What are "bound states" (in quantum physics)?

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I read in Roger Newton's "Inverse Schrödinger Scattering in Three Dimensions" that the inverse problem in one dimension is not unique if there are bound states. Niether in his book nor on Wikipedia is there a good mathematical description of what a "bound state" is, thus I was wondering if someone could explain it to me (being a mathematician in analysis with almost no background in physics)?