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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.

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 [

where

The time-dependent potentials constructed from the well-known family of quasi-exactly solvable sextic anharmonic oscillator potentials

are of the following form [

where

where the function

In this paper, we will construct time-dependent Schrödinger equation for any potential. It means that we will find algebraic solutions namely

The main results are summarized by the following proposition:

Let

with eigenvalue

with time-dependent potential

is given by

We will discuss here an original method to construct time-dependent Hamiltonians which possess algebraic eigenvectors. Let us consider the Schrödinger equation,

with

Note here that this Hamiltonian

As a consequence, the spectral Equation (11) is written as

Let us pose

The next step is to determine the unknown function

which can be rewritten

Manifestly, this equation can be written in terms of

with this expression of the function

Replacing the expression

which can be rewritten

From this equation, the added term

Replacing

where

One can easily remark that

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

with the time-dependent potential

are determined as

where

It means that one has constructed a time-dependent potential from the 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 [

where the Lamé potential is

Replacing the potential

It is easily observed that this last term in

where

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

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 [

where the potential

where

where

In this case, the usual non time-dependent eigenvalue Schrödinger equation is of the form

where

with

From this change of variable, the Equation (43) takes the following form

where

After the change of function as

one can write the matrix time-dependent Schrödinger equation such that the initial potential acquires a supplementary term

which leads to

In the next step, we will calculate the function

Obviously, the two equations of the above system (51) can be linear respectively in

One can solve the first equation (or the second equation) in

From this expression of

In the next, the idea is to find the unknown function

where

From the Equation (46), this equality can be considered

in the above Equation (55) and accordingly one can write

As it has shown in the above method, this expression of

Finally, from the expression of

In this paper, referring to sextic anharmonic potentials considered in Ref. [

Indeed, we have applied this method to construct the time-dependent potential of Lamé equation. Along the same lines of the method, we have constructed a time-dependent potential associated to the matrix polynomial Hamiltonian which was also studied in [