## An analytical approach to the Schrödinger-Newton equations.(English)Zbl 0942.35077

For a single particle of mass $$m$$, the Schrödinger-Newton equations take the form $-{\hbar^2\over 2m} \nabla^2\psi +U\psi= E\psi, \quad \nabla^2 U=4\pi \gamma|\psi|^2 \tag{1}$ where $$\psi$$ is the wavefunction, $$U$$ is the gravitational potential energy, $$E$$ is the energy eigenvalue. If the functions $$S$$ and $$V$$ are introduced by $$\psi=[({\hbar^2\over 8\pi Gm^3})]^{1 \over 2}S$$, $$E-U= {\hbar^2 \over 2m}V$$ (1) reduces to the pair of equations $\nabla^2S=-SV, \quad \nabla^2 V=-S^2 .\tag{2}$ In this paper the authors consider the case when $$S$$ and $$V$$ are functions of radius $$r$$, only. The wavefunction is said to be normalized if $$\int^\infty_0 r^2S^2 dr=2G^{m^3 \over\hbar^2}$$. It is shown that there exists an infinite family of normalizable, finite energy solutions of (2) which are characterized by being smooth and bounded for all values of the radial coordinate. With this paper the authors provide analytical support for earlier numerical integrations, see I. M. Moroz, R. Penrose and P. Tod [Classical Quantum Gravity 15, No. 9, 2733-2742 (1998; Zbl 0936.83037)].

### MSC:

 35J60 Nonlinear elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J10 Schrödinger operator, Schrödinger equation 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)

### Keywords:

existence; energy solutions; characterizations

Zbl 0936.83037
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