I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential V(x)=−x4 has a discrete bound states with positive energy. How can it be?
Answer
More generally, Carl Bender et al. are considering PT-symmetric Hamiltonians of the form
H = p2+x2(ix)ε,ε∈R,
cf. e.g. Refs. 1-3. The Hamiltonian H is not self-adjoint in the usual sense, but self-adjoint in a PT-symmetric sense. OP's case corresponds to ε=2. The trick is to analytically continue the wave function ψ with real 1D position x∈R into the complex position plane x∈C, and prescribe appropriate boundary behaviour in the complex position plane.
See e.g. Refs. 1-3 and references therein for further details and applications. Note that Refs. 1-3 mainly discuss the point spectrum of the operator H.
References:
C.M. Bender, D.C. Brody, and H.F. Jones, Must a Hamiltonian be Hermitian?, arXiv:hep-th/0303005.
C.M. Bender, Introduction to PT-Symmetric Quantum Theory, arXiv:quant-ph/0501052.
C.M. Bender, D.W. Hook, and S.P. Klevansky, Negative-energy PT-symmetric Hamiltonians, arXiv:1203.6590.
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