Exact propagators for complex SUSY partners of real potentials

A.M. Pupasov; B.F. Samsonov

REVIEW

Exact propagators for complex SUSY partners of real potentials

B.F. Samsonov, A.M. Pupasov

Quantum Physics · Submitted 27 Feb 2006 · 23 references

Pith's verdict: Genuinely new formula for propagators of complex SUSY partners, clean case (ii) derivation, but completeness and continuous-spectrum assumptions are asserted, not proved—deserves review once those gaps are addressed.

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arXiv IDquant-ph/0602218 (version 2)

Pith IDWWVPF4JZ, Pith's permanent record of this paper and its review

CitationA.M. Pupasov, B.F. Samsonov. "Exact propagators for complex SUSY partners of real potentials." arXiv:quant-ph/0602218 (2006). https://pith.science/paper/WWVPF4JZ

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Claim paper

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Editor's note

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Let me give you the short version: this is a genuinely new formula, worth a referee's time, but it is less rigorous than it looks and the authors have papered over one load-bearing assumption.

What's new: Theorem 1, equation (22). I know of no earlier statement that the propagator for a complex SUSY partner is obtained by applying the transformation operators to the convolution of the seed propagator with the seed Green's function. The two examples, the complex oscillator partner and the complex one-soliton potential, are new and the second matches Jauslin's heat-kernel result after t → it, which is a strong sanity check. The derivation for case (ii) is a clean formal manipulation: expand in the transformed eigenfunctions, use K0 to evolve, identify the series as G0. That is a neat move.

Where it gets soft. The proof assumes the set {Lψ_n} (plus φ_α in case (ii)) is a basis of L2(R) in the no-conjugate biorthogonal sense. The paper states this as a consequence of Lψ_n(±∞)=0, but that only gives density at best; the eigenfunction expansion (10) requires a Schauder basis property, which is stronger. The stress-test note is right that this is load-bearing. In case (ii) the gap is repairable—any f orthogonal to all Lψ_n must satisfy L^t f=0, and adding φ_α restores the basis—but that argument is not in the paper. In case (i) it is more serious: you also have to show 1/u ∉ L2 and something like a Riesz-basis estimate. Neither is supplied. The continuous-spectrum extension, which Example 2 depends on, is asserted in one sentence; no spectral-singularity check is given. These are all fixable, but they are not cosmetic. The examples also skip a lot of algebra; I could not quickly verify Eq. (31) without sitting down with it.

Citation-wise, the reliance on the authors' own preprint [4] for the case (i) classification is a mild self-citation, not a red flag. The references to the Russian spectral theory literature are appropriate.

Bottom line: this is a useful physics letter, not a rigorous operator-theory paper. A referee should ask for the basis/spectral-singularity assumptions to be stated as assumptions or proved, and for more detail on the continuous-spectrum example. If those are addressed, it's publishable. I would not cite it in my own work in the next year, but colleagues working on exact propagators for non-Hermitian Hamiltonians should know about it. Send it to review.

Referee report

Significance

If Theorem 1 is valid as stated, the paper provides an elegant and practical route to exact time-dependent propagators for a class of complex potentials with real spectra, a class for which direct calculations are otherwise difficult. The derivation is transparent, the worked examples are nontrivial, and the agreement of the soliton example with Jauslin's result after Wick rotation is concrete evidence in favor of the method. No fitted parameters appear, and the main formula follows from standard SUSY factorization and spectral expansions. However, the proof relies on unproved completeness and basis properties of the transformed eigenfunctions, and the extension to continuous spectra used in the second example is asserted without proof. These gaps are load-bearing for the central claim, so the theorem and the second example are not yet fully established.

Major objections (3)

[Proof of Theorem 1, Eqs. (20)–(25)] The proof of Theorem 1 rests on the assumption that the functions φ_n = N_n L ψ_n, augmented by φ_α in case (ii), form a biorthogonal basis of L2(R) in the no-conjugate sense (8)–(9). This property is load-bearing: the expansion (10), the interchange of summation and integration in passing from (24) to (25), and the identification of the series with the Green's function G_0 all require the Schauder-basis property for arbitrary initial data, not merely density of the span. The sentence after Eq. (20) derives completeness from Lψ_n(±∞)=0, but a complete minimal system need not be a basis. In case (ii) the gap is repairable: any f orthogonal to every Lψ_n satisfies L^t f=0 and hence f ∝ 1/u, so adding φ_α recovers a basis. In case (i) one must additionally prove that 1/u ∉ L2 and establish a Riesz-basis estimate for {Lψ_n}; neither is supplied. Please provide these proofs or state the basis property as an explicit hypothesis of Theorem 1.
[Continuous-spectrum remark and Example 2, Eq. (34)] The paragraph after the proof states that 'almost the same proof is valid' when h_0 has a continuous spectrum, provided h_c has no spectral singularities, but no proof is given. Example 2, based on the free-particle Hamiltonian, depends directly on this extension: the spectral expansion of K_c involves generalized eigenfunctions and the free Green's function G_0 has continuous-spectrum contributions and a branch cut. Convergence of the resulting integrals and the absence (or harmless nature) of spectral singularities for the potential (32) must be checked. As written, formula (34) is not rigorously supported. Please either supply the continuous-spectrum proof or explicitly present Example 2 as a formal derivation whose rigorous justification is deferred.
[Case (i) of Theorem 1] The proof of Theorem 1 explicitly treats only case (ii), and the statement that case (i) 'does not have essentially new moments' is not an argument. In case (i) the added bound-state term is absent, so completeness of the unaugmented family {Lψ_n} is a nontrivial claim: α is complex, the transformation does not correspond to adding a normalizable zero mode, and the classification of this case is cited to the authors' preprint [4]. Since Theorem 1 covers both cases, this omission leaves half of the theorem unproved. Please provide a self-contained proof or a precise statement (with proof) of the case-(i) classification.

Minor objections (5)

[After Eq. (20)] The phrase 'Lψ_E(±∞) = 0 provided ψ_E(±∞)' appears to contain a typo; it should presumably read 'provided ψ_E(±∞)=0' or an equivalent decay condition.
[Eq. (34)] The notation erf_± = erf[a√it ± (1/(2√it))(x−y)] is ambiguous; please specify the branch of the square root and state that the error function for complex arguments is understood by analytic continuation.
[General] The term 'complete' is used in two senses: density of the linear span (after Eq. (20)) and the coefficient-expansion property expressed by Eqs. (7)–(9). These are different notions, and the proof requires the latter; please distinguish them explicitly.
[Case (i) classification] The proposition from the authors' preprint [4] that underlies the case-(i) versus case-(ii) distinction is quoted only by reference; including the relevant statement (or a short proof) would make the letter more self-contained.
[Title and text] There are minor typos, e.g., 'potenti als' in the title and 'prefect agreement' in the Conclusions; these should be corrected in the final version.

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The paper

The paper claims

This paper proves that the exact propagator of a complex SUSY partner of a real Hamiltonian is built from the original propagator and Green's function, with one extra term when a level is added.

The reading

Complex (non-Hermitian) Schrödinger Hamiltonians with purely real spectra are candidates for generalized quantum mechanics, but their dynamics are hard to compute because ordinary spectral completeness arguments do not apply. This paper proves that when such a Hamiltonian $h_c$ is a supersymmetric (SUSY) partner of a real Hamiltonian $h_0$, its exact propagator can be written from $h_0$'s propagator and Green's function by applying the intertwining operator, plus one explicit bound-state term when the transformation adds a level. The method turns a previously difficult time-evolution problem into a one-dimensional integral for any complex partner whose real SUSY counterpart is solvable. Two examples are worked out in full: a complex anharmonic extension of the harmonic oscillator and a complex one-soliton potential. A sympathetic reader would value this as a concrete route to exact evolution for non-Hermitian systems with real spectra.

What carries the argument

The machinery is the first-order intertwining (SUSY) operator $L = -u'(x)/u(x) + \partial/\partial x$, built from a solution $u$ of $(h_0-\alpha)u=0$, together with its transposed partner $L^t = -u'(x)/u(x) - \partial/\partial x$. These satisfy $L^t L = h_0 - \alpha$ and $LL^t = h_c - \alpha$, so they map eigenfunctions of $h_0$ to eigenfunctions of $h_c$ with the same energy. The load-bearing analytical device is the no-conjugate biorthogonality and completeness of the transformed set, $\int \varphi_n \varphi_k\,dx = \delta_{nk}$ and $\sum_n \varphi_n(x)\varphi_n(y)=\delta(x-y)$, which makes the propagator equal to the spectral sum (10) and allows the series for $K_c$ to be recognized as the resolvent Green's function $G_0(x,y,\alpha)=\sum_m \psi_m(x)\psi_m(y)/(E_m-\alpha)$.

Extended reading notes

Core claim

The central result is Theorem 1. If $L h_0 = h_c L$ with $L = -u'(x)/u(x) + \partial_x$, then in case (i), when the spectrum of $h_c$ coincides with that of $h_0$, the propagator is $K_c(x,y,t) = L_x L_y \int_{-\infty}^\infty K_0(x,z,t) G_0(z,y,\alpha)\,dz$; in case (ii), when $h_c$ has one extra real level $\alpha$ absent from $h_0$, the same expression is augmented by $\varphi_\alpha(x)\varphi_\alpha(y)e^{-i\alpha t}$. The proof expands the propagator in the biorthogonal basis $\varphi_n = N_n L \psi_n$, where $N_n = (E_n-\alpha)^{-1/2}$, and identifies the resulting series as the Green's function (resolvent kernel) of $h_0$ at $E=\alpha$. The paper also states that the same argument covers continuous spectra provided the complex Hamiltonian has no spectral singularities, and demonstrates the formulas explicitly for complex partners of the harmonic oscillator and of the free particle.

Load-bearing premise

The argument rests on the transformed eigenfunctions $\varphi_n = N_n L \psi_n$ (plus $\varphi_\alpha$ in case (ii)) forming a complete set in $L^2(\mathbb{R})$ with the no-conjugate biorthogonality $\int \varphi_n \varphi_k\,dx = \delta_{nk}$; the paper asserts this but does not prove it in general, and the continuous-spectrum version additionally assumes no spectral singularities.

Editorial extensions

If this is right

For any real Hamiltonian $h_0$ with known propagator and Green's function, every SUSY partner of the two allowed types acquires an exact propagator by one quadrature.
The constructed propagator immediately gives the time evolution $\Phi(x,t)=\int K_c(x,y,t)\phi_0(y)\,dy$ for arbitrary square-integrable initial states of the complex Hamiltonian.
The harmonic-oscillator example yields the propagator (31) for the complex anharmonic potential (28), whose spectrum is all oscillator energies plus the level $E=-1/2$.
The one-soliton example yields formula (34), which after the substitution $t\to it$ agrees with the known heat-equation propagator for the one-soliton potential, resolving the question of whether that replacement is legitimate.
The theorem is declared to extend to Hamiltonians with continuous spectra whenever the complex partner has no spectral singularities in the continuous part.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's completeness premise suggests that the method is safe exactly when the SUSY transformation keeps the transformed eigenfunctions biorthogonal with no conjugation; for partners with spectral singularities, the continuous-spectrum version should fail, and a sharper condition could be extracted from the spectral analysis of non-selfadjoint Sturm-Liouville problems.
The same $L_x L_y$ integral structure could be iterated along a chain of SUSY transformations, producing exact propagators for higher-order complex extensions by nested integrals of the original Green's function and the intermediate transformation functions.
A numerical check of the explicit formulas against direct integration of the time-dependent Schrödinger equation for an asymmetric complex partner would test both the theorem and the completeness assumption; the paper does not include such a check.

Original abstract

A method for calculating exact propagators for those complex potentials with a real spectrum which are SUSY partners of real potentials is presented. It is illustrated by examples of propagators for some complex SUSY partners of the harmonic oscillator and zero potentials.

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The abstract, sentence by sentence

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A method for calculating exact propagators for those complex potentials with a real spectrum which are SUSY partners of real potentials is presented. It is illustrated by examples of propagators for some complex SUSY partners of the harmonic oscillator and zero potentials.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A method for calculating exact propagators for those complex potentials with a real spectrum which are SUSY partners of real potentials is presented.	contribution 0.73	0.91	0.60	asserted 1.00
s1	It is illustrated by examples of propagators for some complex SUSY partners of the harmonic oscillator and zero potentials.	contribution 0.40	0.89	0.08	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 1.00.

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References

Works cited (23)

[4]

B. F. Samsonov, Irreducible second order SUSY transformatio ns between real and complex potentials, 2006, preprint quant-ph/0602101

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M. Znojil, Ed. Pseudo-Hermitian Hamiltonians in Quantum Physics, s pecial issue of Czech. J. Phys. 55 (2005) No 9

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[5]

B. F. Samsonov, J. Phys. A: Math. Gen. 38 (2005) L397; B. F. Samsonov, J. Phys. A: Math. Gen. 38 (2005) L571

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B.F. Samsonov and V.V. Shamshutdinova, J. Phys. A: Math. Gen. 38 (2005) 4715

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V.G. Bagrov and B.F. Samsonov, Phys. Lett. A210 (1996) 60

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L. Solombrino, J. Math. Phys. 43 (2002) 5439; G. Scolarici and L. Solombrino, J. Math. Phys. 44 (2003) 4450

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A. Mostafazadeh, J. Math. Phys. 43 (2002) 205; A. Mostafazadeh, J. Math. Phys. 43 (2002) 2814; A. Mostafazadeh, J. Math. Phys. 43 (2002) 3944; A. Mostafazadeh, J. Math. Phys. 43 (2002) 6343

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C. M. Bender, D. C. Brody and H. F. Jones Phys. Rev. Lett. 89 (2002) 270402

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B. Mielnik and O. Rosas-Ortiz, J. Phys. A: Math. Gen. 37 (2004) 10007; V. G. Bagrov and B. F. Samsonov, Phys. Part. Nucl. 28 (1997) 374 ; V. G. Bagrov and B. F. Samsonov, Theor. Math. Phys., 104 (1995) 356; F. Cooper, A. A. Khare and U. Sukhatme, Phys. Rep. 251 (1995) 2 67 C. V. Sukumar, J. Phys. A: Math. Gen. 18 (1985) 2917

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R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integ rals, McGraw-Hill, New York, 1965

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[22]

A. M. Pupasov and B. F. Samsonov, Symm. Integrabil. Geom.: Me th. Appl. (SIGMA), 1 (2005) paper 020

View on Pith 2005
[23]

H. R. Jauslin, J. Phys. A: Math. Gen. 21 (1988) 2337. 10

View on Pith 1988

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [4]

B. F. Samsonov, Irreducible second order SUSY transformatio ns between real and complex potentials, 2006, preprint quant-ph/0602101

View on Pith Pith review arXiv 2006

[2] [1]

M. Znojil, Ed. Pseudo-Hermitian Hamiltonians in Quantum Physics, s pecial issue of Czech. J. Phys. 55 (2005) No 9

View on Pith 2005

[3] [2]

C.M. Bender, J.-H. Chen, K.A. Milton J. Phys. A: Math. Gen. 39 (20 06) 1657

View on Pith

[4] [3]

A. A. Andrianov, F. Cannata, J.-P. Dedonder and M. Ioﬀe, Int. J. Mod. Phys. A 14 (1999) 2688; F. Cannata, G. Junker and J. Trost, Phys. Lett. A 246 (1998) 21 9; V. Milanovi´ c and Z. Ikoni´ c, Phys. Lett. A 293 (2002) 29; J. S. Petrovi´ c, V. Milanovi´ c and Z. Ikoni´ c, Phys. Lett. A 300(2002) 595. 8

View on Pith 1999

[5] [5]

B. F. Samsonov, J. Phys. A: Math. Gen. 38 (2005) L397; B. F. Samsonov, J. Phys. A: Math. Gen. 38 (2005) L571

View on Pith 2005

[6] [6]

B.F. Samsonov and V.V. Shamshutdinova, J. Phys. A: Math. Gen. 38 (2005) 4715

View on Pith 2005

[7] [7]

V.G. Bagrov and B.F. Samsonov, Phys. Lett. A210 (1996) 60

View on Pith 1996

[8] [8]

I. Ts. Gohberg and M. G. Krein, Introduction in the theory of line ar non-selfadjoint operators, Nauka, Moscow, 1965

View on Pith 1965

[9] [9]

M. A. Naimark, Linear diﬀerential operators, Nauka, Moscow, 1 969

View on Pith

[10] [10]

M. V. Keldysh, Dokl. Akad. Nauk SSSR 77 (1951) 11

View on Pith 1951

[11] [11]

V. B. Lidskiy, Trudy moskovskogo matematicheskogo obsches tva (Proc. Mosc. Math. Soc.) 9 (1960) 45

View on Pith 1960

[12] [12]

B. F. Samsonov and P. Roy, J. Phys. A: Math. Gen. 38 (2005) L 249

View on Pith 2005

[13] [13]

P. A. M. Dirac, Proc. Roy. Soc. A 180 (1942) 1; W. Pauli, Rev. Mod. Phys. 15 (1943) 175; T. D. Lee and G. C. Wick, Nucl. Phys. B 9 (1969) 209

View on Pith 1942

[14] [14]

L. Solombrino, J. Math. Phys. 43 (2002) 5439; G. Scolarici and L. Solombrino, J. Math. Phys. 44 (2003) 4450

View on Pith 2002

[15] [15]

A. Mostafazadeh, J. Math. Phys. 43 (2002) 205; A. Mostafazadeh, J. Math. Phys. 43 (2002) 2814; A. Mostafazadeh, J. Math. Phys. 43 (2002) 3944; A. Mostafazadeh, J. Math. Phys. 43 (2002) 6343

View on Pith 2002

[16] [16]

N. K. Nikolskiy, Lectures on a shift operator, Nauka, Moscow, 1980

View on Pith 1980

[17] [17]

T. Curtright and L. Mezincescu, Biorthogonal quantum syste ms, preprint quant-ph/0507015, 2005

View on Pith Pith review arXiv 2005

[18] [18]

C. M. Bender, D. C. Brody and H. F. Jones Phys. Rev. Lett. 89 (2002) 270402

View on Pith 2002

[19] [19]

B. Mielnik and O. Rosas-Ortiz, J. Phys. A: Math. Gen. 37 (2004) 10007; V. G. Bagrov and B. F. Samsonov, Phys. Part. Nucl. 28 (1997) 374 ; V. G. Bagrov and B. F. Samsonov, Theor. Math. Phys., 104 (1995) 356; F. Cooper, A. A. Khare and U. Sukhatme, Phys. Rep. 251 (1995) 2 67 C. V. Sukumar, J. Phys. A: Math. Gen. 18 (1985) 2917

View on Pith 2004

[20] [20]

P. Morse and H. Feshbach, Methods of Theoretical Physics, v ol 1, McGraw-Hill, New York, 1953. 9

View on Pith 1953

[21] [21]

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integ rals, McGraw-Hill, New York, 1965

View on Pith 1965

[22] [22]

A. M. Pupasov and B. F. Samsonov, Symm. Integrabil. Geom.: Me th. Appl. (SIGMA), 1 (2005) paper 020

View on Pith 2005

[23] [23]

H. R. Jauslin, J. Phys. A: Math. Gen. 21 (1988) 2337. 10

View on Pith 1988