Quasi-Exactly Solvable Time-Dependent Hamiltonians

A generalized method which helps to find a time-dependent Schrödinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.


Introduction
Another direction of investigation of quasi-exactly solvable Schrödinger is the study of time-dependent Hamiltonian.Time-dependence can be set through the potential.A first step is the direction was done in [1].This is related to the quasi-exactly solvable sextic anharmonic oscillator potentials.The Schrödinger equation is now considered with a time-dependent potential ( ) where ( ) The time-dependent potentials constructed from the well-known family of quasi-exactly solvable sextic anharmonic oscillator potentials ( ) ( ) are of the following form [1] ( ) where 0, 0 x t > ≥ , n is a non-negative integer, 0 k ≥ , β is real constant and ( ) u t is an arbitrary function of 0 t ≥ which is positive.If 1 k > , the last term in the above potential ( ) , V x t may be viewed as a centrifugal term in radial equation with x playing the role of radial coordinate.The domain of the definition of the po- tential (4) may be extended to the real line if 0,1 k = . After some algebraic manipulations, one has obtained the algebraic solutions of the Equation (1) of the form where the function ( ) , log 4 2 In this paper, we will construct time-dependent Schrödinger equation for any potential.It means that we will find algebraic solutions namely ( ) of that equation and one can build a time-dependent potential from any non time-dependent one.Note here that the static potential considered can be either quasi-exactly solvable (QES) or simply exactly solvable [2]- [4].It is understood that we will generalize the formalism considered in Ref. [1] where the authors have constructed a time-dependent Schrödinger equation for only one family of quasi-exactly solvable sextic anharmonic oscillator potentials.

Construction of a Time-Dependent Schrödinger Equation
The main results are summarized by the following proposition:

Let ( )
V y be a potential and ( ) y φ be a solution of the eigenvalue equation with eigenvalue λ .Let ( ) t ω be a positive (and derivable) function of t .Then, the solution of the Schrödinger equation with time-dependent potential is given by

Proof of the Proposition
We will discuss here an original method to construct time-dependent Hamiltonians which possess algebraic eigenvectors.Let us consider the Schrödinger equation, with ( ) y φ is an eigenfunction with eigenvalue λ of the Hamiltonian ( ) Note here that this Hamiltonian H (or the potential ( ) V y ) doesn't depend on time t explicitly, it means that t doesn't enter neither in the eigenvalue λ , nor in the eigenfunction ( ) As a consequence, the spectral Equation ( 11) is written as Let us pose and extend the effective potential of the above equation noted by adding a new term ( ) and consider a full Schrödinger equation of the form The next step is to determine the unknown function ( ) , R t x so that one can deduce the time-dependent algebraic solutions ( ) of the Equation ( 15) and relate it to (14).Obviously, the above Equation ( 15) can be developed as follows which can be rewritten Manifestly, this equation can be written in terms of φ (i.e. the first derivative terms of φ must be omitted (must vanish)) only if the following condition is imposed with this expression of the function ( ) , R t x , the Equation (17) takes the following form Replacing the expression by its equivalent one in this above equation, i.e.
2 λω φ as it is given in (14), one can write which can be rewritten ( ) From this equation, the added term ( ) ,  t x ∆ to the initial potential in (15) is easily expressed as , , Replacing ( ) , R t x in this equation by expression (18) and after some algebraic manipulations, one can write ( ) where One can easily remark that ( ) , t x ∆ is real and non-dependent on the eigenvalue λ only if it is expressed as ( ) This is possible due to the following condition Solving the above differential equation and after some algebraic manipulations, one can easily obtain the expression of the function ( ) With this expression of the function ( ) R t , the algebraic solutions of the time-dependent Schrödinger equa- tion , , , with the time-dependent potential are determined as where ( ) t ω is an arbitrary positive function of t and ( ) y φ is the eigenvector of the equation It means that one has constructed a time-dependent potential from the potential ( ) V y which is non time-dependent.This is the generalization of the particular case of potentials considered in Ref. [1].This is a particular case of ours because one can replace the original potential (i.e. the potential which is non time-dependent) in Equation ( 28) by any one which leads to a time-dependent potential associated to the above solutions ( ) as it is given by the Equation (29).These solutions are expressed in terms of the eigenvalues λ of the Schrödinger equation.The values of λ depend on a potential considered, i.e. when the potential is quasi-exactly solvable, only a part of the eigenvalues is found algebraically whereas when the potential considered is exactly solvable, all eigenvalues λ are calculated explicitly.So, we have constructed a generalized formula which helps to find time-dependent potentials, it means that one can deduce for a non time-dependent potential its associated time-dependent one.In the next step, we will use this method established previously, i.e. we will manipulate simply the Equation (28) and Equation (29) respectively to construct the time-dependent Lamé potential and the algebraic solutions of Schrödinger equation.We will also apply the above method to the known QES matrix polynomial operator [5] [6] and interesting remarks will be pointed out.

Example 1: Construction of Time-Dependent Lamé Potential
In this section, along the same lines of the above method, i.e. simply from the Equation (28), we will transform the non time-dependent potential associated to the Lamé equation into the time-dependent one.The Lamé equation is quasi-exactly solvable and the original form is as follows [7] where the Lamé potential is λ is the eigenvalue of the Lamé Hamiltonian and ( ) , sn y k is the Jacobi elliptic function with modulus ( ) . This function is periodic (i.e. the Lamé potential is also periodic) with period

( )
4K k which denotes the complete elliptic integral of the first type, i.e.

V t x ω
in the Equation (28) by the above Lamé potential (32), we find the following time-dependent Lamé potential It is easily observed that this last term in 2 x of (34) isn't periodic so that it spoils the periodicity of the above time-dependent Lamé potential.The above time-dependent Lamé potential (34) can become periodic only if the following condition is satisfied 0 ∆ = , ( ) where c is a real constant.
From the expression of ( ) From the above expressions ( 35) and ( 36), the time-dependent Schrödinger Equation ( 1) is of the following form Referring to the Equation (29) and Equation ( 35), the algebraic solutions of this Schrödinger equation are obtained Note that one can deduce from a non time-dependent potential (for which the eigenvalues λ exist) its corre- sponding time-dependent one by using the general formula established in Equation ( 28) while the algebraic solutions of the Schrödinger equation are found from the Equation (29).

Example 2: Extension to Matrix Time-Dependent Schrödinger Equation
The goal of this section is to construct a matrix time-dependent Schrödinger equation by the above method used to find the time-dependent potential of the non coupled Lamé equation.Let us consider the following matrix Hamiltonian [5] [6] ( ) ( ) where the potential ( ) where 1 3 , σ σ are the Pauli matrices, 2 1 is the matrix identity, 1 2 0 , , p p k are free real parameters and m is an integer.

( )
H y can be written in the matrix form as follows ( ) From the Equation (46), this equality can be considered