I am trying to look for references on estimate of the lower bound of the spectrum of a Schrodinger operator $-\Delta + V$ on a bounded domain in three-dimensional space. For simplicity, we can take the domain to be a ball or torus, and the potential $V$ to be cylindrically radial, i.e. $V = V(r)$, where $r$ is as in the cylindrical coordinates.

Any suggestion is welcome - Thanks in advance!


1 Answer 1


Here are some very classical references:

M. Cwikel. Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math. (2), 106(1):93–100, 1977.

E. Lieb. Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc., 82(5):751–753, 1976.

G. V. Rozenbljum. Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR, 202:1012–1015, 1972.


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