Spectral intertwining relations in exactly solvable quantum-mechanical systems

In exactly solvable quantum-mechanical systems, ladder and intertwining operators play a central role because, if they are found, the energy spectra can be obtained algebraically. In this paper, we propose the spectral intertwining relation as a unified relation of ladder and intertwining operators in a way that can depend on the energy eigenvalues. It is shown that the spectral intertwining relations can connect eigenfunctions of different energy eigenvalues belonging to two different Hamiltonians, which cannot be obtained by previously known structures such as shape invariance. As an application, we find new spectral intertwining operators for the Hamiltonians of the hydrogen atom and the Rosen--Morse potential.


Introduction
What is an important structure commonly found in exactly solvable quantum-mechanical systems? The concept of ladder operators is of great interest in this question because, since Dirac's use [1], it has played a central role in quantum mechanics. As is well known, a ladder operator D for a Hamiltonian H is a linear operator that satisfies the commutation relation where ε is a real constant. In particular, it is called a raising operator if ε > 0 or a lowering operator if ε < 0 as it raises or lowers the energy eigenvalues. In fact, when D acts on an eigenfunction ψ n with energy E n , i.e., Hψ n = E n ψ n , we see that HDψ n = DHψ n + [H, D]ψ n = (E n + ε)Dψ n , (1.2) which shows that Dψ n is an eigenfunction with energy E n + ε. (Note: In this paper, we focus only on bound states, so that the energy eigenfunctions ψ n and energy eigenvalues E n are discretized and labeled by integers n. We assign 0 to the ground state and 1, 2, . . . to the excited states, unless otherwise noted.) Hence, for any n, there exists an integer m such that ψ n+m ∝ Dψ n and E n+m = E n + ε. Since it follows that, if m = 1, ψ n ∝ D n ψ 0 and E n = E 0 + nε, ladder operators allow us to construct the full energy spectrum for the bound states algebraically (Fig. 1). However, we notice that the commutation relation (1.1) always leads to equally spaced energy spectra, while almost all the systems that we know of have energy spectra that are not equally spaced. This motivates us to extend the commutation relation (1.1) in a way to enable us to construct unequally spaced energy spectra algebraically. A remarkable formulation for such ladder operators has been proposed as the closure relation [2][3][4], which is the double commutation relation for a Hamiltonian and a function called a sinusoidal coordinate. Given a sinusoidal coordinate creation/annihilation operators for energy eigenstates can be constructed from it, and the energy spectra generated by them become unequally spaced in general, as shown in Fig. 2.
In this paper we also study intertwining operators in the context of quantum mechanics; they have already been studied in many areas of mathematics and physics. An intertwining operator D in quantum mechanics is a linear operator that satisfies the intertwining relation  Fig. 3. The two concepts of ladder and intertwining operators are basically different. However, when a Hamiltonian is parametrized by a parameter or a set of parameters, we can organize the two concepts in a unified way as spectral intertwining operators. The linear operators connect eigenfunctions of different eigenvalues between two different Hamiltonians, as shown in Figs. 4 and 5. In this paper, we point out that the presence of spectral intertwining operators is important for quantum-mechanical systems being exactly solvable. In supersymmetric quantum mechanics [5], spectral intertwining operators have already been studied as shape invariance [6]. In supersymmetric quantum mechanics , two supersymmetric Hamiltonians are related to each other by the intertwining relations where a † , a are intertwining operators (called supercharges since a † (a) maps eigenfunctions for H − (H + ) into eigenfunctions for H + (H − ) at the same energy level). In addition, when two Hamiltonians have shape invariance, the energy spectrum of the system can be solved exactly by using the supercharges in combination with shape invariance (see Sect. B.1 for details). In the past, many exactly solvable quantum-mechanical systems have been studied by means of shape invariance (see, e.g., Ref. [7]). While the shape invariance is very powerful to construct exactly solvable quantum-mechanical models, it is important to obtain other concepts to characterize such models. Very recently, a coupled system in supersymmetric quantum mechanics was studied in Ref. [8], where the authors discussed the underlying structure for operators that connect eigenstates between two sectors. In previous works, the operators explained above have been formulated in n-independent ways. They have a merit that one does not have to know the n-dependence of the eigenfunctions and eigenvalues. In contrast, we dare to formulate them in an n-dependent way in the present paper because this could allow us to study as many examples as possible or to find new examples that have never been elucidated. We attempt to make a formulation for spectral intertwining operators. Actually, we show several examples in Sect. 3. Previously known formulations such as the closure relation and shape invariance are discussed in the appendices, which will illuminate how special they are in our n-dependent formulation.
This paper is organized as follows: In the next section, we propose the spectral intertwining relations by unifying the relation of ladder and intertwining relations. In Sect. 3, as an example, we show spectral intertwining relations for several Hamiltonians such as the hydrogen atom (Coulomb potential) and the Rosen-Morse potentials. It is known that the Coulomb and Rosen-Morse potentials admit spectral intertwining operators associated with the shape invariance. However, the spectral intertwining operators that we obtain are different from them. Section 4 is devoted to a summary and discussion. In Appendix A, we consider the one-parameter representation for ladder operators, which leads to a modification of the commutation relation (1.1) in a natural way. Several examples that fit into the modified commutation relation are also given. In Appendix B, we discuss the case in which a Hamiltonian is parametrized and a spectral intertwining relation is introduced. It is also shown that the relation introduced can be found in supersymmetric quantum mechanics with shape invariance.

Examples of the spectral intertwining relation
In this section, we give examples of the spectral intertwining relations of the hydrogen atom and the Rosen-Morse potentials. Although these models have shape invariance, the spectral intertwining relations cannot be derived from it.

Hydrogen atom
The Hamiltonian is parametrized by the mass m and the coupling constants and as which recovers the radial part of the Schrödinger equation for the hydrogen atom if m = m e , = m e 0 , and = 0, 1, 2, . . . . Since the Hamiltonian depends on m, , and , the eigenfunctions and eigenvalues also depend on them in general. In the present case, we have where C ,n, are normalization constants and L α n (ξ) are the associated Laguerre functions in ξ. In a heuristic way, we are able to find the operator which satisfies the spectral intertwining relation This relation fits into the spectral intertwining relation (2.2) with (3.7) D n ( , ) maps an eigenfunction ψ n ( n ( , ), ) into an eigenfunction ψ n+1 ( , ). Since we are now interested in the Hamiltonian H(m, , ) , we obtain A further insight is obtained by rescaling the coordinate r in the Hamiltonian (3.1). Rescaling r with a parameter α, we easily find that This implies that the energy eigenfunctions of the Hamiltonian (3.1) rescaled by r → αr are equivalent to those rescaled by m → α 2 m and → α up to normalization, i.e., ψ n ( , ; αr) ∝ ψ n (α , ; r) .
the composite operatorD satisfies the spectral intertwining relation We have found that the system of the hydrogen atom possesses the spectral intertwining relations (3.5) and (3.16), which are related to the parameters , , and m. These operators had already been found in Refs. [9][10][11][12][13] as part of the dynamical group O(4, 2), but they have not been viewed as spectral intertwining operators before.

Summary and discussion
In this paper we have considered the question of what is an important structure commonly found in exactly solvable quantum-mechanical systems. For a parametrized Hamiltonian, we have proposed the spectral intertwining relation (2.2) as a unified relation of ladder and intertwining relations. Since it is formulated in an n-dependent way, the spectral intertwining relations can connect eigenfunctions between two different Hamiltonians. It is emphasized that each of the spectral intertwining relations (3.5), (3.16), (3.23) and (3.31) connects two different Hamiltonians, which cannot be obtained by shape invariance. Thus, our formulation can deal with many examples including previously known ones, and can connect many models in a wider frame.
It has been thought that ladder and intertwining operators premise the (dynamical) symmetry of a system, so that they have been studied in algebraic ways. For example, shape invariance in supersymmetric quantum mechanics can be interpreted by means of extended Lie algebras [14,15]. Hence, as it was pointed out in Refs. [9][10][11][12][13] that the spectral intertwining operators for the hydrogen atom are part of the dynamical group O(4, 2), we expect that the spectral intertwining relation (2.2) is in general related to some sort of symmetry of a system. To see this, it would be interesting to lift a quantum system to a spacetime with Bergmann structure in higher dimensions [16][17][18][19], where the Schrödinger symmetry in the original quantum system can be found explicitly as part of the conformal symmetry in the lifted spacetime. Some other lifts have also been studied in Refs. [20,21]. In this way, the spectral intertwining relation (2.2) would be of interest from a geometric point of view.

A One-parameter representation of ladder operators
In this section, we consider how to modify the commutation relation (1.1). In order to organize unequally spaced energy spectra, we make the ladder operators depend on the energy level n, which can be done with the commutation relation where a is a parameter, ε(a) and E(a) are real constants for every a, and Q(a) is a linear operator depending on a. We note that, if D(a) and ε(a) are independent of the parameter a and Q(a) ≡ 0, it recovers the commutation relation (1.1). Given a linear operator D(a) satisfying the commutation relation (A.1) for a Hamiltonian H, we set a = a n so as to satisfy E n = E(a n ) , D n = D(a n ) , ε n = ε(a n ) , and obtain for an eigenfunction ψ n when D n ψ n < ∞ and D n ψ n 0, which shows that D n ψ n is an eigenfunction with energy E n + ε n . Hence, for any n, there exists an integer m such that ψ n+m ∝ D n ψ n and E n+m = E n + ε n ; i.e., D n acts as a ladder operator for an eigenfunction ψ n . We should note that the value a n must be chosen appropriately so as to satisfy the condition (A.3).
If the value a n is not appropriate, D n ψ n does not become an eigenfunction in general. If m = 1 for all n, the energy spectrum can be constructed entirely in the manner shown in Fig. 2, and we obtain, for n = 1, 2, . . . , where we emphasize again that each a i satisfies the condition (A.3), i.e., E(a i ) = E i (i = 0, 1, 2, . . . ). Thus it is sufficient to obtain the eigenfunction ψ 0 for the ground state to construct the full energy spectrum. In general, we may replace the second term on the right-hand side of Eq. (A.1) with a more general form, e.g., replacing f (H − E n ) with an arbitrary function f (x) with f (0) = 0. However, since we want to get ladder operators as differential operators, it is reasonable to assume that f is polynomial, which leads to the current form in the end.
It should also be noted that, if the energy eigenstates are degenerate, additional quantum numbers should be introduced. Accordingly, spectral intertwining operators can have several parameters. For example, if there exists a linear operator K that commutes with a Hamiltonian H, [H, K] = 0, the commutation relation (A.1) can be modified to (A.5)

A.1 Harmonic oscillator in one dimension
The Schrödinger equation for a harmonic oscillator in one dimension is given by the Hamiltonian where m > 0 and ω are the mass and frequency, respectively. The eigenfunctions and eigenvalues are given by where C n are normalization constants and H n (ξ) are the Hermite polynomials in ξ. The well known ladder operators are given by which satisfy the commutation relations It is well known that the commutation relations (A.10) are derived from H HO = (a + a − + 1/2)ω and the Heisenberg algebra [a − , a + ] = 1. Using the Heisenberg algebra, it is also shown that the linear operatorŝ D ± ≡ ∓a ± a ± satisfy [H HO ,D ± ] = ±2ωD ± , which shows thatD ± are ladder operators that map ψ n into ψ n±2 . As differential operators, they are explicitly written aŝ where H HO is the Hamiltonian (A.6). Since H HO is replaced by the energy eigenvalue E n whenD ± act on an eigenfunction ψ n ,D ± may be expressed as first-order operators in the form where n is an integer. D ± (n) satisfy the commutation relations which fit into the commutation relation (A.1). In the process from (A.11) to (A.12) the second-order operatorsD ± were transformed into the first-order operators D (±) n by replacing the Hamiltonian H HO with the eigenvalues E n . In return for this, D (±) n came to depend on n. They are, of course, just different representations. However, when we want to get expressions of ladder operators as differential operators, the latter is more useful in a generic case. To show this, we review Odake and Sasaki's construction for ladder operators [2][3][4] in the next subsection.

A.2 Odake and Sasaki's construction for ladder operators
It was shown in Refs. [2][3][4] that one can construct ladder operators for a Hamiltonian H if there exists a function η, called a sinusoidal coordinate, which satisfies the closure relation where R 0 (H), R 1 (H) and R −1 (H) are polynomials in H. Given a sinusoidal coordinate η, the ladder operators are formally provided1 byD where α ± are given by R 0 = −α − α + and R 1 = α + + α − . Indeed, it is shown that 16) which leads to the commutation relation [H,D ± ]ψ n = α(E n )D ± ψ n for an eigenfunction ψ n with energy E n , i.e., Hψ n = E n ψ n . ThusD ± are ladder operators that change the energy eigenvalues from E n to 1We point out here that the derivation of (A.16) is independent of the fact that η is a function, which implies that if the closure relation (A.14) is satisfied for a linear operator η, thenD ± provide higher-order ladder operators. E n + α ± (E n ). It is worth commenting that ladder operators constructed from a sinusoidal coordinate have nothing to do with shape invariance (see also Sect. B.1), so that they cannot factorize the Hamiltonian in general.
It is difficult in general to express the ladder operators (A.15) as differential operators because they consist of the fractional terms R −1 (H)/α ± (H), and α ± (H) are not necessarily polynomials in H. To obtain expressions as differential operators, we take the same procedure as that from (A.11) to (A.12), namely, replace the Hamiltonian H by the energy eigenvalues E n . Actually, when the ladder operatorsD ± act on an eigenfunction ψ n with energy E n , i.e., Hψ n = E n ψ n , they are realized bŷ hence we obtain Using these expressions, we can explicitly check that D ± (E n ) satisfy the commutation relations with some operator Q(n). Thus, we find that ladder operators constructed from a sinusoidal coordinate fit into the commutation relation (A.1) by replacing the Hamiltonian with the energy eigenvalues. Moreover, Eq. (A.18) allows us to express the ladder operators as differential operators.