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The following comparison rules for the discrete spectrum of the position-dependent mass
(PDM) Schrödinger equation are established. (i) If a constant mass

Last few decades, quantum mechanical systems with position-dependent mass (PDM) have received considerable attention. The interest stems mainly from the relevance of the PDM background for describing the physics of compositionally graded crystals [

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 [

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 [

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 [

The plan of the paper is as follows. In Section

For the PDM Schrödinger equation, the most general form of the Hamiltonian is given by [

The methods we are going to apply are valid for arbitrary dimension

To derive our main results, we need the Hellmann-Feynman theorem [

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

Suppose that the Hamiltonian

Define the Hamiltonian

Applying the Hellmann-Feynman theorem

Integrating by parts and taking into account that

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, at least one of the ambiguity parameters

The discrete eigenvalues

Let us prove the inequalities for

Integration by parts yields

Let

Moreover, it is now evident from (

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.

The three-dimensional mass distribution of the form

For this case, the discrete energy eigenvalues of the PDM Hamiltonian (

It is Theorems

In order to illustrate these inequalities, we present Figure

Energy for the states (a) (

Now, let us consider the one-dimensional mass distribution

It is worth examining how this sign indefiniteness affects the energy spectrum. To that end, we choose the harmonic-oscillator potential,

From Figure

Energy for the states (a)

In this paper, we have established the comparison theorems for the PDM Schrödinger equation. Our first theorem states that the corresponding eigenvalues of a constant-mass Hamiltonian and of a BenDaniel-Duke PDM Hamiltonian with the same potential are ordered if the constant and position-dependent masses are ordered everywhere. The second theorem concerns PDM Hamiltonians with the different sets of ambiguity parameters: the BenDaniel-Duke, Li-Kuhn, and Gora-Williams Hamiltonians. It is proved that their corresponding eigenvalues are ordered if the Laplacian of the inverse mass distribution

We have applied these theorems to the PDM Coulomb and harmonic-oscillator problems and have been led to the following conclusions. First, the eigenvalues of PDM Hamiltonians other than the BenDaniel-Duke one do not have to be in the strict order with respect to the eigenvalues of the constant-mass Hamiltonian. For instance, from both Figures

The comparison rules we have found out can be employed for analyzing the energy spectra in semiconductor nanodevices; 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 [

The author thanks Dr. O. Yu. Orlyansky for discussions and a careful reading of the paper. The research was supported by Grant N0109U000124 from the Ministry of Education and Science of Ukraine which is gratefully acknowledged.