Properties of shape-invariant tridiagonal Hamiltonians

Hashim A. Yamani; Zouha\"ir Mouayn

arXiv:1812.10749·math-ph·December 31, 2018

Properties of shape-invariant tridiagonal Hamiltonians

Hashim A. Yamani, Zouha\"ir Mouayn

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TL;DR

This paper explores the properties of shape-invariant, tridiagonal Hamiltonians, showing how their spectra and supersymmetric partners can be explicitly determined, and constructs associated coherent states with illustrative examples.

Contribution

It demonstrates that shape invariance in tridiagonal Hamiltonians allows explicit spectral and partner Hamiltonian determination, extending the understanding of their algebraic structure.

Findings

01

Explicit relations for energy spectra of shape-invariant tridiagonal Hamiltonians.

02

Method to determine matrix elements of supersymmetric partner Hamiltonians.

03

Construction of coherent states for these Hamiltonians.

Abstract

It has been established that a positive semi-definite Hamiltonian, $H$ , that has a tridiagonal matrix representation in a basis set, allows a definition of forward (and backward) shift operators that can be used to define the matrix representation of the supersymmetric partner Hamiltonian $H^{(+)}$ in the same basis. \ We show that if, additionally, the Hamiltonian has a shape invariant property, the matrix elements of the Hamiltonian are related in a such a way that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of super-symmetric partner Hamiltonians. Additionally, we derive the coherent states associated with this type of Hamiltonians and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also has tridiagonal matrix representation.

Equations270

H_{n, m} = ⟨ ϕ_{n} ∣ H ∣ ϕ_{m} ⟩ = b_{n - 1} δ_{n, m + 1} + a_{n} δ_{n, m} + b_{n} δ_{n, m - 1}

H_{n, m} = ⟨ ϕ_{n} ∣ H ∣ ϕ_{m} ⟩ = b_{n - 1} δ_{n, m + 1} + a_{n} δ_{n, m} + b_{n} δ_{n, m - 1}

H_{n, m}^{(+)} = ⟨ ϕ_{n} ∣ H^{(+)} ∣ ϕ_{m} ⟩ = b_{n - 1}^{(+)} δ_{n, m + 1} + a_{n}^{(+)} δ_{n, m} + b_{n}^{(+)} δ_{n, m - 1},

H_{n, m}^{(+)} = ⟨ ϕ_{n} ∣ H^{(+)} ∣ ϕ_{m} ⟩ = b_{n - 1}^{(+)} δ_{n, m + 1} + a_{n}^{(+)} δ_{n, m} + b_{n}^{(+)} δ_{n, m - 1},

a_{n} = c_{n}^{2} + d_{n}^{2}, b_{n} = c_{n} d_{n + 1}

a_{n} = c_{n}^{2} + d_{n}^{2}, b_{n} = c_{n} d_{n + 1}

a_{n}^{(+)} = c_{n}^{2} + d_{n + 1}^{2}, b_{n}^{(+)} = c_{n + 1} d_{n + 1} .

a_{n}^{(+)} = c_{n}^{2} + d_{n + 1}^{2}, b_{n}^{(+)} = c_{n + 1} d_{n + 1} .

A ∣ ϕ_{m} ⟩ = c_{m} ∣ ϕ_{m} ⟩ + d_{m} ∣ ϕ_{m - 1} ⟩

A ∣ ϕ_{m} ⟩ = c_{m} ∣ ϕ_{m} ⟩ + d_{m} ∣ ϕ_{m - 1} ⟩

A^{†} ∣ ϕ_{m} ⟩ = c_{m} ∣ ϕ_{m} ⟩ + d_{m + 1} ∣ ϕ_{m + 1} ⟩ .

A^{†} ∣ ϕ_{m} ⟩ = c_{m} ∣ ϕ_{m} ⟩ + d_{m + 1} ∣ ϕ_{m + 1} ⟩ .

H = A^{†} A, H = A A^{†} .

H = A^{†} A, H = A A^{†} .

c_{n}^{2} = - b_{n} \frac{P _{n + 1} ( ε _{0} )}{P _{n} ( ε _{0} )}, d_{n + 1}^{2} = - b_{n} \frac{P _{n} ( ε _{0} )}{P _{n + 1} ( ε _{0} )}

c_{n}^{2} = - b_{n} \frac{P _{n + 1} ( ε _{0} )}{P _{n} ( ε _{0} )}, d_{n + 1}^{2} = - b_{n} \frac{P _{n} ( ε _{0} )}{P _{n + 1} ( ε _{0} )}

ε P_{n} (ε) = b_{n - 1} P_{n - 1} (ε) + a_{n} P_{n} (ε) + b_{n} P_{n + 1} (ε) .

ε P_{n} (ε) = b_{n - 1} P_{n - 1} (ε) + a_{n} P_{n} (ε) + b_{n} P_{n + 1} (ε) .

H^{(+)} (η) = H (η + δ) + R (η)

H^{(+)} (η) = H (η + δ) + R (η)

a_{n}^{(+)} (η) = a_{n} (η + δ) + ε_{1} (η),

a_{n}^{(+)} (η) = a_{n} (η + δ) + ε_{1} (η),

b_{n}^{(+)} (η) = b_{n} (η + δ) .

b_{n}^{(+)} (η) = b_{n} (η + δ) .

c_{n}^{2} (η) + d_{n + 1}^{2} (η) = c_{n}^{2} (η + δ) + d_{n}^{2} (η + δ) + ε_{1} (η)

c_{n}^{2} (η) + d_{n + 1}^{2} (η) = c_{n}^{2} (η + δ) + d_{n}^{2} (η + δ) + ε_{1} (η)

c_{n + 1} (η) d_{n + 1} (η) = c_{n} (η + δ) d_{n + 1} (η + δ)

c_{n + 1} (η) d_{n + 1} (η) = c_{n} (η + δ) d_{n + 1} (η + δ)

ε_{1} (η) = (c_{n}^{2} (η) - c_{n + 1}^{2} (η)) + (d_{n + 1}^{2} - d_{n}^{2})

ε_{1} (η) = (c_{n}^{2} (η) - c_{n + 1}^{2} (η)) + (d_{n + 1}^{2} - d_{n}^{2})

ε_{1} (η) = (c_{0}^{2} (η) - c_{1}^{2} (η)) + d_{1}^{2}

ε_{1} (η) = (c_{0}^{2} (η) - c_{1}^{2} (η)) + d_{1}^{2}

\overline{H}^{(+)} (η) = B B^{†} = T A A^{†} T^{†} = H^{(+)} (η - δ) .

\overline{H}^{(+)} (η) = B B^{†} = T A A^{†} T^{†} = H^{(+)} (η - δ) .

\overline{H}^{(+)} (η) ∣ ϕ_{n} ⟩ = c_{n} (η - δ) d_{n} ∣ ϕ_{n - 1} ⟩ + (c_{n}^{2} (η - δ) + d_{n + 1}^{2}) ∣ ϕ_{n} ⟩ + c_{n + 1} (η - δ) d_{n + 1} ∣ ϕ_{n + 1} ⟩ .

\overline{H}^{(+)} (η) ∣ ϕ_{n} ⟩ = c_{n} (η - δ) d_{n} ∣ ϕ_{n - 1} ⟩ + (c_{n}^{2} (η - δ) + d_{n + 1}^{2}) ∣ ϕ_{n} ⟩ + c_{n + 1} (η - δ) d_{n + 1} ∣ ϕ_{n + 1} ⟩ .

[B, B^{†}] ∣ ϕ_{n} ⟩ = ([c_{n - 1}^{2} (η) + d_{n + 1}^{2}] - [c_{n}^{2} (η) + d_{n}^{2}]) ∣ ϕ_{n} ⟩ .

[B, B^{†}] ∣ ϕ_{n} ⟩ = ([c_{n - 1}^{2} (η) + d_{n + 1}^{2}] - [c_{n}^{2} (η) + d_{n}^{2}]) ∣ ϕ_{n} ⟩ .

[B, B^{†}] = ε_{1} (η - δ) .

[B, B^{†}] = ε_{1} (η - δ) .

∣ ε_{m + 1} (η) ⟩ = \frac{1}{k = 0 \sum m ε _{1} ( η + k δ )} B^{†} \frac{1}{k = 0 \sum m - 1 ε _{1} ( η + k δ )} B^{†} ... \frac{1}{ε _{1} ( η ) + ε _{1} ( η + δ )} B^{†} \frac{1}{ε _{1} ( η )} B^{†} ∣ ε_{0} (η) ⟩

∣ ε_{m + 1} (η) ⟩ = \frac{1}{k = 0 \sum m ε _{1} ( η + k δ )} B^{†} \frac{1}{k = 0 \sum m - 1 ε _{1} ( η + k δ )} B^{†} ... \frac{1}{ε _{1} ( η ) + ε _{1} ( η + δ )} B^{†} \frac{1}{ε _{1} ( η )} B^{†} ∣ ε_{0} (η) ⟩

∣ ε_{m + 1} (η) ⟩ = \frac{1}{k = 0 \sum m ε _{1} ( η + k δ )} B^{†} ∣ ε_{m} (η) ⟩ .

∣ ε_{m + 1} (η) ⟩ = \frac{1}{k = 0 \sum m ε _{1} ( η + k δ )} B^{†} ∣ ε_{m} (η) ⟩ .

H^{(+)} (η - δ) = H (η) + ε_{1} (η - δ) .

H^{(+)} (η - δ) = H (η) + ε_{1} (η - δ) .

H^{(+)} (η - δ) ∣ ε_{m} ⟩ = [H (η) + ε_{1} (η - δ)] ∣ ε_{m} ⟩

H^{(+)} (η - δ) ∣ ε_{m} ⟩ = [H (η) + ε_{1} (η - δ)] ∣ ε_{m} ⟩

ε_{m}^{(+)} (η - δ) ∣ ε_{m} ⟩ = [ε_{m} (η) + ε_{1} (η - δ)] ∣ ε_{m} ⟩ .

ε_{m}^{(+)} (η - δ) ∣ ε_{m} ⟩ = [ε_{m} (η) + ε_{1} (η - δ)] ∣ ε_{m} ⟩ .

ε_{m}^{(+)} (η - δ) = [ε_{m} (η) + ε_{1} (η - δ)]

ε_{m}^{(+)} (η - δ) = [ε_{m} (η) + ε_{1} (η - δ)]

ε_{m}^{(+)} (η) = [ε_{m} (η + δ) + ε_{1} (η)]

ε_{m}^{(+)} (η) = [ε_{m} (η + δ) + ε_{1} (η)]

ε_{m + 1} (η) = ε_{m} (η + δ) + ε_{1} (η) = k = 0 \sum m ε_{1} (η + k δ) .

ε_{m + 1} (η) = ε_{m} (η + δ) + ε_{1} (η) = k = 0 \sum m ε_{1} (η + k δ) .

ε_{1} (η + k δ) = [c_{n}^{2} (η + k δ) - c_{n + 1}^{2} (η + k δ)] + [d_{n + 1}^{2} - d_{n}^{2}] .

ε_{1} (η + k δ) = [c_{n}^{2} (η + k δ) - c_{n + 1}^{2} (η + k δ)] + [d_{n + 1}^{2} - d_{n}^{2}] .

ε_{1} (η + k δ) = [c_{n}^{2} (η) - c_{n + k + 1}^{2} (η)] + [d_{n + 1}^{2} - d_{n}^{2}] .

ε_{1} (η + k δ) = [c_{n}^{2} (η) - c_{n + k + 1}^{2} (η)] + [d_{n + 1}^{2} - d_{n}^{2}] .

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Full text

Properties of shape-invariant tridiagonal Hamiltonians

Hashim A. Yamani∗ and Zouhaïr Mouayn‡

Abstract

It has been established that a positive semi-definite Hamiltonian, $H$ , that has a tridiagonal matrix representation in a basis set, allows a definition of forward (and backward) shift operators that can be used to define the matrix representation of the supersymmetric partner Hamiltonian $H^{\left(+\right)\text{\ }}$ in the same basis. We show that if, additionally, the Hamiltonian has a shape invariant property, the matrix elements of the Hamiltonian are related in a such a way that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of super-symmetric partner Hamiltonians. Additionally, we derive the coherent states associated with this type of Hamiltonians and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also has tridiagonal matrix representation.

PACS: 02.60 Jh (or generally, 02.60.-x), 02.07.-c

Keywords:Supersymmetry, Shape-invariant potentials, Superpotential, Raising and lowering operators, Coherent states.

∗Mabuth-2414, Medina 42362-6959, Saudi Arabia

‡Department of Mathematics, Faculty of Sciences and Technics (M’Ghila)

P.O. Box. 523, Béni Mellal, Morocco.

1 Introduction

With the idea of supersymmetric quantum mechanics (SUSY) the concept of shape invariance was put forward by Gendenshtein [1]. The condition of shape invariance requires from the supersymmetric partner potentials a condition of type $V_{\eta}^{\left(+\right)}\left(x\right)=V_{f\left(\eta\right)}\left(x\right)+R\left(\eta\right)$ where $f\left(\eta\right)$ is a function of $\eta$ and $R\left(\eta\right)$ is independent of $x.$ This is equivalent to the operator relation $H^{\left(+\right)}\left(\eta\right)=H\left(f\left(\eta\right)\right)+R\left(\eta\right)$ which is also an integrability condition [1] that has been proved sufficient to get exact results. Indeed, SUSY and the shape-invariance condition provide an algebraic procedure to determine the entire spectrum of solvable quantum systems, without any need to solve a differential equation. Almost exactly one-dimensional potential problems encountered in quantum mechanics are shape invariant where the parameters are related by a translation $\gamma=f\left(\eta\right)=\eta+\delta$ [2]. Actually, with such potentials is associated the Lie algebra $so\left(2,1\right)$ which was used to recover the spectrum of the model. For a review of supersymmetry, shape invariance and exactly solvable potential see [3].

Traditionally, the literature of SUSY quantum mechanics is cast in terms of the system Hamiltonian’s representation in configuration space [4]. One of the benefits of this representation is to deduce the explicit form, in configuration space, of the partner potential and the so-called superpotential. Alternatively, working with the matrix representation of a Hamiltonian $H$ in a given basis set $\left(\mid\phi\rangle\right)_{n=0}^{\infty}$ has proved to be a viable description of the physical system ever since the beginning years of the development of quantum mechanics. It is by now an established fact that many calculation tools in quantum mechanics employ the language of matrix representation of physical operators in a basis [5]. The authors have already shown [6, 7] that if the matrix representation of the Hamiltonian in the basis is tridiagonal,

[TABLE]

a supersymmetric partner Hamiltonian $H^{\left(+\right)}$ also has a tridiagonal representation in the same basis,

[TABLE]

with the coefficients related as

[TABLE]

Here the two sets of coefficients $\left(c_{n}\right)$ and $\left(d_{n}\right)$ figure in the definition of the forward-shift operator $A$ through specifying its action on a basis vector as

[TABLE]

This means that the action of its adjoint $A^{\dagger}$ on a basis is now

[TABLE]

The Hamiltonian and its supersymmetric partner are related to these operators as

[TABLE]

More explicitly, the parameters $\left(c_{n}\right)$ and $\left(d_{n}\right)$ can be calculated recursively if the tridiagonal matrix elements of the Hamiltonian are known, or alternatively from the relations

[TABLE]

where $\varepsilon_{0}$ is the energy of the ground state. Here, the set $\left(P_{n}\left(\varepsilon\right)\right)$ is a solution of three-term recursion relation

[TABLE]

In this paper, we intent to explore the implications of shape invariance and find the additional properties that the above basic parameters satisfy and which in turn enable us to find solution to the physical quantities associated with the system Hamiltonian. In section 2, we explore the form of various parameters list above and particular the set $\left\{c_{n},d_{n}\right\}$ which plays a fundamental role in the descriptions. We show that we may define a modified version of the operators $A$ and $A^{\dagger}$ which can be interpreted essentially as lowering and raising operators. This allows us to specify the full energy spectrum of the system using only the parameters $\left\{c_{0},d_{1}\right\}.$ This also leads easily to the determination of the supersymmetric partner potential. We also show that the shape-invariance property allows the complete specification of the set of $c_{n}-$ like and the $d_{n}-$ like parameters associated with the hierarchy of supersymmetric partner Hamiltonians. Although in our setup the concept of superpotential plays no role in describing the system, we, nonetheless, write down what the form of a quantity that has all the properties of the familiar superpotential. In section 3, we explicitly construct the coherent states associated with a shape-invariant tridiagonal Hamiltonian and show by examples that it possesses the expected properties. Section 4 is devoted to some concluding remarks. Throughout, we illustrate our results using cases previously studied leaving details to the appendices.

2 Implications of shape-invariance

2.1 Properties of the matrix elements of shape-invariant Hamiltonians

A given positive semi-definite Hamiltonian $H$ (with the ground state $\varepsilon_{0}$ set to zero) satisfying a shape-invariance property is related to its supersymmetric partner Hamiltonian $H^{\left(+\right)}$ via the relation

[TABLE]

where the parameters $\eta,\delta$ and $R$ are property of the given Hamiltonian. Since the spectrum of $H^{\left(+\right)}$ is shifted by an amount of $\varepsilon_{1}$ compared to the spectrum of $H,$ we must have $R\left(\eta\right)=\varepsilon_{1}\left(\eta\right).$ Furthermore, if $H$ has a tridiagonal representation in a basis as in Eq. $\left(1.1\right)$ , then so does $H^{(+)}$ as in Eq. $\left(1.2\right).$ Hence the above symmetry relation translates into the following connections among their matrix elements,

[TABLE]

From the relations $\left(1.3\right)-\left(1.4\right)$ of the parameters $\left(a_{n}^{\left(+\right)},b_{n}^{\left(+\right)}\right)$ and $\left(a_{n},b_{n}\right)$ to the basic coefficients $\left(c_{n},b_{n}\right)$ , the above relations means that

[TABLE]

We now postulate that the coefficients $\left(d_{n}\right)$ are independent of the parameter $\eta.$ This turns out to be the case for many of the known shape-invariant Hamiltonians such as Harmonic oscillator and the Morse Hamiltonians. In that case, $d_{n}\left(\eta\right)=d_{n}\left(\eta+\delta\right)=d_{n}.$ This gives $c_{n+1}\left(\eta\right)=c_{n}(\eta+\delta).$ Putting the above relations together, we get

[TABLE]

We make two remarks regarding the above important result. First, to calculate $\varepsilon_{1}\left(\eta\right)$ there is no need to identify explicitly what the parameter $\eta$ is. The second is that this result is independent of $n$ . Specifically, if we set $n=0,$ we have the simple yet powerful relation

[TABLE]

giving $\varepsilon_{1}\left(\eta\right)$ in terms of only the three parameters $\left\{c_{0}\left(\eta\right),c_{1}\left(\eta\right),d_{1}\right\}.$

As is suggestive in the symmetry property Eq. $\left(2.1.0\right)$ , we will find it convenient to utilize the translation operator $T:=e^{-\delta\frac{\partial}{\partial\eta}}.$ In fact, if $K\left(\eta\right)$ is any operator, then $T^{{\dagger}}K\left(\eta\right)T=K\left(\eta+\delta\right)$ . Here, of course, $T^{{\dagger}}=e^{\delta\frac{\partial}{\partial\eta}}.$ We choose to integrate this property in the basic setup by replacing the operator $A$ by the operator $B=TA$ and its adjoint $B^{{\dagger}}=A^{{\dagger}}T^{{\dagger}}$ . Remarkably, we still have $H\left(\eta\right)=A^{{\dagger}}A=B^{{\dagger}}B.$ On the other hand, we have a new version of the supersymmetric partner Hamiltonian, namely $\overline{H}^{\left(+\right)}\left(\eta\right)=BB^{{\dagger}}.$ It is related to $H^{\left(+\right)}\left(\eta\right)$ simply as

[TABLE]

Because of this relation, it is clear that $\overline{H}^{\left(+\right)}\left(\eta\right)$ also has a tridiagonal matrix representation in the same basis, namely

[TABLE]

Now comparing $BB^{{\dagger}}\mid\phi_{n}\rangle$ and $B^{{\dagger}}B\mid\phi_{n}\rangle$ while utilizing the symmetry properties $d_{n}\left(\eta\right)=d_{n}\left(\eta+\delta\right)=d_{n}$ and $c_{n+1}\left(\eta\right)=c_{n}(\eta+\delta),$ we get

[TABLE]

It is not hard to recognize that the quantity on the right-hand side is just $\varepsilon_{1}\left(\eta-\delta\right)\mid\phi_{n}\rangle.$ We thus obtain the major result that

[TABLE]

This commutation relation suggests that the operators $B^{{\dagger}}$ and $B$ may be considered as raising and lowering operators provided their noncommutativity with $\varepsilon_{1}\left(\eta\right)$ is taken into account. The noncommutativity of the operators $B^{{\dagger}}$ and $B$ with any function $f\left(\eta\right)$ is given by the following two very important relations $f\left(\eta\right)B^{{\dagger}}=B^{{\dagger}}f\left(\eta-\delta\right)$ and $f\left(\eta\right)B=Bf\left(\eta+\delta\right).$

We list in Table 1 results that apply to three shape-invariant Hamiltonians having tridiagonal matrix representation, namely the $l$ -th partial wave Kinetic energy, Harmonic oscillator and Morse Hamiltonians.

2.2 The operator $B^{{\dagger}}$ as a raising operator

In this subsection, we clarify the sense of $B^{{\dagger}}$ behaving as a raising operator. In fact, we will show that

[TABLE]

The proof can be proceed by induction (see Appendix A). An iterative version of the relation is

[TABLE]

The latter one shows even more the sense in which $B^{{\dagger}}$ is considered as a raising operator since it can generate the higher energy eigenstates not only from the ground state as in Eq.(2.2.0) but alternatively from the level just below it. This property will be exploited to construct coherent states using the displacement operator $D\left(z\right)=\exp\left(zB^{{\dagger}}-z^{\ast}B\right)$ as we will show in Section 3. As an example, we carry out explicitly in Appendix B a calculation that shows that the operator $B$ acts as a lowering operator for the harmonic oscillator Hamiltonian.

2.3 The energy spectrum

It is remarkable how the shape-invariance together with minimal extra information leads to full characterization of the system’s energy spectrum. To see this, we note that from Eq.(2.1.0),

[TABLE]

Since the operators $B^{{\dagger}}$ and $B$ are treated as raising and lowering operators, the Hamiltonians share the same spectrum except for the ground state. So, if $H\left(\eta\right)\mid\varepsilon_{m}\rangle=\varepsilon_{m}\left(\eta\right)\mid\varepsilon_{m}\rangle,$ then the above equation gives

[TABLE]

Hence,

[TABLE]

or

[TABLE]

On the other hand, if we write $\varepsilon_{m}^{\left(+\right)}\left(\eta\right)=\varepsilon_{m+1}$ then an interesting new result emerges for tridiagonal Hamiltonians satisfying the shape invariance property, namely,

[TABLE]

Since we can derive $\varepsilon_{m}\left(\eta\right)$ and $\varepsilon_{m}^{\left(+\right)}\left(\eta\right)$ from $\varepsilon_{1}\left(\eta\right),$ and since $\varepsilon_{1}\left(\eta\right)$ can be derived from the three parameters $\left\{c_{0},c_{1,}d_{1}\right\}$ as per Eq.(2.1.6), the system energy spectrum can essentially be derived from these parameters as well. Alternatively, if we know $\left\{\left(c_{n}\right)_{n=0}^{\infty},d_{1}\right\},$ then we can find a very simple expression for $\varepsilon_{m}\left(\eta\right)$ or $\varepsilon_{m}^{\left(+\right)}\left(\eta\right).$ From Eq.(2.1.5) we have

[TABLE]

Also, from Eq.(2.1.3) and Eq.(2.1.4)

[TABLE]

Then from $d_{n}(\eta)=d_{n}(\eta+\delta)=d_{n}$ and $c_{n+1}(\eta)=c_{n}(\eta+\delta)$ we have

[TABLE]

We now use the telescoping property for the last sum. Specifically, we have

[TABLE]

A simple expression results from specializing to the case $n=0,$

[TABLE]

As examples, the parameters listed in Table 1 can be used to show how the above results apply to the listed Hamiltonians.

2.4 Supersymmetric partner potential

If the Hamiltonian satisfies the shape-invariance property of Eq.(2.1.0), then we have in more details

[TABLE]

Since the reference Hamiltonian $H_{0}\left(\eta\right)=-\frac{1}{2}\frac{d^{2}}{dx^{2}},$ we must have $H_{0}\left(\eta\right)$ independent of the parameter $\eta$ . Hence $H_{0}\left(\eta\right)=H_{0}\left(\eta+\delta\right).$ Therefore, the two partner potential are related as

[TABLE]

We verify this relation for three familiar potential in Table 1.

2.5 The coefficients $\left(c_{n},d_{n}\right)$ associated with

the hierarchy of supersymmetric shape-invariant Hamiltonians

For a given positive semi-definite tridiagonal Hamiltonian $H\left(\eta\right)$ (with the ground state energy $\varepsilon_{0}\left(\eta\right)=0$ ), the coefficients $\left(c_{n},d_{n}\right)$ are related to the coefficients $\left(a_{n},b_{n}\right)$ as in Eq.(1.3), namely, $a_{n}=c_{n}^{2}+d_{n}^{2},$ $b_{n}=c_{n}d_{n+1}.$ We then ask the question: what are the coefficients $\left(c_{n}^{\left(+\right)},d_{n}^{\left(+\right)}\right)$ that are associated with the partner Hamiltonian $H^{\left(+\right)}\left(\eta\right)$ , so that we can similarly write

[TABLE]

Since we already know that

[TABLE]

the answer as suggested in [6] is $c_{n}^{\left(+\right)}=d_{n+1}$ and $d_{n}^{\left(+\right)}=c_{n}$ . This is a general result. If we now repeat the same question regarding the spersymmetric partner $H^{\left(++\right)}\left(\eta\right)$ in the next level of hierarchy, we have to be careful to seek the supersymmetric partner of the Hamiltonian

[TABLE]

that has the ground state $\widetilde{\varepsilon}_{0}^{\left(+\right)}=0.$ In general, we do not know the explicit form of the energy $\varepsilon_{1}\left(\eta\right).$ However, for the shape invariant Hamiltonian, the set of parameters $\left\{c_{0},c_{1},d_{1}\right\}$ is enough to completely specify the energy spectrum, as has been demonstrated in the previous subsections. We now show that shape invariance enables us to find these coefficients associated with the hierarchy of supersymmetric partner Hamiltonians.

We now consider, as a starting point the Hamiltonian $\widetilde{H}^{\left(+\right)}\left(\eta\right)$ and seek its super-symmetric partner. From Eq.(2.5.2) and the explicit form of $\varepsilon_{1}$ as given by Eq.(2.1.6), we simply write the following two relations.

[TABLE]

It is easy to conclude that the pair $\left(\widetilde{c}_{n}^{\left(+\right)},\widetilde{d}_{n}^{\left(+\right)}\right)$ is just $(c_{n+1},d_{n})$ . We can now write the explicit form of the pair $\left(c_{n}^{\left(++\right)},d_{n}^{\left(++\right)}\right)$ associated with $H^{\left(++\right)}\left(\eta\right)$ which is the supersymmetric partner Hamiltonian to $\widetilde{H}^{\left(+\right)}\left(\eta\right).$ That is, $c_{n}^{\left(++\right)}=d_{n+1}$ and $d_{n}^{\left(++\right)}=c_{n+1}.$ This immediately leads to following explicit form of the pair $\left(a_{n}^{\left(++\right)},b_{n}^{\left(++\right)}\right)$ as follows: $a_{n}^{\left(++\right)}=c_{n+1}^{2}+d_{n+1}^{2}$ and $b_{n}^{\left(++\right)}=c_{n+2}d_{n+1}.$ Because of the shape-invariance property, we now verify that

[TABLE]

with $\widetilde{R}\left(\eta\right)=\left(\varepsilon_{2}-\varepsilon_{1}\right).$ To do that, we first note that $b_{n}^{\left(++\right)}=\widetilde{b}_{n}^{\left(+\right)}\left(\eta+\delta\right).$ Next, we have

[TABLE]

which is just $\left(\varepsilon_{2}-\varepsilon_{1}\right)$ by Eq.(2.3.5).

If we want to continue the process and find the analogous properties to the next Hamiltonian in the hierarchy, we first go through the intermediary Hamiltonian which has zero energy for its ground state, and then repeat the procedure outlined in detail above. The result is summarized in the Table 2.

2.6 Constructing $\left(c_{n},d_{n}\right)$ from the spectrum of a

shape-invariant Hamiltonian

We have just shown how to deduce the energy spectrum of a shape-invariant Hamiltonian from knowledge of the parameters $\{c_{n},d_{n}\}$ . In this subsection we will show the reverse: that from knowing the energy spectrum of a shape-invariant Hamiltonian together with additional minimal information we can create a tridiagonal representation of the shape-invariant Hamiltonian.

We recall Eq.(2.1.5) that $\varepsilon_{1}(\eta)=[c_{n}^{2}(\eta)-c_{n+1}^{2}]+[d_{n+1}^{2}-d_{n}^{2}]$ which is independent of the index $n$ . Thus, choosing $n=0$ , we also get $\varepsilon_{1}(\eta)=[c_{0}^{2}(\eta)-c_{1}^{2}]+d_{1}^{2}$ From the properties of the hierarchy of shape-invariant Hamiltonians detailed in the previous subsection, we recognize that the relationship between the set $\{c_{n},d_{n}\}$ and $\varepsilon_{1}(\eta)$ is analogous to the relationship between the set $\{c_{n+k},d_{n}\}$ and $\varepsilon_{1}(\eta+k\delta)$ . Therefore, we can write the analogous to Eq.(2.1.6), namely

[TABLE]

Now we compare this with Eq.(2.1.5) when $n=k$ . The result is

[TABLE]

We now sum each side of the above equation from $k=0$ to $k=m$ so as to obtain

[TABLE]

Using the telescoping property to evaluate the right-hand sum, we get

[TABLE]

Therefore, we have the major result that $d_{m+1}$ , is determined by the energy spectrum of the Hamiltonian and the parameter $d_{1}$ . More specifically,

[TABLE]

Furthermore, we already know from Eq.(2.3.6) that

[TABLE]

From the above relations, we can construct a tridiagonal matrix representation of a shape- invariant Hamiltonian from the details of its energy spectrum together with the two parameters $\{c_{0}^{2}(\eta),d_{1}^{2}\}$ .

2.7 The superpotential for shape-invariant tridiagonal Hamiltonian

In the ’differential’ treatment of supersymmetry [4], a quantity, called the superpotential $W(x)$ , is introduced with intimate connection to the ground state wave function, $\psi_{0}(x)$ , of the quantum system as follows:

[TABLE]

Although our treatment for the supersymmetry of tridiagonal Hamiltonians does not need the concept of superpotential, yet we feel the need to show that exist an analogous quantity having the expected property of the superpotential. To do that, we first start from the basic quantities $A,A^{\dagger}$ and their associated representation in the basis. We then define two related operators $\tilde{W}$ and $\tilde{D}$ as follows:

[TABLE]

Conversely, we have

[TABLE]

We note that $\tilde{W}$ is hermitian while $\tilde{D}$ is antihermitian. Also,

[TABLE]

Comparing the form of the Hamiltonian and its supersymmetric partner, we can identify the corresponding expressions as follows:

[TABLE]

This leads to the correspondence

[TABLE]

Taking this into account, and comparing relation Eq.(2.7.2) with the following analogous relation in ’differential’ supersymmetry

[TABLE]

we immediately make the correspondence

[TABLE]

Now we give several plausibility arguments to support the interpretation of $\tilde{W}$ as the superpotential. From the definition of $\tilde{W}$ in Eq.(2.7.1), we can write,

[TABLE]

since, as shown in Appendix C, the operator $A$ annihilates the ground states $|\varepsilon_{0}\rangle$ . This equivalent to

[TABLE]

More explicitly, if we assume that $\tilde{W}$ represent a local function of $x$ , we then have

[TABLE]

We see that this form corresponds exactly to the form of Eq.(2.7.0).

Additionally, we now show that $\tilde{W}$ related to the potentials $V\equiv V^{(-)}$ and $V^{(+)}$ as in the differential form of supersymmetry. We start with the proposition that $V^{(\pm)}\rightarrow\tilde{W}^{2}\pm\tilde{D}\tilde{W}\mp\tilde{W}\tilde{D}$ . Then for any state $|\varphi\rangle$ , we have

[TABLE]

Hence,

[TABLE]

which is the result quoted in the literature if we identify $\tilde{W}$ as the superpotential.

Let’s apply the above results to the Morse Hamiltonian. With

[TABLE]

and Eq(2.7.1) gives us

[TABLE]

On the other hand, Eq.(2.7.9) says has that $\tilde{W}(x)=-\langle x|\widetilde{D}|\varepsilon_{0}\rangle=\frac{-1}{2}\langle x|(A-A^{\dagger})|\varepsilon_{0}\rangle$ . Now $|\varepsilon_{0}\rangle$ is shown in appendix C to have the representation

[TABLE]

where

[TABLE]

Hence,

[TABLE]

Detailed calculation gives

[TABLE]

where $F(\zeta)=\sum\limits_{n=0}^{\infty}Q_{n}(0)L_{n}^{2\gamma}(\zeta)$ . This sum is found to be $\zeta^{D-(\gamma+\frac{1}{2})}\frac{\Gamma(2\gamma+1)}{\Gamma(\gamma+\frac{1}{2}+D)}$ . Now with

[TABLE]

we finally get

[TABLE]

Which is identical to the result of Eq.(2.7.16).

3 The coherent states associated with shape-invariant tridiagonal Hamiltonians

We construct the coherent state $|z\rangle$ by displacing the ground state $|\varepsilon_{0}\rangle$ as

[TABLE]

The exponential operator can be simplified by using the well-known factorization [8]

[TABLE]

By also making use of the commutation relation of Eq.(2.1.10), we can represent the coherent state in terms of the energy eigenstates of the Hamiltonian as

[TABLE]

In subsection 2.2, we have explored how the raising operator acts on the various eigenstates of the system. It follows from Eq.(2.2.0) that the coherent state can be written as

[TABLE]

We now show that these coherent states satisfy the resolution de the identity operator the quantum Hilbert space $\mathcal{H}$ on which the Hamiltonian operator is acting. That is

[TABLE]

where $\varrho\left(\left|z\right|^{2}\right)$ is an auxiliary density function to be determined. By replacing $|z\rangle$ by its expression in (4.7) inside the rank one operator $|z\rangle\langle z|$ and using polar coordinates $z=re^{i\theta},r>0,\theta\in\left[0,2\pi\right)$ then Eq.(4.8) leads to

[TABLE]

More explicitly, we demand that the function $\varrho\left(r^{2}\right)$ solves the equation

[TABLE]

This is equivalent to the moment problem

[TABLE]

for the function $x\longmapsto e^{-x\varepsilon_{1}\left(\eta-\delta\right)}\varrho\left(x\right)$ , which can be solved with the help of some known integral or by making an appeal to a transformation procedure, like Mellin [9] or Fourier.

As an example, we consider the harmonic oscillator Hamiltonian for which $\varepsilon_{1}\left(\eta\right)=2\omega.$ Therefore, the quantity on the right hand side of Eq.(4.11) becomes $\varepsilon$

[TABLE]

This is solved by the choice $\varrho\left(x\right)=4\omega.$

Another example is that of the Morse Hamiltonian where we have here $\varepsilon_{1}\left(\eta\right)=\frac{\alpha^{2}}{2}\left(2D-1\right)$ and $\eta=D,\delta=-1.$ Careful calculation shows that the quantity

[TABLE]

has the value

[TABLE]

This means that the function $\varrho\left(r^{2}\right)$ must satisfy the moment relation

[TABLE]

Of particular interest is the fact that the ground state is a coherent state $|z\rangle$ with $z=0$ since we have shown that $|0\rangle$ is just $|\varepsilon_{0}\rangle$ . This is true in any defining scheme of coherent states. One such scheme is the one that define ”à la Glauber” the coherent state $|z\rangle$ as eigenvector of the operator $B.$ That is $B|z\rangle=z|z\rangle$ . Also, if we define $|z\rangle$ as the eignevector of the operator $A;$ i.e., $A|z\rangle=z|z\rangle,$ then $|0\rangle$ is just $|\varepsilon_{0}\rangle$ . Now if we expand the vector in $|z\rangle$ in terms of the basis $\left\{|\phi_{n}\rangle\right\}_{n=0}^{\infty}$ as

[TABLE]

then the coefficients $\Lambda_{n}\left(z\right)$ can be found recursively as $\Lambda_{n}\left(z\right)=\Lambda_{0}\left(z\right)Q_{n}\left(z\right)$ where $Q_{0}\left(z\right)=1$ and

[TABLE]

This means that we can write the ground state vector in terms of the basis as

[TABLE]

We conclude that the set of coefficients $\left(c_{n},d_{n}\right)$ and the basis $\left\{|\phi_{n}\rangle\right\}_{n=0}^{\infty}$ are sufficient to write the ground state vector. We give two examples of such construction. For the harmonic oscillator Hamiltonian using the relevant quantities listed in Table 1, we easily find that

[TABLE]

With the associated basis, given in Table 1, we have for $\psi_{0}\left(x\right)=\langle x\mid\varepsilon_{0}\rangle$ :

[TABLE]

But we know that for $\left|t\right|<1,$ which is satisfied here, we have a closed form for the sum, namely

[TABLE]

Hence, we have

[TABLE]

On the other hand, we have

[TABLE]

Thus, we finally have

[TABLE]

which is indeed the ground state wave function. It is important to notice that the wave function is, as expected, independent of the free scale parameter $\lambda$ which characterizes the basis not the physical system. We give the details for the ground state of the Morse Hamiltonian in Appendix D.

4 Concluding remarks

We would like to emphasize that several important points regarding the supersymmetric properties of shape-invariant tridiagonal Hamiltonians. First, the basic quantities in our approach are the parameters $\left(c_{n},d_{n}\right)$ which are related to matrix elements of the tridiagonal Hamitonian by Eq. $\left(1.3\right)$ -Eq. $\left(1.4\right)$ . Second, for Hamiltonians with shape-invariance property, an important derived quantity is the energy of the first excited state $\varepsilon_{1}\left(\eta\right)$ which plays a pivotal role in the complete description of the Hamiltonian energy spectrum and that of its supersymmetric partner. Third, the approach adopted here accomodates familiar concepts such as the superpotential although they play no role in the analysis . Finally, the need is evident to catalogue as many cases of Hamiltonians having tridiagonal matrix representation in specific basis. The effort of Alhaidari et al [10] in this regard is commendable.

Appendix A. For the case $n=0$ , consider the vector $B^{(+)}|\varepsilon_{0}\rangle$ . Now

[TABLE]

since $(B^{(+)}B)|\varepsilon_{0}\rangle=(A^{(+)}A)|\varepsilon_{0}\rangle=H|\varepsilon_{0}\rangle=0$ and by using Eq.(2.1.10). This means that the vector $[B^{(+)}|\varepsilon_{0}\rangle]$ is proportional to the energy eigenvector $|\varepsilon_{1}\rangle$ with eigenvalue $\varepsilon_{1}(\eta)$ . Hence $[B^{(+)}|\varepsilon_{0}\rangle]\infty|\varepsilon_{1}(\eta)\rangle$ .

The normalized version of this vector is

[TABLE]

To see this, note that

[TABLE]

We now assume that the result holds for the case $k\leq n$ . We then consider the right-hand side of Eq.(2.2.0) for $n+1$ . We use the induction hypothesis to get :

[TABLE]

Now wee proceed to show that the resulting vector $\frac{1}{\sqrt{\displaystyle\sum_{0}^{n+1}\varepsilon_{1}(\eta+k\delta)}}B^{(+)}|\varepsilon_{n+1}\rangle$ is an eigenvector of the Hamiltonian with eigenvalue $\varepsilon_{n+2}$ . Specifically, we have

[TABLE]

But,

[TABLE]

Therefore,

[TABLE]

It is easy to show that the vector $\left[\sqrt{\displaystyle\sum_{k=0}^{n+1}\varepsilon_{1}(\eta+k\delta)}\right]^{-1}B^{(+)}|\varepsilon_{n+1}\rangle$ is indeed normalized. This completes the proof.

Appendix B. In this Appendix, we show in the case of the tridiagonal harmonic oscillator Hamiltonian, that $B=TA,\,(T=e^{-\delta\partial/\partial\eta})$ yields a lowering operator associated with the harmonic oscillator. We recall the signature property is that this operator has the following effect on the energy eigenstates:

[TABLE]

In particular, it annihilates the ground state since $\varepsilon_{0}=0$ . In fact, since $\varepsilon_{m}=2m\omega$ , we should be able to show that, for the harmonic oscillator,

[TABLE]

We recall from Table 1 that the basis that makes the harmonic oscillator tridiagonal is given by

[TABLE]

On the other hand, the energy eigenstate has the explicit form:

[TABLE]

If we now write

[TABLE]

then $\Gamma_{n,m}=\langle\phi_{n}|\varepsilon_{m}\rangle=\displaystyle\int_{0}^{\infty}\phi_{n}(r)\chi_{m}(r)$ . Detailed performance of this integral yields the following explicit result

[TABLE]

With $B=TA,\,(T=e^{-\delta\partial/\partial\eta})$ , together with the expansion of the energy eigenstate in terms of the basis, we have:

[TABLE]

But we know that $A|\phi_{n}\rangle=c_{n}|\phi_{n}\rangle+d_{n}|\phi_{n-1}\rangle$ . Hence,

[TABLE]

Using Eq.(B.6), we can write explicitly the quantity in parenthesis as

[TABLE]

where we have used the abbreviation $\tau=\left(\frac{\lambda^{2}+\omega}{\lambda^{2}-\omega}\right)^{2}$ and the relation ([11],p.47 )

[TABLE]

Now we use further the relation ([11],p.46 )

[TABLE]

for parameters

[TABLE]

Therefore $c_{n}(\eta)\Gamma_{n,m}(\eta)+d_{n+1}\Gamma_{n+1,m}(\eta)=\sqrt{2m\omega}\Gamma_{n,m-1}(\eta+1);\quad\eta\equiv\nu$ . With $\delta=+1$ , Eq.(B.8) becomes

[TABLE]

This shows that the operator $B=TA,\,(T=e^{-\delta\partial/\partial\eta})$ is the lowering operator for the Harmonic oscillator Hamiltonian.

Appendix C. We can check that $B|\varepsilon_{0}\rangle=0$ as follows. Using the expansion of $|\varepsilon_{0}\rangle$

[TABLE]

then, the action of the operator $A$ on this state is

[TABLE]

Since $c_{n}^{2}=-b_{n}\frac{P_{n+1}(\varepsilon_{0})}{P_{n}(\varepsilon_{0})}$ and $d_{n}^{2}=-b_{n}\frac{P_{n}(\varepsilon_{0})}{P_{n+1}(\varepsilon_{0})}$ , it follows that $A|\varepsilon_{0}\rangle=0$ . Recalling that $B=TA$ , it also follows that $B|\varepsilon_{0}\rangle=0$ .

Appendix D. The ground state ground state is a coherent state $|z\rangle$ with $z=0$ . The eigenvector can be written in terms of the basis as

[TABLE]

where $Q_{0}(z)=1,\,Q_{n}(z)=\prod\limits_{j=0}^{n-1}\left(\frac{z-c_{j}}{d_{j+1}}\right)$ for $n\geq 1$ , and $\Lambda_{0}(z)=\frac{1}{\sqrt{\sum\limits_{n=0}^{\infty}|Q_{n}(z)|^{2}}}$ .

Here we give details of the calculation of the ground state of the Morse Hamiltonian using the relevant parameter in Table l. We can easily find that

[TABLE]

With the basis that renders the Morse Hamiltonian tridiagonal, the ground state wave function has the form

[TABLE]

On the one hand,

[TABLE]

Now we make use of two important relations ([11], p.267):

[TABLE]

and ([11], p.289)

[TABLE]

This means that

[TABLE]

On the other hand,

[TABLE]

But ([11], p.40) :

[TABLE]

Therefore

[TABLE]

Thus,

[TABLE]

Combining all above results together, we finally get the following explicit form of the ground state wave function

[TABLE]

It is to be noticed, again, that the wave function is independent of the free scale parameter $\gamma$ which characterizes the chosen basis.

Table 1. Parameters and results K.E., radial oscillator and Morse hamiltonians.

$K.$ $E$ $Harmonic$ $Oscillator$ $Morse$ $Oscillator$

$H$ $-\frac{1}{2}\frac{d^{2}}{dr^{2}}+\frac{l(l+1)}{2r^{2}}$ $-\frac{1}{2}\frac{d^{2}}{dr^{2}}+\frac{l(l+1)}{2r^{2}}+\frac{1}{2}\omega^{2}r^{2}-(l+\frac{3}{2})\omega$ $-\frac{1}{2}\frac{d^{2}}{dx^{2}}+V_{0}(e^{-2\alpha x}-2e^{-\alpha x})+\frac{1}{2}\alpha^{2}D^{2}$

$\phi_{n}(y)$ $K_{n}(y)^{\nu+\frac{1}{2}}e^{-\frac{y^{2}}{2}}L_{n}^{\nu}(y^{2})$ $K_{n}(y)^{\nu+1/2}e^{-y^{2}/2}L_{n}^{\nu}(y^{2})$ $K_{n}(y)^{\gamma+1/2}e^{-y/2}L_{n}^{(\gamma)}(y)$

$y$ $\lambda r$ $\lambda r$ $\frac{\sqrt{8V_{0}}}{\alpha}e^{-\alpha x}$

$K_{n}$ $\sqrt{\frac{(2\lambda)n!}{\Gamma(n+\nu+1)}}$ $\sqrt{\frac{(2\lambda)n!}{\Gamma(n+\nu+1)}}$ $\sqrt{\frac{(\alpha)n!}{\Gamma(n+2\gamma+1)}}$

Free parameters $\lambda$ $\lambda$ $\gamma$

Other parameter $\nu=l+\frac{1}{2}$ $\nu=l+\frac{1}{2}$ $D=\frac{\sqrt{2V_{0}}}{\alpha}-\frac{1}{2}$

$c_{n}^{2}$ $\frac{\lambda^{2}}{2}(n+\nu+1)$ $\frac{(\lambda-\frac{\omega}{\lambda})^{2}}{2}(n+\nu+1)$ $\frac{\alpha^{2}}{2}\left(n+\gamma+\frac{1}{2}-D\right)^{2}$

$d_{n}^{2}$ $\frac{\lambda^{2}}{2}n$ $\frac{(\lambda+\omega/\lambda)}{2}n$ $\frac{\alpha^{2}}{2}n(n+2\gamma)$

$\varepsilon_{1}(\eta)$ [math] $2\omega$ $\frac{\alpha^{2}}{2}(2D-1)$

$\eta$ $\nu$ $\nu$ $D$

$\delta$ $1$ $1$ $-1$

$\varepsilon_{1}(\eta-\delta)$ [math] $2\omega$ $\frac{\alpha^{2}}{2}(2D+1)$

$\varepsilon_{\mu}^{(+)}(\eta)$ [math] $2(\mu+1)\omega$ $\frac{\alpha^{2}}{2}(\mu+1)(2D-(\mu+1))$

$[B,B^{{\dagger}}]$ [math] $2\omega$ $\frac{\alpha^{2}}{2}(2D-1)$

$V(r)$ $\frac{l(l+1)}{2r^{2}}$ $\frac{l(l+1)}{2r^{2}}+\frac{1}{2}\omega^{2}r^{2}-\left(l+\frac{3}{2}\right)\omega$ $\frac{\alpha^{2}}{2}\left(D+\frac{1}{2}\right)^{2}(e^{-2\alpha x}-2e^{-\alpha x})+\frac{\alpha^{2}}{2}D^{2}$

$V^{(+)}(r)$ $\frac{(l+1)(l+2)}{2r^{2}}$ $\frac{(l+1)(l+2)}{2r^{2}}+\frac{1}{2}\omega^{2}r^{2}-\left(l+\frac{3}{2}\right)\omega$ $\frac{\alpha^{2}}{2}\left(D-\frac{1}{2}\right)^{2}(e^{-2\alpha x}-2e^{-\alpha x})+\frac{\alpha^{2}}{2}D^{2}$

Table 2. Basic parameters for the hierarchy of supersymmetric partner hamiltonians.

[TABLE]

Bibliography11

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] L. Gendenshtein, JETP Lett. 38 (1983) 356
2[2] F. Cooper, A. Khare and U. P. Sukhatme, Phys. Rept. 251 (1995) 267.
3[3] R. Dutt, A. Khare, and U. P. Sukhatme, Am. J. Phys. 56 (1988) 163.
4[4] F. Cooper, A. Khare, and U. P. Sukhtame, Supersymmetry in Quantum Mechanics, World Scientific, Singapore (2001).
5[5] M. S. Swanson, A Concise Introduction to Quantum Mechanics, Morgan and Claypool Publishers (2018)
6[6] H. A. Yamani and Z. Mouayn, J. Phys. A: Math. Theor. 47 (2014) 265203.
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8[8] J. R. Klauder and B-S. Skagerstam, Coherent States, World Scientific, Singapore (1985).

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Properties of shape-invariant tridiagonal Hamiltonians

Abstract

1 Introduction

2 Implications of shape-invariance

2.1 Properties of the matrix elements of shape-invariant Hamiltonians

2.2 The operator B†B^{{\dagger}}B† as a raising operator

2.3 The energy spectrum

2.4 Supersymmetric partner potential

2.5 The coefficients (cn,dn)\left(c_{n},d_{n}\right)(cn​,dn​) associated with

2.6 Constructing (cn,dn)\left(c_{n},d_{n}\right)(cn​,dn​) from the spectrum of a

2.7 The superpotential for shape-invariant tridiagonal Hamiltonian

3 The coherent states associated with shape-invariant tridiagonal Hamiltonians

4 Concluding remarks

2.2 The operator $B^{{\dagger}}$ as a raising operator

2.5 The coefficients $\left(c_{n},d_{n}\right)$ associated with

2.6 Constructing $\left(c_{n},d_{n}\right)$ from the spectrum of a