I asked a question which had a problem that the Hamiltonian was not Hermitian. However, after thinking about it for some time I managed to make the Hamiltonian Hermitian. While it may still have problems I think it can be re-opened at least.

Hamiltonian that can produce random primes?

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    $\begingroup$ If you think that the main issue with that question was that the alleged Hamiltonian was not Hermitian, then you should read my comments again. There are several places where it's unclear what is happening mathematically, and additionally why you expect this to have any physical relevance at all. $\endgroup$ – ACuriousMind Jun 11 '17 at 17:56
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    $\begingroup$ I don't see where the physics lies in the question, it still seems purely mathematical to me. $\endgroup$ – Kyle Kanos Jun 11 '17 at 18:20
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    $\begingroup$ @KyleKanos I believe we've established that mathematical questions which arise in the context of physics are on topic here, and this question appears to fit that description - after all, the key question is what physical system corresponds to the Hamiltonian. Of course the question was closed for being unclear, which is independent of whether it is on topic. $\endgroup$ – David Z Jun 11 '17 at 21:17
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    $\begingroup$ @David Yes, those are on topic, but I don't see how it applies to the given question at all. $\endgroup$ – Emilio Pisanty Jun 11 '17 at 21:43
  • $\begingroup$ @DavidZ yes, in the context of physics math questions can be on topic. That said, please identify for​ me what physics underlies this question because I don't at all see it. $\endgroup$ – Kyle Kanos Jun 11 '17 at 22:18
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    $\begingroup$ @Kyle (and Emilio) as I mentioned in my previous comment, asking what physical system corresponds to a Hamiltonian (which is the only real question I can find in the post), is the physical context that keeps the question from being off topic as pure math, as far as I'm concerned. $\endgroup$ – David Z Jun 11 '17 at 22:40

One can think of $k$ as the measure of smaller solutions it is composed of. In fact, one can impose the condition $y(0)= \ln(p + \hat \epsilon)$ where $p$ is some arbitrary prime. Then, $k$ becomes the number of prime factors.

The last part is rather unclear. The number of prime factors of what?

Now making the change of variables in $ e^y y' = k \hat \epsilon$ such that $e^y = \frac{1}{z}\frac{d z}{dx} - \frac{1}{z} -I_{2 \times 2} \lambda x $,

To me, this is unclear too. A change of variables is of the form $y=f(z)$, and I cannot rewrite your expression into this form. Please try to fix this.

NOTE- This is invertible as: [...]

You should remove this part. It doesn't add anything to your question.

Making the change of variables $ z = \exp{((k \hat \epsilon+ \lambda ) x^2/2 + cx )}\exp(S)$

I don't understand what this is supposed to mean. You didn't define $S$ anywhere.

We add an additional boundary condition that removes the additional solution and $\lambda$ being an arbitrary constant:

Where is the additional boundary condition? I can't find it anywhere... Also, where does the equation $\frac{1}{z} \frac{d^2 z}{d x^2} - \lambda = k \hat \epsilon$ come from, and how is it related to anything you wrote before?

Also, you cannot take $$ (I_{2 \times 2} \frac{d^2 }{d x^2} - k \hat \epsilon) \cdot z = \lambda z $$ and "Taking dot product with a $2 \times 1$ matrix $\psi_A(z)$:" because the derivatives $\frac{d^2 }{d x^2}$ will act on $\psi_A(z)$ as well (because $z=z(x)$). So you either write everything in terms of $z$ (the function and the derivatives) or everything in terms of $x$. Mixing both variables makes no sense.

I don't see any logical connection from one paragraph to the next. Equations seem to pop in and out with no justification. Some mathematical manipulations are incorrect. You introduce constants and never define them. Finally, and most importantly, how is this a physics question?

Sorry, but the question is just as unclear as before. The hermiticity of $H$ is irrelevant.

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    $\begingroup$ I agree with much of this, in particular with the next-to-last paragraph. The question is long and not very focused; there are many parts that don't seem to have any logical connection to the main question being asked. $\endgroup$ – David Z Jun 11 '17 at 21:20
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    $\begingroup$ To add another inconsistency to the list, the equation $$\begin{pmatrix} 0 & H^\dagger \\ H & 0 \end{pmatrix} \begin{pmatrix} 0 \\ \phi_A \end{pmatrix} = E \begin{pmatrix} 0 \\ \phi_A \end{pmatrix},$$ on which OP seems to depend for "hermiticity", is patently and obviously inconsistent. $\endgroup$ – Emilio Pisanty Jun 11 '17 at 21:45

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