Spectral intertwining relations in exactly solvable quantum-mechanical systems
KKOBE-COSMO-16-14, KOBE-TH-16-08June 19, 2017
Spectral intertwining relations in exactly solvablequantum-mechanical systems
Tsuyoshi Houri , Makoto Sakamoto and Kentaro Tatsumi Department of Physics, Kobe University, Kobe 657-8501, Japan
Abstract
In exactly solvable quantum-mechanical systems, ladder and intertwining operators play a centralrole because, if they are found, the energy spectra can be obtained algebraically. In this paper, wepropose the spectral intertwining relation as a unified relation of ladder and intertwining operators ina way that can depend on the energy eigenvalues. It is shown that the spectral intertwining relationscan 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 theRosen–Morse potential. Email address: [email protected] Email address: [email protected] Email address: [email protected] a r X i v : . [ qu a n t - ph ] J un ontents A.1 Harmonic oscillator in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 7A.2 Odake and Sasaki’s construction for ladder operators . . . . . . . . . . . . . . . . . . . 8A.2.1 Calogero–Sutherland model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
B Intertwining operators and parameter shifting 10
B.1 Shape invariance in supersymmetric quantum mechanics . . . . . . . . . . . . . . . . . 10
What is an important structure commonly found in exactly solvable quantum-mechanical systems? Theconcept of ladder operators is of great interest in this question because, since Dirac’s use [1], it has playeda central role in quantum mechanics. As is well known, a ladder operator D for a Hamiltonian H is alinear operator that satisfies the commutation relation [ H , D ] = ε D , (1.1)where ε is a real constant. In particular, it is called a raising operator if ε > or a lowering operator if ε < as it raises or lowers the energy eigenvalues. In fact, when D acts on an eigenfunction ψ n withenergy E n , i.e., H ψ n = E n ψ n , we see that H D ψ n = DH ψ n + [ H , D ] ψ n = ( E n + ε ) D ψ n , (1.2)which shows that D ψ n is an eigenfunction with energy E n + ε . (Note: In this paper, we focus only onbound states, so that the energy eigenfunctions ψ n and energy eigenvalues E n are discretized and labeledby integers n . We assign to the ground state and , , . . . to the excited states, unless otherwise noted.)Hence, for any n , there exists an integer m such that ψ n + m ∝ D ψ n and E n + m = E n + ε . Since it followsthat, if m = , ψ n ∝ D n ψ and E n = E + n ε , ladder operators allow us to construct the full energyspectrum 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 haveenergy spectra that are not equally spaced. This motivates us to extend the commutation relation (1.1) in away to enable us to construct unequally spaced energy spectra algebraically. A remarkable formulation forsuch ladder operators has been proposed as the closure relation [2–4], which is the double commutationrelation for a Hamiltonian and a function called a sinusoidal coordinate. Given a sinusoidal coordinatecreation/annihilation operators for energy eigenstates can be constructed from it, and the energy spectragenerated 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 havealready been studied in many areas of mathematics and physics. An intertwining operator D in quantummechanics is a linear operator that satisfies the intertwining relation H D = DH , (1.3)1here H and H are two different Hamiltonians. From this relation, it can be found that for aneigenfunction ψ , n for H obeying H ψ , n = E , n ψ , n , H (cid:0) D ψ , n (cid:1) = D (cid:0) H ψ , n (cid:1) = E , n (cid:0) D ψ , n (cid:1) , (1.4)which shows that D ψ , n is an eigenfunction for H . Thus the intertwining operator D maps eigenfunctionsfor H into eigenfunctions for H , as shown in Fig. 3.The two concepts of ladder and intertwining operators are basically different. However, when aHamiltonian is parametrized by a parameter or a set of parameters, we can organize the two concepts in aunified way as spectral intertwining operators . The linear operators connect eigenfunctions of differenteigenvalues between two different Hamiltonians, as shown in Figs. 4 and 5. In this paper, we point outthat the presence of spectral intertwining operators is important for quantum-mechanical systems beingexactly solvable. In supersymmetric quantum mechanics [5], spectral intertwining operators have alreadybeen studied as shape invariance [6]. In supersymmetric quantum mechanics , two supersymmetricHamiltonians are related to each other by the intertwining relations H + a † = a † H − , H − a = aH + , (1.5)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 haveshape invariance, the energy spectrum of the system can be solved exactly by using the superchargesin combination with shape invariance (see Sect. B.1 for details). In the past, many exactly solvablequantum-mechanical systems have been studied by means of shape invariance (see, e.g., Ref. [7]). Whilethe shape invariance is very powerful to construct exactly solvable quantum-mechanical models, it isimportant to obtain other concepts to characterize such models. Very recently, a coupled system insupersymmetric quantum mechanics was studied in Ref. [8], where the authors discussed the underlyingstructure for operators that connect eigenstates between two sectors.In previous works, the operators explained above have been formulated in n -independent ways. Theyhave a merit that one does not have to know the n -dependence of the eigenfunctions and eigenvalues. Incontrast, we dare to formulate them in an n -dependent way in the present paper because this could allowus to study as many examples as possible or to find new examples that have never been elucidated. Weattempt to make a formulation for spectral intertwining operators. Actually, we show several examples inSect. 3. Previously known formulations such as the closure relation and shape invariance are discussed inthe 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 relationsby unifying the relation of ladder and intertwining relations. In Sect. 3, as an example, we show spectralintertwining relations for several Hamiltonians such as the hydrogen atom (Coulomb potential) and theRosen–Morse potentials. It is known that the Coulomb and Rosen–Morse potentials admit spectralintertwining operators associated with the shape invariance. However, the spectral intertwining operatorsthat 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 thecommutation relation (1.1) in a natural way. Several examples that fit into the modified commutationrelation are also given. In Appendix B, we discuss the case in which a Hamiltonian is parametrized and aspectral intertwining relation is introduced. It is also shown that the relation introduced can be found insupersymmetric quantum mechanics with shape invariance. When a Hamiltonian H ( ν ) is parametrized by a parameter ν (or a set of parameters), the Schrödingerequation is given by H ( ν ) ψ n ( ν ) = E n ( ν ) ψ n ( ν ) . (2.1)2or such a Hamiltonian, we propose a spectral intertwining relation as the n -dependent commutationrelation H ( ν ) D n ( ν ) = D n ( ν ) (cid:0) H ( ν n ( ν )) + ε n ( ν n ( ν )) (cid:1) + Q n ( ν n ( ν )) (cid:0) H ( ν n ( ν )) − E n ( ν n ( ν )) (cid:1) , (2.2)where ν and n are parameters, Q n ( ν n ( ν )) are some operators, and ε n ( ν n ( ν )) , E n ( ν n ( ν )) and ν n ( ν ) areconstants depending on ν and n . We call D n ( ν ) a spectral intertwining operator .Given D n ( ν ) satisfying the commutation relation (2.2) and an eigenfunction ψ n ( ν n ( ν )) for H ( ν n ( ν )) ,we obtain H ( ν ) D n ( ν ) ψ n ( ν n ( ν )) = (cid:0) E n ( ν n ( ν )) + ε n ( ν n ( ν )) (cid:1) D n ( ν ) ψ n ( ν n ( ν )) , (2.3)which shows that D n ( ν ) ψ n ( ν n ( ν )) is an eigenfunction for the Hamiltonian H ( ν ) with energy E n ( ν n ( ν )) + ε n ( ν n ( ν )) . Hence, for any n , there exists an integer m such that ψ n + m ( ν ) ∝ D n ( ν ) ψ n ( ν n ( ν )) , (2.4) E n + m ( ν ) = E n ( ν n ( ν )) + ε n ( ν n ( ν )) , (2.5)when (cid:107) D n ( ν ) ψ n ( ν n ( ν )) (cid:107) < ∞ and D n ( ν ) ψ n ( ν n ( ν )) (cid:44) . If m = for all n , as before, we get the diagramshown in Fig. 5. (For nontrivial examples of m (cid:44) , see Ref. [4].) As mentioned in the previoussection, these operators connect eigenfunctions of different eigenvalues between two Hamiltonians. Thedifference from the previous ones is that such spectral intertwining operators depend not only on ν , whichparametrizes the theories, but also on n , which parametrizes the eigenfunctions and eigenvalues. In thenext section, we provide some examples of Hamiltonians that admit such spectral intertwining operators. In this section, we give examples of the spectral intertwining relations of the hydrogen atom and theRosen–Morse potentials. Although these models have shape invariance, the spectral intertwining relationscannot be derived from it.
The Hamiltonian is parametrized by the mass m and the coupling constants (cid:49) and (cid:96) as H ( m , (cid:49) , (cid:96) ) ≡ m (cid:20) − r ddr (cid:18) r ddr (cid:19) + (cid:96) ( (cid:96) + ) r − (cid:49) r (cid:21) , ≤ r < ∞ , (3.1)which recovers the radial part of the Schrödinger equation for the hydrogen atom if m = m e , (cid:49) = m e (cid:49) ,and (cid:96) = , , , . . . . Since the Hamiltonian depends on m , (cid:49) , and (cid:96) , the eigenfunctions and eigenvaluesalso depend on them in general. In the present case, we have ψ n ( (cid:49) , (cid:96) ) = C (cid:49) , n ,(cid:96) r (cid:96) exp (cid:16) − (cid:49) rn + (cid:96) + (cid:17) L (cid:96) + n (cid:18) (cid:49) rn + (cid:96) + (cid:19) , (3.2) E n ( m , (cid:49) , (cid:96) ) = − (cid:49) m ( n + (cid:96) + ) , n = , , . . . , (3.3)where C (cid:49) , n ,(cid:96) 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 D n ( (cid:49) , (cid:96) ) ≡ r ddr − (cid:49) rn + (cid:96) + + n + (cid:96) + , (3.4)3hich satisfies the spectral intertwining relation H ( m , (cid:49) , (cid:96) ) D n ( (cid:49) , (cid:96) ) = D n ( (cid:49) , (cid:96) ) H ( m , (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) + (cid:18) H ( m , (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) + (cid:49) n ( (cid:49) , (cid:96) ) m ( n + (cid:96) + ) (cid:19) , (3.5)where (cid:49) n ( (cid:49) , (cid:96) ) = (cid:49) n + (cid:96) + n + (cid:96) + . (3.6)This relation fits into the spectral intertwining relation (2.2) with ε n ( m , (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) = , Q n ( m , (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) = , E n ( m , (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) = − (cid:49) n ( (cid:49) , (cid:96) ) m ( n + (cid:96) + ) . (3.7) D n ( (cid:49) , (cid:96) ) maps an eigenfunction ψ n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) into an eigenfunction ψ n + ( (cid:49) , (cid:96) ) . Since we are nowinterested in the Hamiltonian H ( m , (cid:49) , (cid:96) ) , we obtain ψ n + ( (cid:49) , (cid:96) ) ∝ D n ( (cid:49) , (cid:96) ) ψ n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) . (3.8)A further insight is obtained by rescaling the coordinate r in the Hamiltonian (3.1). Rescaling r witha parameter α , we easily find that H ( m , (cid:49) , (cid:96) ; α r ) = α H ( m , α (cid:49) , (cid:96) ; r ) = H ( α m , α (cid:49) , (cid:96) ; r ) . (3.9)This implies that the energy eigenfunctions of the Hamiltonian (3.1) rescaled by r → α r are equivalent tothose rescaled by m → α m and (cid:49) → α (cid:49) up to normalization, i.e., ψ n ( (cid:49) , (cid:96) ; α r ) ∝ ψ n ( α (cid:49) , (cid:96) ; r ) . (3.10)Since the rescaling of the coordinate r can be represented by the operator S ( α ) = exp [( ln α ) r d / dr ] , wehave S ( α ) ψ n ( (cid:49) , (cid:96) ; r ) = ψ n ( (cid:49) , (cid:96) ; α r ) . (3.11)Combining (3.8), (3.10) and (3.11) with α n ( (cid:96) ) (cid:49) ≡ (cid:49) n ( (cid:49) , (cid:96) ) = ( n + (cid:96) + )/( n + (cid:96) + ) (cid:49) , we obtain ψ n + ( (cid:49) , (cid:96) ) ∝ D n ( (cid:49) , (cid:96) ) S ( α n ( (cid:96) )) ψ n ( (cid:49) , (cid:96) ) . (3.12)Since we have H ( m , (cid:49) , (cid:96) ) D n ( (cid:49) , (cid:96) ) = D n ( (cid:49) , (cid:96) ) H ( m , α n ( (cid:96) ) (cid:49) , (cid:96) ) + (cid:16) H ( m , α n ( (cid:96) ) (cid:49) , (cid:96) ) − E n (cid:0) m / α n ( (cid:96) ) , (cid:49) (cid:1) (cid:17) , (3.13) H ( m , α n ( (cid:96) ) (cid:49) , (cid:96) ) S ( α n ( (cid:96) )) = α n ( (cid:96) ) S ( α n ( (cid:96) )) H ( m , (cid:49) , (cid:96) ) = S ( α n ( (cid:96) )) H (cid:0) m / α n ( (cid:96) ) , (cid:49) , (cid:96) (cid:1) , (3.14)the composite operator ˜ D n ( (cid:49) , (cid:96) ) ≡ D n ( (cid:49) , (cid:96) ) S ( α n ( (cid:96) )) (3.15)satisfies the spectral intertwining relation H ( m , (cid:49) , (cid:96) ) ˜ D n ( (cid:49) , (cid:96) ) = ˜ D n ( (cid:49) , (cid:96) ) H (cid:0) m / α n ( (cid:96) ) , (cid:49) , (cid:96) (cid:1) + S ( α n ( (cid:96) )) (cid:16) H (cid:0) m / α n ( (cid:96) ) , (cid:49) , (cid:96) (cid:1) − E n (cid:0) m / α n ( (cid:96) ) , (cid:49) (cid:1) (cid:17) . (3.16)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 (cid:96) , (cid:49) , and m . These operators had already beenfound in Refs. [9–13] as part of the dynamical group O ( , ) , but they have not been viewed as spectralintertwining operators before. 4 .2 Rosen–Morse potential The 1D Hamiltonian with the spherical (or trigonometric) Rosen–Morse potential is given by H sph. ( (cid:49) , (cid:96) ) = − d dx + (cid:96) ( (cid:96) + ) sin x − (cid:49) cot x , ≤ x ≤ π , (3.17)where ≤ (cid:96) and ≤ (cid:49) are parameters. The eigenfunctions and eigenvalues are given by ψ n ( (cid:49) , (cid:96) ; x ) = C (cid:49) ,(cid:96), n ( sin x ) n + l + e − (cid:49) n + (cid:96) + x P ( a + ( (cid:49) ,(cid:96), n ) , a − ( (cid:49) ,(cid:96), n )) n ( i cot x ) , (3.18) E n ( (cid:49) , (cid:96) ) = + ( n + (cid:96) + ) − (cid:16) (cid:49) n + (cid:96) + (cid:17) , (3.19)where C (cid:49) ,(cid:96), n are the normalization constants, P ( α,β ) n ( ξ ) are the Jacobi functions in ξ and a ± ( (cid:49) , (cid:96), n ) are a ± ( (cid:49) , (cid:96), n ) = −( n + (cid:96) + ) ± i (cid:49) n + (cid:96) + . (3.20)The spectral intertwining operators are given by D n ( (cid:49) , (cid:96) ) = ˜ a + − ˜ a − i sin x + ˜ a + + ˜ a − x − sin x ddx , (3.21)where ˜ a ± ( (cid:49) , (cid:96), n ) = a ± ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96), n ) = a ± (cid:18) (cid:49) n + (cid:96) + n + (cid:96) + , (cid:96), n (cid:19) , (cid:49) n ( (cid:49) , (cid:96) ) ≡ (cid:49) n + (cid:96) + n + (cid:96) + . (3.22)They satisfy the spectral intertwining relation H sph. ( (cid:49) , (cid:96) ) D n ( (cid:49) , (cid:96) ) = D n ( (cid:49) , (cid:96) ) (cid:16) H sph. ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) + ( n + (cid:96) + ) + (cid:17) − x (cid:16) H sph. ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) − E n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) (cid:17) , (3.23)which fits into Eq. (2.2) by setting ε n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) ≡ ( n + (cid:96) + ) + , Q n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) ≡ − x . (3.24)Next, we consider the 1D Hamiltonian with the hyperbolic Rosen–Morse potential H hyp. ( (cid:49) , (cid:96) ) = − d dx + (cid:96) ( (cid:96) + ) sinh x − (cid:49) coth x , ≤ x < ∞ , (3.25)where ≤ (cid:96) and ( (cid:96) + ) ≤ (cid:49) are parameters. The eigenfunctions and eigenvalues are given by ψ n ( (cid:49) , (cid:96) ; x ) = C (cid:49) ,(cid:96), n ( sinh x ) n + (cid:96) + e − (cid:49) n + (cid:96) + x P ( b + ( (cid:49) ,(cid:96), n ) , b − ( (cid:49) ,(cid:96), n )) n ( coth x ) , (3.26) E n ( (cid:49) , (cid:96) ) = −( n + (cid:96) + ) − (cid:16) (cid:49) n + (cid:96) + (cid:17) , (3.27)where C (cid:49) ,(cid:96), n are the normalization constants and P ( α,β ) n ( ξ ) are the Jacobi functions in ξ . We note thatthe system (3.25) has finitely many discrete eigenstates ψ n ( (cid:49) , (cid:96) ; x ) with ≤ n < √ (cid:49) − (cid:96) − . For thehyperbolic potential, b ± are given by b ± ( (cid:49) , (cid:96), n ) = −( n + (cid:96) + ) ± (cid:49) n + (cid:96) + . (3.28)The spectral intertwining operators are given by D n ( (cid:49) , (cid:96) ) = ˜ b + − ˜ b − x + ˜ b + + ˜ b − x − sinh x ddx , (3.29)5here ˜ b ± ( (cid:49) , (cid:96), n ) = b ± ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96), n ) = b ± (cid:18) (cid:49) n + (cid:96) + n + (cid:96) + , (cid:96), n (cid:19) , (cid:49) n ( (cid:49) , (cid:96) ) ≡ (cid:49) n + (cid:96) + n + (cid:96) + . (3.30)They satisfy the spectral intertwining relation H hyp. ( (cid:49) , (cid:96) ) D n ( (cid:49) , (cid:96) ) = D n ( (cid:49) , (cid:96) ) (cid:16) H hyp. ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) − ( n + (cid:96) + ) − (cid:17) − x (cid:16) H hyp. ( (cid:49) n ( g , (cid:96) ) , (cid:96) ) − E n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) (cid:17) , (3.31)which fits into Eq. (2.2) by setting ε n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) ≡ − ( n + (cid:96) + ) − , Q n ( (cid:49) n ( (cid:49) , (cid:96) ) , (cid:96) ) ≡ − x . (3.32)It is known that these Rosen–Morse potentials have shape invariance which leads to the spectralintertwining relation with respect to (cid:96) . In contrast, here, we have obtained the spectral intertwiningrelation with respect to (cid:49) and (cid:96) , which cannot be derived from shape invariance. In this paper we have considered the question of what is an important structure commonly found inexactly solvable quantum-mechanical systems. For a parametrized Hamiltonian, we have proposed thespectral intertwining relation (2.2) as a unified relation of ladder and intertwining relations. Since it isformulated in an n -dependent way, the spectral intertwining relations can connect eigenfunctions betweentwo 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 connectmany models in a wider frame.It has been thought that ladder and intertwining operators premise the (dynamical) symmetry of asystem, so that they have been studied in algebraic ways. For example, shape invariance in supersymmetricquantum mechanics can be interpreted by means of extended Lie algebras [14, 15]. Hence, as it waspointed out in Refs. [9–13] that the spectral intertwining operators for the hydrogen atom are part ofthe dynamical group O ( , ) , we expect that the spectral intertwining relation (2.2) is in general relatedto some sort of symmetry of a system. To see this, it would be interesting to lift a quantum system toa spacetime with Bergmann structure in higher dimensions [16–19], where the Schrödinger symmetryin the original quantum system can be found explicitly as part of the conformal symmetry in the liftedspacetime. Some other lifts have also been studied in Refs. [20, 21]. In this way, the spectral intertwiningrelation (2.2) would be of interest from a geometric point of view. Acknowledgements
The authors would like to thank Masato Nozawa and Satoru Odake for useful comments. This work wassupported in part by JSPS KAKENHI Grant No. JP14J01237 (T.H.) and Grants-in-Aid for ScientificResearch (No. 15K05055 and No. 25400260 (M.S.)) from the Ministry of Education, Culture, Sports,Science and Technology (MEXT) in Japan. 6
One-parameter representation of ladder operators
In this section, we consider how to modify the commutation relation (1.1). In order to organize unequallyspaced energy spectra, we make the ladder operators depend on the energy level n , which can be donewith the commutation relation [ H , D ( a )] = ε ( a ) D ( a ) + Q ( a ) ( H − E ( a )) , (A.1)where a is a parameter, ε ( a ) and E ( a ) are real constants for every a , and Q ( a ) is a linear operatordepending on a . We note that, if D ( a ) and ε ( a ) are independent of the parameter a and Q ( a ) ≡ , itrecovers 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 ) , n = , , . . . (A.2)and obtain for an eigenfunction ψ n when (cid:107) D n ψ n (cid:107) < ∞ and D n ψ n (cid:44) , H D n ψ n = ( E n + ε n ) D n ψ n , n = , , . . . (A.3)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 = for all n , the energy spectrum can be constructed entirely in the manner shown in Fig. 2, and we obtain, for n = , , . . . , ψ n ∝ D n − · · · D ψ , E n = E + n − (cid:213) i = ε i , (A.4)where we emphasize again that each a i satisfies the condition (A.3), i.e., E ( a i ) = E i ( i = , , , . . . ) . Thusit is sufficient to obtain the eigenfunction ψ 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 generalform, e.g., replacing f ( H − E n ) with an arbitrary function f ( x ) with f ( ) = . However, since we wantto get ladder operators as differential operators, it is reasonable to assume that f is polynomial, whichleads to the current form in the end.It should also be noted that, if the energy eigenstates are degenerate, additional quantum numbersshould be introduced. Accordingly, spectral intertwining operators can have several parameters. Forexample, if there exists a linear operator K that commutes with a Hamiltonian H , [ H , K ] = , thecommutation relation (A.1) can be modified to [ H , D ( a , b )] = ε ( a , b ) D ( a , b ) + Q ( a , b )( H − E ( a )) + Q ( a , b )( K − λ ( b )) . (A.5) A.1 Harmonic oscillator in one dimension
The Schrödinger equation for a harmonic oscillator in one dimension is given by the Hamiltonian H HO ≡ − m d dx + m ω x , −∞ < x < ∞ , (A.6)where m > and ω are the mass and frequency, respectively. The eigenfunctions and eigenvalues aregiven by ψ n = C n exp (cid:18) − m ω x (cid:19) H n (cid:16) √ m ω x (cid:17) , (A.7) E n = (cid:18) n + (cid:19) ω , n = , , , . . . , (A.8)7here C n are normalization constants and H n ( ξ ) are the Hermite polynomials in ξ . The well knownladder operators are given by a ± ≡ ∓ (cid:114) m ω (cid:18) ddx ∓ m ω x (cid:19) , (A.9)which satisfy the commutation relations [ H HO , a ± ] = ± ω a ± . (A.10)It is well known that the commutation relations (A.10) are derived from H HO = ( a + a − + / ) ω and theHeisenberg algebra [ a − , a + ] = . Using the Heisenberg algebra, it is also shown that the linear operators ˆ D ± ≡ ∓ a ± a ± satisfy [ H HO , ˆ D ± ] = ± ω ˆ D ± , which shows that ˆ D ± are ladder operators that map ψ n into ψ n ± . As differential operators, they are explicitly written as ˆ D ± ≡ x ddx ∓ m ω x ± H HO ω + , (A.11)where H HO is the Hamiltonian (A.6). Since H HO is replaced by the energy eigenvalue E n when ˆ D ± act onan eigenfunction ψ n , ˆ D ± may be expressed as first-order operators in the form D (±) n ≡ x ddx ∓ m ω x ± (cid:18) n + (cid:19) + , (A.12)where n is an integer. D ± ( n ) satisfy the commutation relations [ H HO , D (±) n ] = ± ω D (±) n + ( H HO − E n ) , (A.13)which fit into the commutation relation (A.1).In the process from (A.11) to (A.12) the second-order operators ˆ D ± were transformed into thefirst-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 toget expressions of ladder operators as differential operators, the latter is more useful in a generic case. Toshow this, we review Odake and Sasaki’s construction for ladder operators [2–4] in the next subsection. A.2 Odake and Sasaki’s construction for ladder operators
It was shown in Refs. [2–4] that one can construct ladder operators for a Hamiltonian H if there exists afunction η , called a sinusoidal coordinate, which satisfies the closure relation [ H , [ H , η ]] = [ H , η ] R ( H ) + η R ( H ) + R − ( H ) , (A.14)where R ( H ) , R ( H ) and R − ( H ) are polynomials in H . Given a sinusoidal coordinate η , the ladderoperators are formally provided by ˆ D ± ≡ [ H , η ] − ηα ∓ ( H ) + R − ( H ) α ± ( H ) , (A.15)where α ± are given by R = − α − α + and R = α + + α − . Indeed, it is shown that [ H , ˆ D ± ] = ˆ D ± α ± ( H ) , (A.16)which leads to the commutation relation [ H , ˆ D ± ] ψ n = α ( E n ) ˆ D ± ψ n for an eigenfunction ψ n with energy E n , i.e., H ψ n = E n ψ n . Thus ˆ D ± are ladder operators that change the energy eigenvalues from E n to We point out here that the derivation of (A.16) is independent of the fact that η is a function, which implies that if theclosure relation (A.14) is satisfied for a linear operator η , then ˆ D ± provide higher-order ladder operators. n + α ± ( E n ) . It is worth commenting that ladder operators constructed from a sinusoidal coordinate havenothing to do with shape invariance (see also Sect. B.1), so that they cannot factorize the Hamiltonian ingeneral.It is difficult in general to express the ladder operators (A.15) as differential operators because theyconsist of the fractional terms R − ( H )/ α ± ( H ) , and α ± ( H ) are not necessarily polynomials in H . To obtainexpressions 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 operators ˆ D ± act onan eigenfunction ψ n with energy E n , i.e., H ψ n = E n ψ n , they are realized by ˆ D ± ψ n = (cid:18) [ H , η ] − ηα ∓ ( E n ) + R − ( E n ) α ± ( E n ) (cid:19) ψ n , (A.17)hence we obtain D ± ( E n ) ≡ [ H , η ] − ηα ∓ ( E n ) + R − ( E n ) α ± ( E n ) . (A.18)Using these expressions, we can explicitly check that D ± ( E n ) satisfy the commutation relations [ H , D ± ( E n )] = α ± ( E n ) D ± ( E n ) + Q ( n )( H − E n ) , (A.19)with some operator Q ( n ) . Thus, we find that ladder operators constructed from a sinusoidal coordinate fitinto 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. A.2.1 Calogero–Sutherland model
For example, we shall look at the Calogero–Sutherland model [22, 23] for a single particle, which is aparticular case of the Pöschl–Teller potential [24]. The Hamiltonian is given by H CS ( g ) = − (cid:20) d dx − g ( g − ) sin x (cid:21) = − (cid:18) ddx + g tan x (cid:19) (cid:18) ddx − g tan x (cid:19) + g , (A.20)where g > and < x < π . The eigenfunctions and eigenvalues are given by ψ n ( g ) = C n ( sin x ) g P ( g − / , g − / ) n ( cos x ) , (A.21) E n ( g ) = ( n + g ) , n = , , . . . , (A.22)where C n are normalization constants and P ( α,β ) n ( ξ ) are the Jacobi polynomials in ξ . In Ref. [3], it wasshown that the sinusoidal coordinate is given by η = cos x , and the closure relation (A.14) becomes [ H CS , [ H CS , cos x ]] = [ H CS , cos x ] + cos x (cid:18) H CS − (cid:19) , (A.23)where R = , R = H CS − / and R − = , which leads to α ± ( H CS ) = / ± √ H CS . Hence, the ladderoperators (A.15) are given by ˆ D ± = sin x ddx ± cos x (cid:112) H CS , (A.24)where we have used the fact that [ H CS , cos x ] = ( sin x ) d / dx + ( cos x )/ . Replacing the Hamiltonian bythe eigenvalues (A.22), we obtain the ladder operators D ± ( E n ) = sin x ddx ± ( n + g ) cos x . (A.25)Since D ± ( E n ) satisfy the commutation relation (A.19) with α ± ( E n ) = / ±( n + g ) and Q ( n ) = η = x ,they change the eigenvalues from E n to E n ± . 9 Intertwining operators and parameter shifting
We consider Eq. (2.1). Since the parameter ν can be thought of as parametrizing theories, we mayconsider the intertwining relation (1.3) with H = H ( ν ) and H = H ( ν ) for two values ν and ν . Moregenerally, it is reasonable to consider a one-parameter family of intertwining operators D ( ν ) dependingon ν by setting ν = ν and ν = f ( ν ) , where f ( ν ) is a function of ν . The function f ( ν ) describes how theparameter ν in eigenfunctions and eigenvalues is shifted. Thus, the intertwining relation (1.3) is modifiedto the relation H ( ν ) D ( ν ) = D ( ν ) (cid:16) H ( f ( ν )) + ε ( f ( ν )) (cid:17) , (B.1)where we have added the term ε ( f ( ν )) to the right-hand side in order to describe shifts of energyeigenvalues. Given this relation, it is shown that, for an eigenfunction ψ n ( f ( ν )) for H ( f ( ν )) with energy E n ( f ( ν )) , we obtain H ( ν ) (cid:16) D ( ν ) ψ n ( f ( ν )) (cid:17) = (cid:16) E n ( f ( ν )) + ε ( f ( ν )) (cid:17) D ( ν ) ψ n ( f ( ν )) , (B.2)which shows that D ( ν ) ψ n ( f ( ν )) is an eigenfunction for H ( ν ) with energy E n ( f ( ν )) + ε ( f ( ν )) when (cid:107) D ( ν ) ψ n ( f ( ν ))(cid:107) < ∞ and D ( ν ) ψ n ( f ( ν )) (cid:44) . Since D ( ν ) map eigenfunctions for H ( f ( ν )) into eigenfunc-tions for H ( ν ) , there exists an integer m for any n such that ψ n + m ( ν ) ∝ D ( ν ) ψ n ( f ( ν )) , (B.3) E n + m ( ν ) = E n ( f ( ν )) + ε ( f ( ν )) . (B.4)Thus, we call such operators that connect eigenfunctions between two Hamiltonians spectral intertwiningoperators.Like ladder operators, spectral intertwining operators are also useful to construct full or partial energyspectra. For instance, if m = , D ( ν ) connects ψ n ( f ( ν )) to ψ n + ( ν ) as shown in Fig. 4. Hence, ψ n ( ν ) for n = , , . . . are constructed from ψ ( ν n ) by ψ n ( ν ) ∝ D ( ν ) D ( ν ) D ( ν ) . . . D ( ν n − ) ψ ( ν n ) , n = , , . . . , (B.5)where ν n = f ( ν n − ) = f ( f ( ν n − )) = f ( f (· · · f ( f ( ν )) · · · )) . The energy spectrum is given by E n ( ν ) = E ( ν n ) + n (cid:213) i = ε ( ν i ) , n = , , . . . , (B.6)In this way, the full energy spectrum can be constructed if the eigenfunctions for the ground states ψ ( ν ) are obtained for all ν . B.1 Shape invariance in supersymmetric quantum mechanics
In supersymmetric quantum mechanics [5], two Hamiltonians are related to each other by the intertwiningrelations (1.5). Moreover, if the two Hamiltonians are parametrized by a common parameter ν and havethe shape invariance [6] H − ( ν ) = H + ( f ( ν )) + ε ( f ( ν )) , (B.7)such a system becomes exactly solvable (see, e.g., Ref. [7] for a review of supersymmetric quantummechanics). We show this in terms of spectral intertwining operators.Substituting (B.7) into the intertwining relations (1.5), we obtain H + ( ν ) a † ( ν ) = a † ( ν ) (cid:16) H + ( f ( ν )) + ε ( f ( ν )) (cid:17) , (B.8) H + ( ν ) a ( f − ( ν )) = a ( f − ( ν )) (cid:16) H + ( f − ( ν )) − ε ( ν ) (cid:17) , (B.9)which fit into the spectral intertwining relation (B.1).10 eferences [1] P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, 1930) p. 257.[2] S. Odake and R. Sasaki, Physics Letters B , 112 (2006), arXiv:quant-ph/0605221 [quant-ph] .[3] S. Odake and R. Sasaki, Journal of Mathematical Physics , 102102 (2006), arXiv:quant-ph/0605215[quant-ph] .[4] S. Odake, Journal of Mathematical Physics , 113503 (2016), arXiv:1606.02836 .[5] E. Witten, Nuclear Physics B , 513 (1981).[6] L. Gendenshtein, Pis’ma v Zh. Eksp. Teor. , 299 (1983).[7] F. Cooper, A. Khare, and U. Sukhatme, Physics Reports , 267 (1995), arXiv:hep-th/9405029 .[8] B. G. B. Cameron L. Williams, Nikhil N. Pandya and D. J. Kouri, (2017), arXiv:1701.02767v1 .[9] R. Musto, Physical Review , 1274 (1966).[10] R. H. Pratt and T. F. Jordan, Physical Review , 1276 (1966).[11] I. A. Malkin and V. I. Man’ko, Soviet Journal of Experimental and Theoretical Physics Letters ,146 (1966).[12] A. O. Barut and H. Kleinert, Physical Review , 1541 (1967).[13] A. O. Barut and H. Kleinert, Physical Review , 1180 (1967).[14] A. B. Balantekin, Physical Review A , 4188 (1998), arXiv:quant-ph/9712018 .[15] A. Gangopadhyaya, J. V. Mallow, and U. P. Sukhatme, Physical Review A , 4287 (1998).[16] C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Physical Review D , 1841 (1985).[17] C. Duval, P. Horváthy, and L. Palla, Physics Letters B , 39 (1994), arXiv:hep-th/9401065 .[18] C. Duval, P. A. Horváthy, and L. Palla, Physical Review D , 6658 (1994), arXiv:hep-th/9404047 .[19] M. Cariglia, C. Duval, G. Gibbons, and P. Horváthy, Annals of Physics , 631 (2016),arXiv:1605.01932 .[20] A. Karamatskou and H. Kleinert, International Journal of Geometric Methods in Modern Physics , 1450066 (2014).[21] M. Cariglia, Annals of Physics , 642 (2015), arXiv:1506.00714 .[22] F. Calogero, Journal of Mathematical Physics , 419 (1971).[23] B. Sutherland, Physical Review A , 1372 (1972).[24] G. Pöschl and E. Teller, Zeitschrift fur Physik , 143 (1933).11 𝑬 𝑬 𝑬𝑬 𝑫𝑬 𝑫𝑫𝑫𝑯
Figure 1: The ladder structure correspondingto the commutation relation (1.1). The energyspectrum is equally spaced. 𝑬 𝑬 𝑬 𝑬𝑬 𝑫 𝒂 𝑫 𝒂 𝑫 𝒂 𝑯 Figure 2: The ladder structure correspondingto the commutation relation (A.1). The energyspectra can be unequally spaced in general.12 𝑯 𝑯 𝜓 +,-./ 𝑫 𝑬 𝑬 𝑬 𝑬 …… 𝑬 𝑬 𝑬 𝑬 …… 𝜓 +,-.+ 𝜓 +,- 𝜓 +,-1+ 𝜓 /,-./ 𝜓 /,-.+ 𝜓 /,- 𝜓 /,-1+ 𝑫𝑫𝑫
Figure 3: A sketch of the intertwining relation (1.3). The operator D connects eigenfunc-tions for H into eigenfunctions for H in the same energy level. 𝜓 𝜈 + 𝑯(𝝂 )𝜓 + (𝜈 ) 𝜓 𝜈 + 𝜓 / 𝜈 + 𝑯 𝝂 𝜓 / (𝜈 )𝜓 (𝜈 ) 𝑬 𝜓 𝜈 / 𝜓 + 𝜈 + 𝜓 / 𝜈 / 𝑯 𝝂 𝜓 𝜈 𝜓 𝜈 𝜓 / 𝜈 𝑯 𝝂 𝜓 𝜈 / 𝜓 + 𝜈 / 𝜓 (𝜈 ) 𝜓 + 𝜈 𝑬 𝝂𝑬 𝝂𝑬 𝝂𝑬 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 𝑫 𝝂 Figure 4: A diagram describing how the spectral intertwining operators D ( ν ) satisfyingEq. (B.1) map energy eigenfunctions for H ( ν i ) , where ν i = f ( f (· · · f ( f ( ν )) · · · )) for i = , , . . . . Since the shifts of energy eigenvalues depend only on ν , all the arrowsbetween H ( ν i − ) and H ( ν i ) are parallel. 13 / 𝜈 / (𝜈)𝑯(𝜈) 𝑬 𝝂𝑬 𝝂𝑬 𝝂 𝜓 (𝜈) 𝜓 𝜈 (𝜈) 𝜓 + 𝜈 + (𝜈)𝑯 𝜈 (𝜈)𝜓 + (𝜈)𝜓 / (𝜈) 𝑬 𝑯 𝜈 + (𝜈) 𝑯 𝜈 / (𝜈)𝜓 (𝜈) 𝑬 𝝂 𝑫 𝝂𝑫 𝝂 𝑫 𝝂 Figure 5: A diagram describing how the spectral intertwining operators D n ( ν ) act on theenergy eigenfunctions ψ n ( ν n ( ν )) for H ( ν n ( ν )) for n = , , , . . ., . . .