determine the spectrum of the operator $$-\Delta-5\phi^4,\phi=3^{1/4}(1+|x|^2)^{-\frac{1}{2}},x\in R^3$$
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1$\begingroup$ Could you explain the choice of the constants and what you expect? $\endgroup$– Giorgio MetafuneCommented Mar 15, 2021 at 18:51
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1$\begingroup$ The spectrum is purely absolutely continuous on $[0,\infty)$, and there are negative eigenvalues. Normally, one would not expect to be able to find these explicitly. $\endgroup$– Christian RemlingCommented Mar 15, 2021 at 19:37
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$\begingroup$ @Christian Remling How you see the presence of negative eigenvalues? $\endgroup$– Giorgio MetafuneCommented Mar 15, 2021 at 19:55
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$\begingroup$ @GiorgioMetafune: Actually, it's not clear immediately, I confused something (though my guess still is that there is negative spectrum, potential -5/3 near $x=0$ feels negative enough, but one would have to look at it more closely, using min-max). $\endgroup$– Christian RemlingCommented Mar 15, 2021 at 20:20
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