Analytically Solvable PT-Invariant Periodic Potentials

Associated Lam\'e potentials V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\cn^2 (x,m)}/{\dn^2(x,m)} are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation $x \to ix+\beta$, where $\beta$ is any nonzero real number. These PT-invariant potentials are defined by $V^{PT}(x) \equiv -V(ix+\beta)$, and have a different real period from $V(x)$. They are analytically solvable potentials with a finite number of band gaps, when $a$ and $b$ are integers. Explicit expressions for the band edges of some of these potentials are given. For the special case of the complex potential V^{PT}(x)=-2m\sn^2(ix+\beta,m), we also analytically obtain the dispersion relation. Additional new, solvable, complex, PT-invariant, periodic potentials are obtained by applying the techniques of supersymmetric quantum mechanics.Comment: 12 pages, 3 figure

PT-Invariant Periodic Potentials with a Finite Number of Band Gaps

We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant generalized associated Lam\'e potentials V^{PT}(x)=-a(a+1)m \sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2 (y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}, where $y \equiv ix+\beta$, and there are four parameters $a,b,f,g$. This work is a substantial generalization of previous work with the associated Lam\'e potentials V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)} and their corresponding PT-invariant counterparts $V^{PT}(x)=-V(ix+\beta)$, both of which involving just two parameters $a,b$. We show that for many integer values of $a,b,f,g$, the PT-invariant potentials $V^{PT}(x)$ are periodic problems with a finite number of band gaps. Further, usingsupersymmetry, we construct several additional, new, complex, PT-invariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lam\'e potential problem and Heun's differential equation.Comment: 30 pages, 0 figure

Extending PT symmetry from Heisenberg algebra to E2 algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.Comment: 8 pages, 7 figure

In situ study of oxidation states of platinum nanoparticles on a polymer electrolyte fuel cell electrode by near ambient pressure hard X-ray photoelectron spectroscopy

We performed in situ hard X-ray photoelectron spectroscopy (HAXPES) measurements of the electronic states of platinum nanoparticles on the cathode electrocatalyst of a polymer electrolyte fuel cell (PEFC) using a near ambient pressure (NAP) HAXPES instrument having an 8 keV excitation source. We successfully observed in situ NAP-HAXPES spectra of the Pt/C cathode catalysts of PEFCs under working conditions involving water, not only for the Pt 3d states with large photoionization cross-sections in the hard X-ray regime but also for the Pt 4f states and the valence band with small photoionization cross-sections. Thus, this setup allowed in situ observation of a variety of hard PEFC systems under operating conditions. The Pt 4f spectra of the Pt/C electrocatalysts in PEFCs clearly showed peaks originating from oxidized Pt(II) at 1.4 V, which unambiguously shows that Pt(IV) species do not exist on the Pt nanoparticles even at such large positive voltages. The water oxidation reaction might take place at that potential (the standard potential of 1.23 V versus a standard hydrogen electrode) but such a reaction should not lead to a buildup of detectable Pt(IV) species. The voltage-dependent NAP-HAXPES Pt 3d spectra revealed different behaviors with increasing voltage (0.6 → 1.0 V) compared with decreasing voltage (1.0 → 0.6 V), showing a clear hysteresis. Moreover, quantitative peak-fitting analysis showed that the fraction of non-metallic Pt species matched the ratio of the surface to total Pt atoms in the nanoparticles, which suggests that Pt oxidation only takes place at the surface of the Pt nanoparticles on the PEFC cathode, and the inner Pt atoms do not participate in the reaction. In the valence band spectra, the density of electronic states near the Fermi edge reduces with decreasing particle size, indicating an increase in the electrocatalytic activity. Additionally, a change in the valence band structure due to the oxidation of platinum atoms was also observed at large positive voltages. The developed apparatus is a valuable in situ tool for the investigation of the electronic states of PEFC electrocatalysts under working conditions

Platinum thickness dependence of the inverse spin-Hall voltage from spin pumping in a hybrid YIG/Pt system

We show the first experimental observation of the platinum (Pt) thickness dependence in a hybrid YIG/Pt system of the inverse spin-Hall effect from spin pumping, over a large frequency range and for different rf powers. From the measurement of the dc voltage ($\Delta\textrm{V}$) at the resonant condition and the resistance ($R$) of the Pt layer, a strong enhancement of the ratio $\Delta\textrm{V}/R$ has been observed, which is not in agreement with previous studies on the NiFe/Pt system. The origin of this behaviour is still unclear and cannot be explained by the spin transport model that we have used.Comment: 4 pages, 3 figure

All Hermitian Hamiltonians Have Parity

It is shown that if a Hamiltonian $H$ is Hermitian, then there always exists an operator P having the following properties: (i) P is linear and Hermitian; (ii) P commutes with H; (iii) P^2=1; (iv) the nth eigenstate of H is also an eigenstate of P with eigenvalue (-1)^n. Given these properties, it is appropriate to refer to P as the parity operator and to say that H has parity symmetry, even though P may not refer to spatial reflection. Thus, if the Hamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has PT symmetry. This shows that PT symmetry is a generalization of Hermiticity: All Hermitian Hamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric Hamiltonians of this form are Hermitian