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I am trying to understand how to compute the spectra of one-dimensional Schrödinger operators $$ \mathcal{L}:=-\partial_x^2+V, $$ where $V$ is a bounded function in the whole line. I am particularly interested in understanding the discrete spectra for the case $V:=1-2\,\text{sech}^2(x)$, which is a "localized" function but it has no sign neither decay. I think this case shouldn't be difficult, but I do not know how to compute it.

Is there any good reference where I can learn how to do it? Or can someone give me some hints? I would really appreciate it.

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put on hold as off-topic by Carlo Beenakker, Alexandre Eremenko, András Bátkai, LSpice, arsmath Jul 16 at 13:59

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