Should the Force Term in Anharmonic Oscillators Only Include Odd Powers of x?

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The discussion centers on the necessity of including only odd powers of the variable x in the force term of anharmonic oscillators. It is established that the highest power of x must be odd with a negative coefficient to ensure bound solutions. Conversely, even powers of x may yield bound solutions at low energy near the origin, but will lead to unbounded solutions at higher energy levels, resulting in a runaway non-restoring force. Graphing polynomials is recommended to visualize these behaviors.

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Is it necessary that force term in anharmonic oscillator should contain only third power in dependent variable(say x)? or any other higher power in dependent variable.
 
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The highest power of x in the force equation will have bound solutions if the power is odd (and the coefficient out front is negative). If the highest power of x is even, you will have the potential to have bound solutions at low energy (around the origin), but once you get to high enough energy the solutions won't be bounded anymore (i.e. at some point the force will be a runaway non-restoring force). Go graph some polynomials
to see this.
 

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