the eigenvalue equation for a bound state is H(a)|a = E(a)|a , where E(a) is the energy and |a the normalized associated eigenstate, then Note that the proof relies on the self-adjointness of H(a) and does not change for PDM Hamiltonians.

Comparison theorems
First, let us formulate the theorem that confronts the energy eigenvalues of the constant-mass and BenDaniel-Duke PDM Hamiltonians with the same potentials.

Theorem 1
Suppose that the Hamiltonian with a real potential V (x) and a constant-mass m 0 has discrete eigenvalues E satisfy provided that these eigenvalues exist.
Proof: Define the Hamiltonian which turns into H Applying the Hellmann-Feynman theorem (2), we get where the integration is performed over the whole space and the asterisk denotes complex conjugation.
Integrating by parts and taking into account that ψ {n} (x; a) and ∇ψ {n} (x; a) must vanish at infinity, we obtain It is a positive (negative) that completes the proof. Note that an alternative proof can be given by applying the variational characterization [31] of the discrete part of the Schrödinger spectrum.
It is now tempting to compare the eigenvalues of the constant-mass Hamiltonian with those of PDM Hamiltonians other than the BenDaniel-Duke one. However, in that case we encounter an obstacle that becomes clear if we first find out how the eigenvalues of different PDM Hamiltonians are ordered. This is done in the following theorem.

Theorem 2
The discrete eigenvalues E satisfy provided that these eigenvalues exist.
Proof: Let us prove the inequalities for E and make use of the Hellmann-Feynman theorem (2), to obtain Integration by parts yields Let ∇ 2 (1/m(x)) ≤ 0 for all x, then E {n} (a) is an increasing function and we get that completes the proof. For the case of E {n} , the proof is identical since the factor ∇ 2 (1/m(x)) arises in this case as well.
It is now evident from (9) and (18)

Applications
In this section, we consider two specific PDM problems, which are discussed in literature, and show how the comparison theorems explain the peculiarities of their energy spectra.

Case 1
The three-dimensional mass distribution of the form with r = |x| and nonnegative κ, has been shown [16] to give rise to an exactly- where l = 0, 1, ... and n = l+1, l+2, ... are the orbital and principal quantum numbers, respectively. In contrast to the constant-mass Coulomb problem, the system has only a finite number of discrete levels, so that the allowed values of l and n are restricted Such a restriction implies that in presence of the PDM the energy eigenvalues may be closer to continuum and thus larger than the ordinary Coulomb eigenenergies In order to illustrate these inequalities, we present figure 1 where we plot the energy for the ground state (n = 1, l = 0) and the first radially excited state (n = 2, l = 0), as a function of the deforming parameter κ. In figure 1 the solid lines correspond to the constant-mass case whereas the broken curves represent the PDM cases with different ambiguity parameters. The circles indicate the points at which the bound states disappear according to (22). From figure 1 we see that, for all allowed κ, it holds E (BD) ≥ E (0) , as it was proved, and also E (LK) ≥ E (0) , but we observe both

Case 2
Now let us consider the one-dimensional mass distribution which is found to be useful for studying quantum wells [3]. Applying Theorem 1 to this PDM profile, we get the inequality E (BD) ≤ E (0) that justifies the shift of electron and hole binding energies to lower values which was observed in [3] when the spatial dependence of mass was included. On the other hand, Theorem 2 does not apply since the quantity ∇ 2 (1/m(x)) = κ(6κx 2 − 2)/m 0 (1 + κx 2 ) 3 has an indefinite sign.
It is worth examining how this sign indefiniteness affects the energy spectrum. To that end, we choose the harmonic-oscillator potential, V (x) = 1 2 m 0 ω 2 x 2 , for which the accurate numerical solution of the PDM Schrödinger equation with the mass distribution (23) is available [15]. In figure 2 we plot the corresponding energy of the ground and the fifth excited states, as a function of κ, for the three PDM Hamiltonians with different ambiguity parameters. The energies have been calculated withh = m 0 = ω = 1, by using the shooting method, and are in agreement with those computed in [15] where the results obtained with the same m 0 and ω, and κ = 0.1 are reported. We therefore think that for establishing further comparison rules within the PDM framework one should restrict the potential profile to, e.g., a spherically-symmetric case, the way the generalized comparison theorems for the ordinary Schrödinger equation have been obtained [27].
The comparison rules we have found out can be employed for analyzing the energy spectra in semiconductor nano devices; an example of application to the quantum well system was sketched in the previous section. In this connection, it is worthwhile to extend the present approach to periodic heterostructures, which allow the direct fit of PDM binding energies to experiment [25]. Then we will have to abandon the requirement of vanishing of the wave function at infinity which the proof of our theorems relies on. What comparison rules might be formulated in that case is an interesting open question.