Definition of quantum anharmonicity I have been reading research papers in mathematical physics for some months now, and I've seen the the term "anharmonic oscillator" quite frequently. At first I assumed that given a Schrodinger equation
$$\frac{d^2u}{dx^2}+(E-V(x))u=0$$
where $E$ is the energy, and $V(x)$ is the potential function. If $V(x) = x^2 +$ higher order polynomial terms, then this gives rise to the anharmonic oscillator since the higher order terms ensure that the potential will deviate from the "harmonic path". However, I've recently seen potentials of the form $$V(x) = \frac{1}{x^3}+\frac{1}{x^4}+\frac{1}{x^5}$$ described anharmonic oscillator as well. I just wish to know what is a good definition of anharmonic oscillators and anharmonicity?
 A: One has to be careful with the given potential.  To start with it must be shown that
$$h=-(d/dx)^2+V(x),$$
defines a unique self-adjoint operator $H$, i.e., is essentially self-adjoint. In case
$$V(x)=ax^2+bx^3+cx^4$$
with $c>0$ this is indeed the case. In fact the resolvent of $H$ is compact (these matters are discussed in the books by Reed and Simon), so $H$ has discrete spectrum. In case $c=0$ and $b≠0$ then $h$ is not  bounded from below, which, apart from the self-adjointness matter, makes physically no sense.
As to
$$V(x)=\frac{1}{x^3}+\frac{1}{x^4}+\frac{1}{x^5},$$
this potential is far too singular in $x=0$ to lead to a correct Hamiltonian. Maybe you can indicate in which context you encountered them?
In summary, a polynomial potential with even highest order term and positive coefficient $V_{2n}=a_{2n}x^{2n}$, where $a_{2n}$ is positive, is acceptable. The analogue of the quartic potential plays an important role in quantum field theory.
A: I've always treated anharmonic oscillators to mean the potential has the form
$$
V(x)=\gamma x^2 + \beta_ix^i
$$
with $i$ being any value except 2, including negative values as well. Anharmonicity then follows as the deviation of the eigenvalue of $V(x)$ above from the harmonic solution.
For example, the paper you link above, Case 1 has an energy eigenvalue of 
$$
E_n'=\hbar\omega\left(n+\frac32+\frac{c}{\sqrt{2d}}\right)
$$
So then the anharmonicity would be
$$
\Delta E=\left|E_n - E_n'\right|=\hbar\omega\left(1+\frac{c}{\sqrt{2d}}\right)
$$
