Sadly this is a 6th order equation rather than a second order equation.

So it seems we need to know through …

When calculating commutators that give ground state energies, the only surviving terms are of the form

These, of course, yield higher order terms, but the constant term given is simply

.

When looking at the second order, we also know that

So our commutators will carry the extra terms…

times an additional .

So it looks like the quartic ground state will be the summation of

or something like this.

The problem is how fast the will diverge. It appears that this will be difficult to resume if the diverge at all.

Again, according to Bender and Wu, the overall terms grows asymptotically like

which means that appear to diverge but at some algebraic rate, perhaps .

Not sure…

When calculating commutators that give ground state energies, the only surviving terms are of the form

These, of course, yield higher order terms, but the constant term given is simply

.

When looking at the second order, we also know that

So our commutators will carry the extra terms…

times an additional .

So it looks like the quartic ground state will be the summation of

or something like this.

The problem is how fast the will diverge. It appears that this is in good standing for resummability if the grow larger than this.

Again, according to Bender and Wu, the overall terms grows asymptotically like

which means that might diverge incredibly fast.

When calculating commutators that give ground state energies, the only surviving terms are of the form

These, of course, yield higher order terms, but the constant term given is simply

.

When looking at the second order, we also know that

So our commutators will carry the extra terms…

times an additional .

So it looks like the quartic ground state will be the summation of

or something like this.

The problem is how fast the will diverge. It appears that this is in good standing for resummability if the grow larger than this.

Again, according to Bender and Wu, the overall terms grows asymptotically like

which means that might diverge incredibly fast.

In this case will be our ground state correction for .

One thing we can see easily is that .

In fact we know from Bender and Wu (ca.1963) .

Let me write these three term energy values as

where corresponds to the correction, corresponds to the power of the overall perturbation, and corresponds to the energy level of the eigenstate.

So here we have:

when and otherwise.

when and when otherwise.

if is odd.

when .

Furthermore, we know

.

The question becomes whether we can define some function (or sequence of functions) which may function as “differentials” so that

.

means and .

]]>In this case we may be able to convert from being in terms of and

to being in terms of and . Luckily (let’s call for the moment)

We can mover the piece furthest to the right into

But

and if then

which makes our commutators a little easier.

Is there a way to setup a spectral sequence structure on the perturbed eigenvalues?

Consider that we need to compute these values…

1) lambda^k (in the power series)

2) x^n (the perturbation in the potential)

3) the m^th energy level for the x^n potential.

Therefore we’re looking for the terms E^k_{n,m}.

What’s known so far are

i) E^0_{n,m} all n,m

ii) E^1_{n,m} all n,m (solved by A.)

iii) E^k_{0,m} all k,m

iv) E^k_{1,m} all k,m

v) E^2_{4,m} all m (solved by A.)

What’s also known is that

E^k_{n,m} implies E^{k+1}_{n,0}

we get the next ground state if we know the entire previous solution.

Can we set up some sort of differential structure here?

Perhaps also we won’t have a d^2=0 but perhaps d^j = 0 for some other j…

Maybe

d^j : E^k_{n,m} rightarrow E^{k+1}_{n,m-j} or something like this…