arXiv:1407.4633 · Canadian Journal of Physics

Isospectral hermitian counterpart of complex nonhermitian Hamiltonian p2 – gx4 + a/x2

Asiri Nanayakkara, Thilagarajah Mathanaranjan

Published Apr 22, 2013• 1 citation

Abstract

We show that the nonhermitian Hamiltonians H = p2 – gx4 + a/x2 and the conventional hermitian Hamiltonians h = p2 + 4gx4 + bx [Formula: see text] are isospectral if a = (b2 – 4gℏ2)/16g and a ≥ –ℏ2/4. This new class includes the equivalent nonhermitian–hermitian Hamiltonian pair, p2 – gx4 and [Formula: see text] found by Jones and Mateo six years ago as a special case. When a = (b2 – 4gℏ2)/16g and a < –ℏ2/4 although h and H are still isospectral, b is pure imaginary, and h is no longer the hermitian counterpart of H.

DOI: 10.1139/cjp-2012-0562

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arXiv:1407.4633 · Canadian Journal of Physics

Isospectral hermitian counterpart of complex nonhermitian Hamiltonian p2 – gx4 + a/x2

Asiri Nanayakkara, Thilagarajah Mathanaranjan

Published Apr 22, 2013• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1139/cjp-2012-0562

Related papers

Equivalent Hermitian Hamiltonians for some non-Hermitian Hamiltonians

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Six years ago, by using operator techniques and path-integral methods, it was shown that the complex non-Hermitian

PT

-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}$ is equivalent to a conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}\ensuremath{-}2\ensuremath{\hbar}\sqrt{g}x$. Further it was revealed that the linear term in the Hermitian Hamiltonian is anomalous in the sense that it has no classical analog. In this paper we show that the complex non-Hermitian

\mathit{PT}

-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}+4i\ensuremath{\hbar}\sqrt{g}x$ and the conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}+6\ensuremath{\hbar}\sqrt{g}x$ have the same eigenspectra. In this case, the anomalous terms in both Hamiltonians are different from the previous one and vanish in the semiclassical limit. Further these equivalent Hamiltonians have zero-energy ground states. The exact ground-state wave functions and supersymmetric partner potentials are derived.

A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator

η

are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.

The generalized Swanson Hamiltonian

H_{GS} = w (\tilde{a}\tilde{a}^\dag+ 1/2) + \alpha \tilde{a}^2 + \beta \tilde{a}^{\dag^2}

with

\tilde{a} = A(x)d/dx + B(x)

, can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as

[\tilde{a},\tilde{a}^\dag]=

constant. However, the main objective of this paper is to show that though the commutator of

\tilde{a}

and

\tilde{a}^\dag

is constant, the generalized Swanson Hamiltonian is not necessarily isospectral to the harmonic oscillator. Reason for this anomaly is discussed in the frame work of position dependent mass models by choosing

A(x)

as the inverse square root of the mass function.

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian

\mathcal{P}\mathcal{T}

-symmetric wrong-sign quartic Hamiltonian $H=\frac{1}{2}{p}^{2}\ensuremath{-}g{x}^{4}$ has the same spectrum as the conventional Hermitian Hamiltonian $\stackrel{\texttildelow{}}{H}=\frac{1}{2}{p}^{2}+4g{x}^{4}\ensuremath{-}\sqrt{2g}x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian

\mathcal{P}\mathcal{T}

-symmetric Hamiltonian. This anomaly in the Hermitian form of a

\mathcal{P}\mathcal{T}

-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into

H

. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $\ensuremath{-}{\ensuremath{\phi}}^{4}$ quantum field theory in higher-dimensional space-time are discussed.