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For Hamitonian operator like this form $-\Delta^2+V(x)$, the ground state is always non-degeneracy in $n$-dim if the potential is continuous and bounded from below and let $-\Delta^2+V(x)$ be essentially self-adjoint. You can see the proof in Page 51 James Glimm and Arthur Jaffe's Quantum Physics. Or see the proof.

If you don't limit the Hamitonian to this form( $-\Delta^2+V(x)$), then if you put magnetic field then it's easy to construct the degeneracy ground state. see Landau level.

For Hamitonian operator like this form $-\Delta +V(x)$, the ground state is always non-degeneracy in $n$-dim if the potential is continuous and bounded from below and let $-\Delta +V(x)$ be essentially self-adjoint. You can see the proof in Page 51 James Glimm and Arthur Jaffe's Quantum Physics. Or see the proof.

If you don't limit the Hamitonian to this form( $-\Delta +V(x)$), then if you put magnetic field then it's easy to construct the degeneracy ground state. see Landau level.

For Hamitonian operator like this form $-\Delta^2+V(x)$, the ground state is always non-degeneracy in $n$-dim if the potential is continuous and bounded from below and let $-\Delta^2+V(x)$ be essentially self-adjoint. You can see the proof in Page 51 James Glimm and Arthur Jaffe's Quantum Physics. Or see the proof.

If you don't limit the form of $-\Delta^2+V(x)$, then if you put magnetic field then it's easy to construct the degeneracy ground state. see Landau level.

For Hamitonian operator like this form $-\Delta^2+V(x)$, the ground state is always non-degeneracy in $n$-dim if the potential is continuous and bounded from below and let $-\Delta^2+V(x)$ be essentially self-adjoint. You can see the proof in Page 51 James Glimm and Arthur Jaffe's Quantum Physics. Or see the proof.

If you don't limit the Hamitonian to this form( $-\Delta^2+V(x)$), then if you put magnetic field then it's easy to construct the degeneracy ground state. see Landau level.

For Hamitonian operator like this form $-\Delta^2+V(x)$, the ground state is always non-degeneracy. You can see the proof in Page 51 James Glimm and Arthur Jaffe's Quantum Physics. Or see the proof.

If you don't limit the form of $-\Delta^2+V(x)$, then if you put magnetic field then it's easy to construct the degeneracy ground state. see Landau level.

For Hamitonian operator like this form $-\Delta^2+V(x)$, the ground state is always non-degeneracy in $n$-dim if the potential is continuous and bounded from below and let $-\Delta^2+V(x)$ be essentially self-adjoint. You can see the proof in Page 51 James Glimm and Arthur Jaffe's Quantum Physics. Or see the proof.

If you don't limit the form of $-\Delta^2+V(x)$, then if you put magnetic field then it's easy to construct the degeneracy ground state. see Landau level.