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Knowing the potential, we can find the spectrum of the Schrödinger operator. The converse question is: Knowing the spectrum, can we reconstruct the potential? As an example, a harmonic potential has an equally spaced spectrum. But is the converse true?

This is, of course, similar to the 'hearing the shape of the drum' problem, which has a negative answer. But we also should notice that in classical mechanics, if the potential is symmetric, we can recover it from the oscillation period as a function of the energy of the particle. This is due to ingenious work by Abel.

Danu
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Jiang-min Zhang
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The answer is no, I am afraid. As you may well know, the self-adjoint Laplace operator $-\Delta$ on $L^2(\mathbb{R})$ has purely absolutely continuous spectrum $\mathbb{R}^+$.

Now let $V\in L^{\infty}(\mathbb{R},\mathbb{R}^+)$ be an arbitrary bounded positive function. Then $-\Delta_x +V(x)$, where $V$ acts as a multiplicative operator is self-adjoint and has spectrum $\mathbb{R}^+$.

yuggib
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