4 March 2002 Physics Letters A 294 (2002) 287–291 www.elsevier.com/locate/pla Pseudo-Hermiticity of Hamiltonians under gauge-like transformation: real spectrum of non-Hermitian Hamiltonians Zafar Ahmed Nuclear Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India Received 1 February 2002; accepted 1 February 2002 Communicated by P.R. Holland Abstract We report that it is the pseudo-Hermiticity of Hamiltonians under the gauge-like transformation that underlies the reality of the spectrum and orthogonality of states for the non-Hermitian Hamiltonians type H β =[p + iβν(x)] 2 /2m + V(x), which could be both PT -symmetric and non-PT -symmetric. Notably, the eigenstates of H β , when it is PT -symmetric, are real and do not satisfy the PT -orthogonality condition.  2002 Elsevier Science B.V. All rights reserved. The Hermiticity of a Hamiltonian was supposed to be the necessary condition for the real spectrum until the year 1998 [1]. A conjecture due to Bender and Boettcher [1] has relaxed this condition by intro- ducing the concept of PT -symmetry of the Hamil- tonian. Here P denotes parity operator (space reflec- tion) and T denotes time-reversal. Let χ = PT , then PT -symmetry implies χHχ -1 = H . Such a Hamil- tonian has been conjectured to possess a real discrete spectrum if the eigenstates also regard the said sym- metry, i.e., χΨ n (x) = (-1) n Ψ n (x). This situation is referred to as PT -symmetry being unbroken or exact. Otherwise, spontaneous breaking of PT -symmetry takes place and the eigenvalues are complex conjugate pairs. The last few years have recorded a lot of numeri- cally [1,2] solved and analytically solvable [3–9] (also see references therein) examples in support of the E-mail address: zahmed@apsara.barc.ernet.in (Z. Ahmed). conjecture. Ref. [9] contains a fully tractable model of a PT -symmetric potential which exhibits both the instances of PT -symmetry broken and unbroken with respect to a parameter about its critical value. Supersymmetric [5,10] and group theoretic methods [8] have been utilized for PT -symmetric Hamiltoni- ans. The eigenstates for a PT -symmetric Hamiltonian become complex and hence new orthogonality con- ditions have been suggested [9,11,15]. Real energy band structure arising due to complex, periodic, PT - symmetric potentials has also been reported [12,13]. However, despite the overwhelming evidence, PT - symmetry could not be seen as a necessary condition for the real spectrum of a non-Hermitian Hamiltonian. As a matter of fact, some non-PT -symmetric poten- tials have been known [3,8] to have a real spectrum. More recent conceptual developments on these topics can be seen in [14–16]. Currently, in a very interesting work [17] Mosta- fazadeh introduces the concept of pseudo-Hermiticity and points out that all the PT -symmetric Hamil- 0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0375-9601(02)00124-X