In addition to @tomleslie 's solutions there is one real one between 0 and 1. I have also made the mininum number of changes to convert from the old linalg package to the LinearAlgebra package. To more consistently find the roots, you could use the NextZero routine in the RootFinding package.

I can't see any typos, so I don't know where the 0.5 root could come from.

Download qm_maple_-_periodic_potentials.mw
 

As @acer suggested, they have different compressions (and formats). Corel PhotoPaint reports the original 125 KB file is 1 bit B&W, 300 dpi, CCITT group 3 subformat (compression), and the 133 KB one written by Maple is 72 dpi, 8 bit grayscale and LZW compression. LZW is a lossless compression, but it is interesting that the output resolution is lower.

Edit: aside from the resolution issue, the above is as indicated on the ImageTools[Formats] help page.

There was nothing wrong with  the actual code, but it hadn't been entered after a prompt (in a worksheet) and was somehow two separate regions. If you enter 1-D code as you did, you need to format it yourself. You get some nicer formatting with a code edit region if you are going to enter a lot of code.

For this site, uploading a worksheet as you did is probably best.

proc.mw

Edit: Using a startup code edit region for initialization and procedure definitions can be a nice way to do it.

The help for CayleyTableGroup specifies: The single required argument is the Cayley table (or multiplication table): an n by n
 Matrix or Array, m, describing the group with n elements, where the [i, j] entry contains the positive integer k such that the product of the ith and j h group elements is the kth group element.

So this needs to be a matrix of integers.

Download Cayley.mw

I don't get an error message for the plot in Maple 2015, just an empty plot. But eval(S,w=1) gives -1.853089027*10^329399 so your numbers seem to be unreasonably large.

Download realroot.mw

[Edited based on Carl's comment]. At least for real parameters (where the Matrix is Hermitian), there must be four eigenvectors.Not setting phi=Pi circumvents the OPs error message, and 4 eigenvectors are found, but then substituting phi=Pi leads to an error.

Download Hermitian2.mw

When legend is a list there need to be multiple curves, but you had one curve with 3 points. One way to do this is:

display(pointplot([28,.6481496576],color=blue),pointplot([28,.648149657615473],color=orange),pointplot([28,.6512873548],color=red),style=point,labels = ["k", "y(k)"], legend = ["10-digit precision", "15-digit precision", "Floating-point iteration"] ,legendstyle = [font = ["HELVETICA", 9], location = right]);
 

Not entirely certain I know what you want, but this may be a start.

[Edit: I changed pi (just a symbol) to Pi (the circle constant thing)]

At the beginning I define i to be sqrt(-1) as you want and D to be a regular variable rather than the differential operator.

Maple gives the answer as a sum over roots of a quartic polynomial. To get a simpler formula, you will need to give values to some variables or parhaps tell Maple their signs (using assuming)

Download Linear_System.mw

As well as Joe's corrections, you need to assign to y (y:=1). 
 

Download ww.mw

I suspect you want gcdex(x^9-a, b*x^6-c, x); but the answer is 1. You may need to specify the numerical value of more of the coefficients.

Try

T := unapply('fsolve'(m(t, c) = -1, t = 0 .. (1/2)*Pi), c, numeric);

In general use unapply to make functions from expressions earlier in the worksheet. The numeric option means that plot and other functions evaluated with a symbolic value will not throw an error.

So the constant _C3 means another condition is required. Presumably you want the solution to drop off at y=infinity.

pdetest suggests your solution does not solve the pde.
 

Download pde.mw

Use := to assign the right-hand side to the left. Some other issues also:

Download vectors.mw

Try entering in 1D Maple input, e.g. at a > prompt use ctrl-M then `≤`

You will see that `≤` is rendered as less-than-or-equals, as it would be in HTML.