I have started coming across square roots [itex](H+kI)^{\frac 12}[/itex] of slight modifications of Schrodinger operators [itex]H[/itex] on [itex]L^2(\mathbb R^d)[/itex]; that is, operators that look like this:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

H = -\Delta + V(x),

[/tex]

where [itex]\Delta[/itex] is the [itex]d[/itex]-dimensional Laplacian and [itex]V[/itex] corresponds to multiplication by some function. But how do we go about defining [itex](H+kI)^{1/2}[/itex]? My understanding was that we defined functions of self-adjoint operators by using the spectral theorem, but that only holds for bounded Borel functions, right? And [itex]f(x) = \sqrt x[/itex] certainly isn't bounded. And because of the [itex]\Delta[/itex], the operator [itex]H[/itex] isn't even bounded. So what do we do?

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# Defining the square root of an unbounded linear operator

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