Miloslav Znojil. Theory Group, Nuclear Physics Institute AS CR, CS Řež, Czech Republic

Transcription

1 What is PT symmetry? Miloslav Znojil Theory Group, Nuclear Physics Institute AS CR, CS Řež, Czech Republic Abstract The recently proposed complexification of Hamiltonians which keeps the spectra real (and is usually called PT symmetry) is re-interpreted as a certain natural linearalgebraic alternative to Hermiticity. PACS Bz, Ca Fd March 20, 200, whatis.tex file, J. Phys. A (lett.)

2 Formalism of quantum mechanics is often illustrated by the harmonic oscillator H (HO) = p 2 + q 2. Its eigenstates form a complete basis in Hilbert space. The Hamiltonian itself is Hermitian and commutes with the parity P. This means that we can split the basis in two subsets and write H (HO) = n (+) ε (+) n n (+) + n=0 m=0 m ( ) ε ( ) m m ( ) where P n (±) = ± n (±) and ε n (±) = 4n + 2. In the generalized, non-hermitian quantum mechanics as proposed by Bender et al [], the similar illustrative role is played by the imaginary cubic H (IC) (ω) = p 2 + ω q 2 + iq 3. Without a final rigorous mathematical proof, this Hamiltonian seems to generate the real, semi-bounded and discrete energies, at least in a certain range of energies and ω [2]. One may generalize the IC example and contemplate H (toy) = p 2 + W (x) + iu(x) with any symmetric real well W (x) = W ( x) and with its purely imaginary antisymmetric complement iu(x) = iu( x). In place of the current Hermiticity of the Hamiltonians, the latter class of models satisfies a weaker condition which, presumably, implies the reality of the spectrum under certain circumstances. The condition is called PT symmetry and means just the commutativity [H (toy) (ω), PT ] = 0 () where T performs complex conjugation. The mathematical essence of the empirically discovered relation between the spectrum and symmetry () is not clarified yet [3]. In the present note we intend to contribute to the discussion by noticing that there exists a quite close relationship between the Hermiticity and PT symmetry conditions H (HO) = [ H (HO)] + and H (toy) = PT [ H (toy)] PT [ H (toy)], respectively. In our non-hermitian model H (toy) we shall assume that the spectrum remains real. This means that the related Schrödinger equation H (toy) ψ(x) = E ψ(x) (2) is also satisfied by the functions PT ψ(x) = ψ ( x) and PT ψ(x) + ψ(x). In fact, the latter state has the spatially symmetric real part, the spatially antisymmetric imaginary part and is fully characterized by its positive PT parity, PT ψ(x) = +ψ(x). (3)

3 Such a behaviour (or, in effect, normalization) is particularly transparent and will be postulated everywhere in what follows. In a preparatory step, let us recollect the above-mentioned harmonic oscillators { n (±) } and/or any other orthonormalized basis with the property of the well defined parity. Its completeness enables us to expand the functions (3), ψ = n=0 n (+) h (+) n + i m=0 m ( ) h ( ) m. (4) This converts our differential Schrödinger equation (2) with H (toy) = H (+) + iu(x) into its infinite-dimensional linear-algebraic representation k=0 k=0 m (+) H (+) k (+) h (+) k m ( ) H (+) k ( ) h ( ) k + j=0 j=0 m (+) U j ( ) h ( ) j = E h (+) m, m ( ) U j (+) h (+) j m = 0,,.... = E h ( ) m (5) By our assumptions the basis { n (±) } is complete on the real line, x (, ). As long as we need the mere decoupled representation (4) of the wave functions, the use of the vectors { n (σ) } with both the parities σ = ± can be criticized as prodigal. Both these sets are complete on the half-axis of x (0, ). In the other words, we can choose one of them (say, the one with σ = ) and treat the second one as defined in terms of the unitary matrix of overlaps G km, m (+) = k ( ) G km. k=0 This is our key technical step. Our Hermitian sub-hamiltonian H (+) = p 2 + W (x) commutes with the parity P and can be diagonalized by a suitable unitary U, m ( ) H (+) k ( ) = j=0 U mj d j U + jk. Abbreviation Ω mj = m ( ) U j (+) enables us to re-write our linear algebraic set of equations (5) in the partitioned matrix form G+ U ( d E I) U + G Ω + Ω U ( d E I) U + h (+) h ( ) = 0. (6) 2

4 Here, d denotes our diagonalized sub-hamiltonian with elements dkk = d k. The unitary operators U and G are invertible and we eliminate h ( ) = U d E I U + h (+). This can be inserted in the rest of eq. (6), giving our final matrix Schrödinger equation ( d E I + A ) d E I A+ g = 0 (7) where we abbreviated U + G Ω + U = A and U + G h (+) = g. It resembles the Feshbach s introduction of the effective Hamiltonians [4] where, in effect, only the sign of the nonlinearity is opposite. Such a comparison also establishes the immediate nonlinearization connection between the Hermiticity and PT symmetry. At this point, one would be tempted to normalize our solutions, say, by the usual requirement [ N (+)] 2 = in k=0 [g k ] g k = k=0 [ ] (+) 2 [ ] h k = N (+) 2 (8) or, mutatis mutandis, in the analogous formulae for h ( ). Rather unexpectedly, such an option would be badly misleading. The alternative recipe ψ P ψ = [ N (+)] 2 [ ] N ( ) 2 = ± (9) proves much more natural. Its right-hand side changes sign due to our overall normalization convention (3). This sign can be interpreted as a quasiparity of ψ (cf., e.g., ref. [5] for a more detailed illustration). Originally, normalization (9) has been proposed and used in perturbation theory on a purely intuitive basis [6]. Its additional support by non-perturbative arguments results from solvable examples [7]. Still, the explanation of its choice becomes most transparent in our present notation. Firstly, we notice that our reduced Schrödinger equation (7) is insensitive to the change of the sign of the matrix A [i.e., of the matrix Ω in eq. (6) and of the (spatially antisymmetric) function U(x) in eq. (5)]. This is one of the heuristic reasons why one 3

5 inserts the parity P in our normalization recipe (9). A return to our full Schrödinger equation (6) reveals, however, that the above equivalence transformation (Ω Ω) = [ h (±) k ±h (±) ] k changes the solutions thoroughly. Fortunately, as a byproduct, this facilitates the determination of the left eigenvectors of our Hamiltonians, ( h (+), h ( ) ) G+ U ( d E I) U + G Ω + Ω U ( d E I) U + = 0. (0) Hermitian conjugation is to be replaced by its quasi-parity-dependent innovation ψ ± ψ P = ψ. At the two different energies E E 2, the comparison of the left and right PT symmetric equations H ψ = E ψ and ψ 2 H = ψ 2 E 2 leads to the orthogonality ψ 2 ψ = 0. This represents our ultimate reason why the normalization (9) is correct. A marginal remark is inspired by perturbation theory where the PT symmetric Hamiltonians H and solutions E (etc) are represented by the respective power series H = H (0) + λ H () +... and E = E (0) + λ E () +... (etc). In the language of the non-degenerate theory we arrive at the recursive formulae of the type E () ψ (0) P ψ (0) = a known quantity. Against all odds, the indeterminate character of the scalar product on the lefthand side does not induce any difficulties. On the contrary, it reflects the underlying physics because the point where ψ (0) P ψ (0) 0 is precisely the boundary of the applicability of the non-degenerate formalism. At this boundary the unperturbed levels merge in a way which is best illustrated by the PT symmetric square well on a finite interval [8] and/or by the PT symmetric anharmonic oscillator on the whole real line [9]. 4

6 In a non-perturbative, numerically oriented context, our Schrödinger equations (7) are to be solved, presumably, via getting rid of the nonlinearity in their energy dependence. We could recommend the use of a trial energy F, inserted in the denominators. This gives a linearized Schrödinger equation ( d + A ) d F I A+ ˆ g(f ) = Ê(F ) ˆ g(f ). () At any real parameter F we get the auxiliary spectrum {Ên(F )} which is safely real because our new equation is Hermitian. A return to our original, nonlinear eigenvalue problem (7) is then mediated by the selfconsistent re-definition of our auxiliary parameter, F = Ên(F ). Such a recipe can generate real energies, indeed. Unfortunately, one cannot exclude a priori the parallel existence of their complex partners. From this point of view, each particular Hamiltonian must be studied separately. In a less numerical setting, our attention is to be paid to the models where A is finite-dimensional or can be efficiently truncated. This can rely upon a variationallyinspired philosophy as well as upon the underlying physics. One revitalizes once more the Feshbach s idea of an effective reduction of the Schrödinger equation via its projection on an exceptional, model subspace of the Hilbert space [4]. A lot of information is already offered by the most elementary two-dimensional illustration with a b 0... c g 0... A = a.. c Starting from our Schrödinger equation H(E) g = 0 we can profit from our explicit knowledge of its secular determinant in the two-dimensional case, det H(E) = (d 0 E)(d E) (ag bc)2 + b 2 + c a2 d 0 E (d E) + g2 d E (d 0 E) + (d 0 E)(d E). (2) b g. 5

7 After a further simplification a = g = 0 we get the four closed formulae for the energies, E (±) 0 = ( ) d 0 + d ± (d 0 d ) 2 2 4c 2, E (±) = ( ) d 0 + d ± (d 0 d ) 2 2 4b 2. (3) All four of them remain real for the sufficiently small b and c. Vice versa, both these pairs become complex (complex conjugated) beyond certain critical strength of the non-hermiticity. Due to the two-dimensional form of our toy matrix A 0, the quadruplet (3) of the energy roots replaces the doublet of their two Hermitian unperturbed (i.e., b = c = 0) predecessors E = d 0 and E = d. The symbolic manipulation experiments indicate that in any N dimensional model one always arrives at an analogue of eq. (2), det H(E) = P 2N(E) Q N (E) = 0. (4) The polynomial P 2N (E) in the numerator is of the 2Nth degree at most. The denominator is elementary, Q N (E) = det( d E) = (d 0 E)(d E)... (d N E). Such a structure of the secular determinant det H(E) facilitates the graphical localization of the energies (cf., e.g., [0]) and re-confirms that the N dimensional problem (7) generates 2N different bound states at most. One has to study more thoroughly the explicit forms of P 2N (E). In the case of the unbroken PT symmetry (cf. the title of this letter!), all the energies have to be real, ImE n = 0. For an illustration of an ease of a spontaneous breakdown of this rule, let us finally consider a special interaction A dominated by a separable matrix of rank one, A u v. The separability simplifies our eigenvalue condition at any N, + N j,k=0 u 2 j v2 k Q N (E) = 0. (5) (d j E)(d k E) The vectors u and v may be chosen as containing only a single nonzero component. For example, using u 0 0 and v 0 0 we discover that all the detached excitations E n = d n with n =, 2,..., N remain unchanged. Equation (5) only replaces 6

8 the original E 0 = d 0 by the two roots which form a complex conjugate pair, E (±) 0 = d 0 ± i u 0 v 0. For the more general separable A of rank one, we can study the energies within a chosen interval, say, E (d n, d n+ ). Let us assume for simplicity that at some particular n, the sum in (5) is strongly dominated by its two eligible nearest-neighbor terms, n+ u 2 + j d j E j=n ( n+ k=n v 2 k d k E ) = 0. (6) Moreover, let us postulate that u n u n+ while, on the contrary, v n v n+. In the leading-order approximation we then get and solve a quadratic algebraic equation (6). With some suitable constants ρ ε it reads X 2 + ρ 2 X + ε 2 = 0 in the leading-order approximation. It defines the product X = (d n E)(d n+ E). One of the roots is small as required, X (correct) ε 2, and its minus sign re-confirms our assumptions. This root finally defines the two energies E in a way paralleling the closed formulae (3). A generic, visual re-interpretation of the latter approximate construction reveals a squeezed parabola-type curve y(e) = det H(E) which intersects the horizontal line y(e) = 0. The points of intersection coincide with the ends of the interval (d n, d n+ ) in the limit ε 0. With a steady growth of the small ε > 0, we observe a steady upward movement of our parabolic curve y(e). Its two energy zeros move into the interior of the interval (d n, d n+ ). The smooth and growing deviation from the Hermitian starting point A = 0 ends at a certain critical A (crit) where the two energies merge. Next, they form a conjugate pair which moves further in the complex plane. The PT symmetry of the system becomes spontaneously broken. The phenomenon of this type has been detected by the various methods in the spectra of many different PT symmetric Hamiltonians []. 7

9 References [] Bender C M and Boettcher S 998 Phys. Rev. Lett ; Bender C M, Milton K A and Meisinger P N 999 J. Math. Phys ; Znojil M and Tater M 200 J. Phys. A: Math. Gen with further references [2] Calicetti E, Graffi S and Maioli M 980 Commun. Math. Phys. 75 5; Alvarez G 995 J. Phys. A: Math. Gen ; Delabaere E and Trinh D T 2000 J. Phys. A: Math. Gen [3] Buslaev V and Grecchi V 993 J. Phys. A: Math. Gen ; Andrianov A A, Cannata F, Dedonder J P and Ioffe M V 999 Int. J. Mod. Phys. A ; Khare A and Mandel B P 2000 Phys. Lett. A ; Mezincescu G A 2000 J. Phys. A: Math. Gen [4] Feshbach H 958 Ann. Phys. (N.Y.) [5] Znojil M 999 Phys. Lett. A [6] Fernandez F, Guardiola R, Ros J and Znojil M 998 J. Phys. A: Math. Gen [7] Bender C N and Turbiner A 993 Phys. Lett. A ; Znojil M 2000 J. Phys. A: Math. Gen. 33 L6 and 4203 and 456 and 6825; Lévai G and Znojil M 2000 J. Phys. A: Math. Gen [8] Znojil M 200 PT symmetric square well (arxiv: quant-ph/003), submitted [9] Bender C M, Berry M, Meisinger P N, Savage V M and Simsek M 200 J. Phys. A: Math. Gen. 34 L3 8

10 [0] Znojil M 200 Imaginary cubic oscillator in p representation (arxiv: mathph/00027), submitted [] Bender C M and Boettcher S 998 J. Phys. A: Math. Gen. 3 L273; Cannata F, Junker G and Trost J 998 Phys. Lett. A ; Delabaere E and Pham F 998 Phys. Lett. A and 29; Fernandez F, Guardiola R, Ros J and Znojil M 999 J. Phys. A: Math. Gen Bagchi B, Cannata F and Quesne C 2000 Phys. Lett. A [2] Znojil M 200 Annihilation and creation operators anew (arxiv: hepth/002002), submitted Acknowledgement Work inspired by discussions concerning H (IC) (especially with D. Bessis in Saclay in 992), the Feshbach s projection method (e.g., with I. Marek in Plovdiv in 2000) and ref. [2] (with C. Quesne, recently, via ). Partially supported by the GA AS, grant Nr. A