Flow of S-matrix poles for elementary quantum potentials
FFlow of S -matrix poles for elementary quantum potentials ∗ B. Belchev, S.G. Neale, M.A. Walton
Department of Physics and Astronomy, University of LethbridgeLethbridge, Alberta, Canada T1K 3M4 [email protected], [email protected], [email protected]
September 8, 2017
Abstract
The poles of the quantum scattering matrix ( S -matrix) in the complex mo-mentum plane have been studied extensively. Bound states give rise to S -matrixpoles, and other poles correspond to non-normalizable anti-bound, resonance andanti-resonance states. They describe important physics, but their locations canbe difficult to find. In pioneering work, Nussenzveig performed the analysis for asquare well/wall, and plotted the flow of the poles as the potential depth/heightvaried. More than fifty years later, however, little has been done in the wayof direct generalization of those results. We point out that today we can findsuch poles easily and efficiently, using numerical techniques and widely availablesoftware. We study the poles of the scattering matrix for the simplest piece-wise flat potentials, with one and two adjacent (non-zero) pieces. For the finitewell/wall the flow of the poles as a function of the depth/height recovers theresults of Nussenzveig. We then analyze the flow for a potential with two inde-pendent parts that can be attractive or repulsive, the two-piece potential. Theseexamples provide some insight into the complicated behavior of the resonance,anti-resonance and anti-bound poles. PACS: 03.65.-w, 03.65.Nk, 02.60-x, 02.60-Cb ∗ This research was supported in part by an NSERC Undergraduate Summer Research Award (SN)and an NSERC Discovery Grant (MW). a r X i v : . [ qu a n t - ph ] O c t Introduction
The scattering matrix has a rich, interesting history. It has even, in the past, beenpostulated to provide a fundamental physical viewpoint [1]. During the first half ofthe 20th century quantum field theory was ‘plagued’ by infinities. Many were skep-tical of the prospects of quantum field theory to explain physical reality. As a resultthe S -matrix became of central importance and was extensively studied. Once renor-malization techniques resolved the problems with the infinities and more and moreelementary particle phenomena were successfully explained using quantum field the-ory, the S -matrix lost its fundamental role. It remains important in high energy physicsand other fields, however, and is still an interesting object of study. For one, it pro-vides a useful perspective on the quantum physics of local interactions. At a morefundamental level, the poles of the S -matrix provide a unified description of stableand decaying states. Also, they are actively studied by mathematicians in relation tovarious aspects of spectral theory [2].It is well known that bound states correspond to poles of the scattering matrix.However, the S -matrix has other poles that are not associated with normalizable states,but still encode important and interesting physics [3, 4]. In this paper we consider thepositions of all the poles of the S -matrix for certain elementary potentials, and theirflow under deformation of the potential parameters.What is the physical significance of the S -matrix poles that are not associatedwith bound states? In the late 1920’s Gamow proposed an explanation of α -decayin terms of solutions of Schrödinger’s stationary equation with complex eigenvalues,that satisfied a purely outgoing boundary condition, i.e. far enough from the originthe solutions were outgoing plane wave. These solutions could be thought of as wavefunctions, i.e. as the position representations of certain generalized state vectors. Thosesame Gamov vectors turned out to correspond to the poles of the S -matrix and theresidues of the propagator. What Gamow sought to apply to α -decay was in generala way of describing bound and quasi-stable states that emphasizes their similarities.The concept of a quasi-stable state is a fundamental one, and so it has been appliedextensively, and in all areas of physics. The physical effects of the S -matrix poles unrelated to bound states are undeniablein scattering (see [9], e.g.). Strictly speaking, however, the quasi-stable states associ-ated with those poles, the resonance and anti-bound states, are not true states. Thenon-normalizability of their wave functions is one marked difference from the physicalbound states. This non-unifying characteristic, however, can be tamed somewhat, ina mathematical way, by regularizing the integrations, interpreting the probability ina time-dependent setting [11], and/or continuing to complex potentials (see [12], e.g.).Upon continuation to complex potentials, one can also relate the different poles to eachother (bound state poles to anti-bound state poles, e.g.) [13].We should note that, in spite of their non-normalizability, the wave functions asso- Of course, that continues to this day. To mention some recent applications, they have been usedto calculate tunneling ionization rates [5], to understand the phenomenon of diffraction in time [6],to describe the decay of cold atoms in (quasi-)one-dimensional traps [7], and are directly relevant torecent condensed-matter experiments [8]. For brevity and simplicity, however, we will continue to refer to them as states, rather than as‘states’ or virtual states, and rely on the context to make the distinction. See [10] for a discussion. This was first done by Zel’dovich; see [4]. S -matrix ([14], e.g.) and propagators ([15], e.g.) and otherphysical quantities. As a result, various ways of determining and approximating themhave been developed (see [16], e.g.). We will not consider them here, however.We focus on the S -matrix. Remarkably, it can be determined by its poles–theirlocations and residues [3]. The pole locations, and their flow on deformation of thepotential, are therefore clearly of interest. In the simplest case of a square well or wall(barrier), the movements of the poles in the complex momentum plane were studied in[17]. The depth of a simple square well and the height of a square barrier were varied tochange the position of the S -matrix poles and generate their flow. More recently, certainof Nussenzveig’s results were reproduced using simple, elegant graphical methods in[18], and different poles were related by complex continuation in [13].More than fifty years after Nussenzveig’s work, however, little has been done in theway of direct generalization. Even for the next-simplest potentials, numerical methodsmust be used to find the S -matrix poles. But today, it is not difficult to performthe analysis in [17] on a personal computer, using widely-available software (such asMathematica, Maple, MATLAB, e.g.). To illustrate this point, we will reproduce thoseresults here and generalize them to the next-simplest cases. We will plot the trajectoriesof all poles for two types of elementary piece-wise flat potentials–attractive/repulsive(following [17]) and a two-piece combination of both. For short, we will refer to thepotentials as one-piece, or wall or well or wall/well, and two-piece, or well+wall, etc.The above potentials are interesting because they are simple, but still generic. Theirsimplicity permits a relatively easy numerical treatment of the pole structure and acomplete description of the pole positions in the complex momentum plane, with a flowthat is a function of only a few parameters. Most importantly, unlike certain specialpotentials such as the Dirac delta and the Coulomb potential, they are generic enoughto exhibit all the possible families of poles.We should mention that another way to generalize the Nussenzveig flow is to studypotentials for which the S -matrix can be found exactly (see [19], e.g.). Such solvablepotentials are special, however, and so may show non-generic properties. Here, weprefer to generalize the wall/well in a very direct, simple way that we hope will notintroduce any special features into the flow. Let a spin-zero, massive particle move on the real line with coordinate x . The Hamilto-nian of the system H = p / m + V ( x ) has a potential V ( x ) that vanishes fast enough at infinity for scattering to be possible. The stationary Schrödinger equation (cid:2) − (cid:126) ∂ x / m + V ( x ) (cid:3) ψ = Eψ. (1)determines the wave functions of the stationary states and their energies. Let us definethe momentum variable (wave number) k so that k = 2 mE/ (cid:126) , as it will have an In fact Nussenzveig did more–a spherically symmetric rectangular potential well/wall for zeroangular momentum was worked out. Some results were also given for higher angular momenta. We will adopt the same convention as [4], ch. 3, i.e. faster than /x . That guarantees asymptoticsolutions of the type e ± ikx . Potentials studied.
The dashed line shows the square well/wall potential withvariable depth/height
U > / U < . The solid line represents a 2-piece flat potential with anattractive part and repulsive part, the well+wall. Figure 2:
Types of S -matrix poles in a typical configuration. Different types of polescorrespond to wave functions with different boundary conditions. m = (cid:126) = 1 in anticipation of the numericalresults and to lighten the notation. Anytime we consider scattering we deal with planewaves, whether as states or as boundary conditions, which makes it more convenientto work with the momentum k instead of the energy E .Some of the poles of the S -matrix can be identified with the bound states of thesystem: their locations in the k -plane correspond to the bound state energies. Theremaining poles turn out to be physically significant, as well. They correspond to theso-called anti-bound and resonance states , and half of the latter are sometimes called anti-resonance states . The poles will be the central objects of interest of this paper.We therefore provide a short summary of relevant results.Let us consider scattering by a potential that vanishes outside of a finite interval [ a, b ] , where a < , b > . The asymptotic wave function then has the form ψ ( x ) → A ± e ikx + B ∓ e − ikx , x → ±∞ , x / ∈ [ a, b ] . (2)The map that relates the incoming ( − ) and outgoing (+) coefficients is the S -matrix: (cid:18) A + B + (cid:19) = S (cid:18) A − B − (cid:19) (3)Now, let us define the Jost functions φ ± , as the two independent solutions that behaveasymptotically as e ± ikx , respectively, at ±∞ . The nonzero Wronskian shows that φ ± ( x, k ) ∗ = φ ± ( x, − k ) is independent of φ ± ( x, k ) and therefore we can express φ + via φ − and its complex conjugate, and vice versa: φ + ( x, k ) = α ( k ) φ − ( x, − k ) + β ( k ) φ − ( x, k ) . (4)One can write a similar equation to express φ − with different coefficients. Evaluatingthe Wronskians of each side with the appropriate choice of φ ± ( x, ± k ) , reveals thatthose coefficients can be expressed using α ( k ) and β ( k ) [3].The components of S are related to the well known transmission and reflectioncoefficients (amplitudes) that are usually defined via the wave functions for a waveincident from the left or right: ψ L ( x ) = (cid:40) e ikx + R + e − ikx for x (cid:28) a ,T + e ikx for x (cid:29) b ; ψ R ( x ) = (cid:40) T − e − ikx for x (cid:28) a ,e − ikx + R − e ikx for x (cid:29) b . (5)Let us then consider a wave incident from the left. Notice that ψ L ( x, k ) = φ − ( x, − k ) + R + ( k ) φ − ( x, k ) and ψ L ( x, k ) = T + ( k ) φ + ( x, k ) satisfy the Schrödinger equation. More-over, expressing φ + in terms of φ − using (4) shows T + = 1 /α ( k ) , R + ( k ) = β ( k ) /α ( k ) . (6)Using ψ R as the wave function and proceeding just as we did with ψ L reveals that thetransmission coefficient T − coincides with T + , so we define T ( k ) := T − ( k ) = T + ( k ) .Also, in agreement with the conservation of probability, R − only differs from R + by Resonances occur in pairs, at momenta ± k − ik , where k , k > . To distinguish the two polesof such a pair, those in the fourth (third) quadrant are known as (anti-)resonances. Then it clearly decreases ‘fast enough’ at infinity. ψ L and ψ R as the wave function in (2) and writing the correspondingvalues of the coefficients A ± and B ± , via (3), for each case allows us to express the S -matrix in the form S = (cid:18) T R + R − T (cid:19) , (7)a more familiar form, convenient for practical calculations.One can show (e.g. [20]) that the Jost functions are analytic functions of k in theupper-half k -plane. The functions α ( k ) and β ( k ) , therefore, are also holomorphic forIm ( k ) > . It is clear now that the poles of the S -matrix are fully determined by thezeros of α ( k ) .Now let us assume that α ( k ) =0, for k somewhere in the upper plane. From (4)it follows that for k = k the Jost functions are linearly dependent. Now, recall thatthe Jost functions are solutions of the Schrödinger equation and since they are linearlydependent for k we have a solution that vanishes asymptotically at both infinities,i.e. a normalizable, bound state exists with energy k . However, the Hamiltonian H is assumed to be self-adjoint, and therefore has only real eigenvalues. Since | α ( k ) | =1 + | β ( k ) | , α ( k ) (cid:54) = 0 for any x ∈ R and thus the energy can only be real for k purelyimaginary.In addition α ∗ ( k ) = α ( − k ) (from φ + ( x, k ) ∗ = φ + ( x, − k ) ) and the Schwarz reflectionprinciple imply that the analytic continuation of α ( k ) to the lower part of the complexplane is symmetric across the Im ( k ) -axis.The complexification of the momentum, and therefore the energy, is more than amathematical “trick”, however. It has sound physical meaning. In the introduction wementioned wave functions corresponding to complex eigenvalues and the correspondingGamow vectors. The real parts of the complex eigenvalues z = E − i Γ / correspond tothe physical energies, and their imaginary parts correspond to the decay widths, so that / Γ is the lifetime of the decaying state [22]. The Gamow vectors satisfy Schrödinger’sequation with outgoing boundary conditions (see, e.g. [3], [23] and [22]).Our analysis so far does not imply anything about the existence of poles in thelower complex k -plane. The only requirement is that the poles have to be symmetricwith respect to reflection across the Im ( k ) -axis. In particular, they can still lie on it,much like the poles associated with the bound states. If the potential is such that the S -matrix has poles in the lower plane, is there a pole-state correspondence as in thebound state case?As already mentioned, the states signalled by resonances are the Gamow vectors.This is easy to see in the simplest case of a square well. As Zavin and Moiseyev[18] show, outgoing boundary conditions lead to the same equations that determinethe poles of the S -matrix. In addition, the anti-bound states can be obtained usingincoming wave boundary conditions.Let us point out that all poles except those on the positive Im ( k ) -axis lead to non-normalizable wave functions. A way of dealing with the spatial divergence is considering The Schwarz reflection principle states that for a holomorphic function in the upper complexplane, continuous and real valued on the real line, one can write an analytic continuation for thewhole C plane such that f ( z ∗ ) = f ∗ ( z ) . ψ z ( x, t ) = e − iEt e − Γ t/ ψ z ( x ) , (8)which obviously decay with time, as long as Γ > . In terms of the wave number k we have Γ = − Re ( k ) Im ( k ) >0 only for Re ( k ) >0, i.e. for resonance poles. The anti-resonances, with Re ( k ) <0, correspond to the time-reversed behaviour of the resonances.The interpretation of the remaining divergent wave functions, those of the anti-boundstates, with Re ( k ) =0, remains obscure, though they are closely related to the boundstates and resonances. In addition, as indicated by Nussenzveig even the resonancepoles’ interpretation breaks down for | Γ /E | (cid:29) .The complex eigenvalues do not contradict the well known reality of the eigenvaluesof self-adjoint operators. The Hamiltonian, like any operator, is defined not just byits action but also by its domain. Strictly speaking then, two operators with the sameaction can have two different domains, which makes them two distinct operators. Thedomain for which we have determined the self-adjointness of the Hamiltonian, doesnot contain those complex energy states, since they are not normalizable in the usualsense. If we include the Gamow vectors in the domain the operator will no longer beself-adjoint.To conclude this preliminary section let us make a connection with another im-portant quantity – the resolvent. Resonances are also studied through its poles. Thereason is that, while the S -matrix does not exist for all Hamiltonians, the resolvent isa more general mathematical object and as such can be defined for much wider classof operators. And when scattering is possible, the poles of the S -matrix coincide withthose of the resolvent.Consider the time evolution of the state vector as determined by the propagator, theunitary operator of the form U ( t , t ) = e − i ( t − t ) H/ (cid:126) for t -independent Hamiltonian H . Its Laplace transform defines the resolvent operator of the Hamiltonian, R H ( E ) = ( H − E ) − = (cid:88) α ∈ I | E α (cid:105)(cid:104) E α | E α − E , (9)where I ⊂ R is a discrete or continuous interval depending if the energies are part ofthe discrete or continuous spectrum, respectively. Comparing (1) and (9) shows thateigenvalues of the Hamiltonian generate poles of the resolvent. The resolvent, however,treats all the complex eigenvalues on an equal footing.The resolvent and the S -matrix, as a function of k = √ mE/ (cid:126) , have the samesingularities for all physical purposes, as we mentioned. This is to be expected since R H ( E ) is related to the Green’s function G ( x, y ) : the Schwarz operator kernel of R H ( E ) is the Green’s function G ( x, y ) , satisfying ( H − E ) G ( x, y ) = δ ( x − y ) (see, e.g. [24]). Forscattering a natural assumption is that the initial state was the wave function definedin the far past, i.e. at t = −∞ , and similarly the final state, at t = + ∞ . Thus theresolvent is also related to the scattering matrix via S = U ( −∞ , + ∞ ) and a Laplacetransform. For our purposes, however, finding the pole content of the scattering matrix is sim-pler than finding that of the resolvent. For piece-wise flat potentials it can be done by For the mathematical rigour that we ignore, the reader may refer to the mathematical literature.Quite comprehensible lecture notes are available by Tang and Zworski [25], for example. S -matrix allows for easier calculations. A particle moves freely along the real line except in [ − a, a ] ; it interacts by the piece-wiseflat potential V ( x ) = (cid:40) − U, x ∈ [ − a, a ] , , x ∈ ( −∞ , − a ) ∪ ( a, ∞ ) . (10)For positive/negative values of U we have a potential well/wall, as shown in Fig. 1.The matching conditions for the wave function and its derivative are given by ψ ( ± a −
0) = ψ ( ± a + 0) , ψ (cid:48) ( ± a −
0) = ψ (cid:48) ( ± a + 0) . (11)We apply (11) to the wave function in the well ψ U := A exp( iKx ) + B exp( − iKx ) and(5), where K = √ k + 2 U , which allows us to determine the wave function on thewhole real line. This way we also determine the scattering matrix via (7) : S = 2 e − ika kK kK cos(2 Ka ) − i ( k + K ) sin(2 Ka ) (cid:32) ( k − K ) sin(2 Ka )2 ikK ( k − K ) sin(2 Ka )2 ikK (cid:33) . (12)The poles of the S -matrix must obey the nonlinear algebraic equation: k √ k + 2 U cos(2 aK ) − i ( k + U ) sin(2 aK ) = 0 . (13)They will depend on a and U (and if we consider the asymmetric interval, also on b ;see Sect. 4 below). In this section we consider the symmetric interval b = − a = 1 . indimensionless units. Fixing the width of the well, we will have a one-parameter familyof solutions of (13). In the complex momentum plane it generates a flow, a collectionof curves representing the positions of the poles as a function of the depth/height U ,which we will discuss below.Using Mathematica, Maple or other similar software, it is easy to find all the sin-gularities k n . We used Mathematica’s built-in functions to find all solutions in abound region around the origin. Ideally, one can obtain the full flow by increasingthe depth/height incrementally and generating a pole configuration at each step. Theresulting points can then be plotted, displaying the flow. One can plot as many pointsas desired, making the plot more detailed or covering wider range of parameters. Weintroduce a depth/height cutoff in order for the algorithm to be finite. The cutoff ischosen so that the plots include the interesting features. At least in theory, one candecrease the step that changes the parameters to achieve very high level of detail. Inpractice, however, the average computer has the capability to generate only a limitednumber of configurations. For sufficient detail we plotted small portions of the flow ata time and then combined them. 8igure 3: Flow trajectories of the anti-resonance poles for the potential well/wallwith variable depth/height.
Only the trajectories for
Re( k ) < are shown since there is acomplete symmetry under reflection across the Im( k ) -axis. The arrows indicate the directionof the flow for − U varying from −∞ to ∞ (a deep well to a tall wall). The singularities of the S -matrix are almost always simple poles with a typicalconfiguration shown in Fig. 2. As expected from the general properties of the scat-tering matrix, discussed in the previous section, the singularities lie on the imaginarymomentum axis as well as in the lower complex plane. Following Nussenzveig [17] weplot the flow (i.e. the positions of the resonances in the complex plane) for depth/height U ∈ ( −∞ , ∞ ) , treating the square potential well/wall, i.e. the well and wall together.First, let us consider the poles with nonzero real part, i.e. the resonances (or reso-nance poles/states ). Recall that resonances occur in pairs, at two momenta with equaland negative imaginary parts, and equal and opposite real parts. Those in the thirdquadrant are known as anti-resonances . The fourth-quadrant poles are referred to sim-ply as resonances, when no confusion is likely, and sometimes as “physical resonances”when distinctions have to be clear. The flow is shown in Fig. 3 for the anti-resonances, For a discrete set of U values there is a double pole. Flow trajectories of the bound and anti-bound states for a potential well(wall) with depth (height) -U>0 (U>0).
The hollow points trace the positions of thebranch points. The dashed line indicates the coalescence point k = − i/a . Figure 5:
Coalescence of S -matrix poles. Two anti-bound poles merge into a double pole,that consequently gives rise to a resonance + anti-resonance pair. ( k ) -axis. When the well is very deep the poles are located close to the horizontal line Im( k ) = − /a . As the well becomes shallower the poles move monotonously down-wards along the trajectories as indicated by the arrows in Fig. 3. In the case of a wallof increasing height, the poles travel upwards, cross Im( k ) = − /a and diverge to theleft.The horizontal line Im( k ) = − /a bounds the pole trajectories for the well. More-over and as the depth increases a resonance + anti-resonance pair of poles travels to k = − i/a from each side of the Im( k ) -axis and coalesces to form a double pole, conse-quently creating an anti-bound pair, as in Fig. 5. This only happens for a countableset of values U n , n ∈ Z , that can be found numerically. The point k = − i/a is a coa-lescence point for the flow and it represents a transition of a pair of resonance statesinto a pair of anti-bound states. For a very deep well the imaginary part gets closer tothe boundary, accounting for the characteristic flattening of the pole distribution for U (cid:28) .The discrete subset of values of the depth/height U n , for which a double pole exists,corresponds to each of the disjoint sections of the flow: U corresponds to the trajectoryclosest to the Im ( k ) -axis (and its 4th quadrant counterpart), U to the second closest,etc. This is easy to see if we notice that once the resonance reaches the boundaryand disappears there is no other resonance on that trajectory, as the whole resonanceconfiguration just travels up and to the right along fixed branches of the flow.Another way of labeling the trajectories is by their asymptotic behaviour. We findfrom (13) that the pole locations k n → ± πn/ a − i ∞ , n ∈ N , as U → . The rateat which the poles move also increases as U → . In fact, as pointed out by Regge[21], very small changes in the depth result in very large shifts in the locations of theresonances. Despite their sensitivity, however, they follow a smooth trajectory andwhile the flow rate of the poles is large for shallower wells, the trajectory is well definedfor all values of U . As | k n | → ∞ , we verify that resonances are absent in the freeparticle case.For large n , the locations of the resonance + anti-resonance poles can be approxi-mated [3]. For a finite range constant potential we have Re k n ∼ n and Im k n ∼ − ln n as n → ∞ . It can also be shown that the distance between the asymptotic values ofthe branches of the flow lim U →∞ ( k n − k n − ) is independent of n .As discussed, the parameter U can be taken into negative values to reproduce thesquare wall as a deformation of the square well. The flow generated for U < comesfrom − i ∞ and shares vertical asymptotes with the flow for U > . For that reasonall the branches are parametrized by their asymptotic values, as before. The shape ofthe trajectories is shown in Fig. 3. In the infinite wall limit ( U → −∞ ) we find theasymptotic values | k n | → ∞ . Clearly, for an “impenetrable” barrier, resonances do notexist.Since the flow does not converge to a single point as in the well case, no boundand anti-bound states will be created by the resonances. For the attractive potentialthe resonance states are attracted towards each other to produce bound states. Thelocation of the attractor (or coalescence point k = − i/a ) is independent of the potentialstrength (at least for the cases we consider) but depends on its width. Conversely, therepulsive potential pushes the resonances away, preventing them from coalescing, and11o from subsequently producing a bound state.Now let us turn our attention to the singularities with Re( k ) = 0 . They correspondto the bound states (Im( k ) > and the anti-bound states (Im( k ) < . For a potentialwell, consider a typical bound-state–not the ground state, nor the first excited state.The corresponding pole location has (Im( k ) > , and as the well-depth decreases, (Im( k ) decreases monotonically. When (Im( k ) becomes negative, the state changesfrom bound to anti-bound. For a small range of U thereafter, it has the strange propertyof an “energy” that becomes more negative as the well becomes shallower. Then itmeets another anti-bound state ascending from − i ∞ at the k = − i/a coalescencepoint. There a pair of anti-bound states “annihilate” giving birth to a resonance +anti-resonance pair of poles (Fig. 5). The decrease in the number of bound states istied to the decrease in the number of anti-bound states. The exceptions are the lasttwo bound-state poles–one having as a limit and the other rapidly traveling to − i ∞ .For very shallow wells, a surviving bound state always exists. The last bound statebecomes an anti-bound when the well turns into a wall, and heads to − i ∞ as the wallgrows, with the Im ( k ) axis as an asymptote.One can see that as the well becomes shallower the highest bound state will turninto a zero-energy state. Zero-energy states have infinite spread, i.e. their de Brogliewave length becomes infinite. They represent a critical point: the normalizable wavefunctions become divergent anti-bound states just above the well. From (13), we seethat zero-energy states are admitted when U takes the values n π / a , n ∈ N .Besides the bound and anti-bound state flows, Fig. 4 also depicts the trajectories ofthe branch points, distinguished by the hollow points. The branch cut is determined by √ k + 2 U , i.e. Im ( k ) ∈ ( −∞ , −√ U ) ∪ ( √ U , ∞ ) . As we can see all the bound statesare contained within the parabola, their energy cannot be smaller than the depth of thewell, since the wave functions for E < U min are not normalizable. As the anti-boundstates are non-normalizable anyway, they are not affected by the branch cut, as theanti-bound states for − < U < illustrate in Fig. 4. An interesting observation isthat the bound/anti-bound pairs are always contained within the branch points for allvalues of U .For the finite barrier, i.e. U < , only resonance (and anti-resonance) poles occur.Recall that when the resonance flow converges, a pair of trajectories collide from eachside of the imaginary axis which results in the creation of a bound/anti-bound pair.For the potential wall the resonance flow diverges, so there are no bound states, andtherefore, no anti-bound states either. Now let us investigate how the presence of both attractive and repulsive parts affectsthe flow of the poles of the S -matrix. Again, we consider the simplest case possible, apotential that has two constant non-zero pieces, the well+wall shown in Fig. 1. V ( x ) = (cid:40) − U, x ∈ [ a, ,H, x ∈ (0 , b ] , , x ∈ ( −∞ , a ) ∪ ( b, ∞ ) . (14)12igure 6: Flow trajectories of the anti-resonance states.
The flow is for a potentialconsisting of a fixed well and variable well changing into a wall (fixed U , varying H ). When either U or H change sign, the corresponding piece of the potential changescharacter, from a well to a wall, or vice-versa. Therefore, to illustrate the changesto the pole distribution, we will provide 2 graphs: Fig. 6 depicts the flow for thewell+well/wall, and Fig. 7 the wall+well/wall.The matching at the discontinuities provides the analytical expression for the S -matrix. Using the ansatz (5) we find the scattering matrix in the form of (7): T = 2 e i ( a − b ) k kκK/ T R + = − e iak (cid:2) κ ( K − k ) sin( aK ) cos( bκ )+ (15) + ik ( κ − K ) sin( aK ) sin( bκ ) + K ( K − κ ) cos( aK ) sin( bκ ) (cid:3) / T R − = − ie − ibk (cid:2) κ ( k − K ) sin( aK ) cos( bκ ) −− ik ( κ − K ) sin( aK ) sin( bκ ) + K ( κ − k ) cos( aK ) sin( bκ ) (cid:3) / T Flow trajectories of the anti-resonance states.
The flow is for a potentialconsisting of a fixed wall and variable well changing into a wall (fixed H , varying U ). with κ = √ k − H, K = √ k + 2 U and T = sin( bκ ) (cid:2) − i ( κ + k ) K cos( aK ) + K ( κ + K ) sin( aK ) (cid:3) + (16) κ cos( bκ ) (cid:2) kK cos( aK ) + i ( k + K ) sin( aK ) (cid:3) . The poles are then determined by the equation T = 0 . The resonance flows areshown in Figs. 6 and 7. In the first case we have a potential that consists of a fixedwell of depth U combined with a variable-depth ( H < ) well transitioning into a wall( H > ). The second figure (Fig. 7) shows the “reverse” case, where we fix the barrierand vary the well with depth − U < into a barrier with height − U > .Let us concentrate on the first case for now. The flows are now separated intotwo distinct families. For a fixed well + varied well we have the familiar behavior for14igure 8: Resonance poles and the reflection coefficient.
The well+wall potentialhas a well with fixed depth with the barrier increasing in height. Increasing the wall height“unzips” the line of resonances into two lines.
Figure 9:
Resonance poles and the reflection coefficient.
The well+wall potential hasa well with fixed depth and a wall with fixed height, but increasing width. The “unzipping” alsooccurs in this case.
Flow of the bound and anti-bound poles for the well+wall potential.
Thepotential consists of a well with fixed depth and a well transforming into a wall with variabledepth/height (fixed well + variable well/wall).
Figure 11:
Flow of the bound and anti-bound poles for the well+wall potential.
Thepotential consists of a wall with fixed height and a well transforming into a wall with variabledepth/height (fixed wall + variable well/wall).
Magnified flow for the 2-piece potential.
Trajectories of the bound, andanti-bound states for the fixed well part (dashed line) and fixed wall (solid line) potentialsaround the origin. very deep (symmetric) well–all the poles tend to move towards k = − i/a , with thecharacteristic annihilation of resonances occurring as before. However the presence ofthe fixed well destroys the monotonous behavior of the flow and generates the oscillatingpattern seen in both Fig. 6 and Fig. 7. The oscillating pattern shows that despite thefact that | U | (cid:28) | H | , there is an observable influence of the finite part even when thevariable part of the potential is of much greater magnitude. H → shows separation of the resonances into two families: rapidly diverging, dueto the increasing wall, and slowly moving (finite well) ones. As we will confirm later,each family can be associated with one of the two pieces of the potential. The slowlymoving half of the resonances need to recover the configuration for a finite well (of / the width) so they remain finite for all H . The transition from H < to H > resultin the local minima of Im ( k ) observed in Fig. 6.As the well turns into a wall we observe rapid growth of new branches from − i ∞ .Those consist of poles that can be associated with the varied part of the potential.Unlike the symmetric well, however, the flow does not share asymptote with its H < counterpart. Again we observe two types of resonance poles. On one hand we observe thatsome of the poles follow diverging trajectories heading to the left with Re ( k ) → −∞ and Im ( k ) → . The locations of the rest of the poles move upwards until theyreach maxima and then descend with decreasing rate to reach a configuration thatchanges very slightly with the varied height of the barrier. However we observedsimilar behavior for the 1-piece potential barrier. This seems to suggest that indeed We have to truncate the values of H , in order for the algorithm to terminate, but we did notobserve convergence between the H < and H > branches of the flow. lim H →∞ | k | → ∞ correspond to the barrier and the rest – to the well. Furtherconfirmation can be found in Fig. 8 – it shows “snap shots” of the locations of the poles.The pole configuration exhibits a zipper-like behavior due to the separation of the polesalong the two distinct flow branches, as discussed. The “top” poles are pushed awayby the branch cut of √ k − H (introduced by the varied well/barrier) and does notaffect the “bottom” poles. In addition Fig. 9 shows that widening the barrier, for fixed H does not affect the “bottom” poles while significantly increases the count of the “top”poles.The pole separation is quite surprising since the behaviour of the resonances is de-termined by a non-linear relation and yet they behave in a remarkably linear fashion.Only when the two pieces are of comparable size can the resonances not be distin-guished without information about their trajectories. However, labeling the poles willbe incorrect since the poles corresponding to the fixed well and those correspondingto the varied well barrier can exchange roles and only make sense when | U | (cid:28) | H | .Indeed the poles that survive the limit H → are on the same trajectory as the polesthat vanish in the limit H → ∞ .Note that Fig. 9 indicates, as mentioned earlier, that the “top” resonances corre-spond to local minima of the reflection probability. Thus, the name resonance polesis justified–they are fully transmitted modes responsible for a transmission resonance.For low potential barriers the peaks begin to widen and overlap, as in the case of the“bottom” poles, gradually destroying the visual correspondence observed in Fig. 9. Thelast two figures also illustrate the physical significance of the resonance poles, as wellas the branch cuts. The resonances indicate maxima in the transmission coefficients,while the brunch cuts act as an impenetrable barrier for the resonance states associatedwith the varied portion of the potential.Now let us consider the bound states. Fig. 10 shows that when two wells interactthe matching distorts the trajectories and they have inflexion points for certain valuesof the potential. This “complex plane” scattering is not present in the case of Fig. 11since the barrier does not have bound states. While there are also zero-energy statesfor a discrete set of potentials, they cannot be found in closed terms, due to the morecomplicated matching conditions.There is a noticeable difference around U = 0 –instead of the least-bound stateturning changing into an anti-bound one and then disappearing to infinity, we havean ascending anti-bound state annihilating it for a very small height of the wall. Themost-bound state survives the transition from a well to a wall due to the presence ofthe fixed well to the left.The features around the zero potentials are interesting enough to be plotted ingreater detail in Fig. 12. Apparently, the absence of bound states associated with thebarrier part leaves the familiar monotonous behavior of the bound/anti-bound statesunchanged. We studied the flow of S -matrix poles in the plane of complex momentum for thesimple potential(s) depicted in Fig. 1. Our results are summarized by the remainingFigures. 18he most elementary potential considered was the square well/wall, and our Figs.2-5 recover the seminal results of Nussenzveig [17]. Some of Nussenzveig’s conclusionswere also confirmed by less direct, graphical methods in [18] (see also [13]).New is a similar study of the “two-piece” potential, consisting of two adjacentwell/walls, with strengths controlled by independent parameters ( U and H in Fig.1). The flow of S -matrix poles was calculated as a function of various potential pa-rameters, one at a time, and plotted in Figs. 6-12.The Appendix also verifies that the “one-piece” and “two-piece” flows are consistentwith the pole structure of a δ -function potential and a δ (cid:48) -function potential, respec-tively, if the parameters are controlled in the appropriate way. The calculations are asimple, yet interesting check of our methods. Figs. 13-15 depict the relevant flows.Unsurprisingly, the flows for the more complicated potentials are more complicated.Certain features can be understood simply, however, and we hope that our extension ofthe old analysis of Nussenzveig [17] will help lead to further insight that can be appliedgenerally. Appendix: δ - and δ (cid:48) -sequence potentials We can perform a simple, interesting check of our results by recovering the well-known textbook examples of the Dirac delta function (or δ ) potential and its derivative, δ (cid:48) , as limits of the “one-piece” and “two-piece” potentials, respectively.Consider first V ( x ) = − λδ ( x ) . We realize this δ -potential as the a → limit ofa δ -sequence of the “one-piece" potential: − U [ θ ( x + a ) − θ ( x − a )] , with the strength λ = 2 aU kept fixed. To see what we should find, we solve the Schrödinger equation for x ∈ R / { } and demand that the wave function is square-integrable to show ψ ∝ e − λ | x | .Integrating equation (1) over an infinitesimal interval ( − ε, ε ) around yields ψ (cid:48) ( ε ) − ψ (cid:48) ( − ε ) = − mλψ (0) , (17)which determines the energy E = − mλ / of the bound state. A straightforwardcalculation shows the transmission and reflection amplitudes t = (1 − mλ/ik ) − , r = ( ik/mλ − − (18)have a pole only at the bound state energy, i.e. no resonances exist for the δ -potential.This is also confirmed by the fact that the probabilities for transmission | t | and re-flection | r | are monotonous functions, i.e. there are no transmission maxima.The graphical representation of the flow leads to the same conclusions. The polespectrum of the square well flows asymptotically to that of the δ -potential. Setting aU = 1 and taking the limit U → ∞ produces the trajectories of the resonance polesshown on the left side of Fig. 15. The resonances approach complex infinity as thedepth increases and no finite resonances exist in the δ -limit. Similarly, a single boundstate survives (Fig. 13) in agreement with the above discussion. The final anti-boundstate rapidly diverges to − i ∞ leaving no trace.19igure 13: Flow of the bound/anti-bound states of the δ -sequence potential. Asingle bound state survives as the potential approaches the delta-function limit.
Figure 14:
Flow of the bound state of the δ (cid:48) -sequence potential. A single bound statesurvives; however, it is a zero-energy state.
Flow of the resonance states in the δ -sequence (left) and δ (cid:48) -sequence(right) potentials. All the resonances move away from the origin, completely disappearingin the two limits.
Now let us consider the derivative of the δ -potential as arising from − U [ θ ( x + a ) − θ ( x ) + θ ( x − a )] , (19)a special case of the “two-piece potential”. We can show that the Schrödinger equationwith δ (cid:48) -potential has the same solutions as for the δ -potential but the matching hasto be done differently. Assuming the wave function is continuous at x = 0 , we onceagain integrate the Schrödinger equation around zero in order to obtain a matchingcondition for the derivative: ψ (cid:48) ( − ε ) = ψ (cid:48) ( ε ) . (20)This is quite different from the δ -potential, since the matching condition demandsa wave function that is also smooth at the origin, which is only possible for k =0 . Matching for a plane wave incident from the left and right shows the absence ofresonances and anti-resonances as well.While somewhat trivial, this potential illustrates another zero-feature limit, thatof the well+wall. The flows generated by the limit are shown in Fig. 14 for thebound/anti-bound poles and Fig. 15 (together with the δ case), in agreement with theanalytic arguments. 21 eferences [1] J. Mehra, H. Rechenberg, The Historical Development of Quantum Theory, vol.6, part 2 (Springer-Verlag, New York, 2001);G. Chew, The Analytic S -Matrix (W.A. Benjamin, New York, 1966)[2] M. Reed, B. Simon, Methods of Modern Mathematical Physics vol.4 - Analysis ofOperators (Academic Press, 1978)[3] R.G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, 1982);V.I. Kukulin, V.M. Krasnopol’sky, J. Horáček, Theory of Resonances (Kluwer,Dordrecht, 1989);S. Albeverio, L. S. Ferreira, L. Streit, Lecture Notes in Physics: Resonances –Models and Phenomena (Springer-Verlag, 1984);A. Bohm, M. Gadella, Dirac Kets, Gamow Vectors and Gel’fand Triplets: TheRigged Hilbert Space Formulation of Quantum Mechanics, Lecture Notes inPhysics, Vol 348 (Springer, 1989);M. Razavy, Quantum Theory of Tunneling (World Scientific, Singapore, 2003)[4] A. Baz, A. Perelomov, Ya.B. Zel’dovich, Scattering, Reactions and Decay in Non-relativistic Quantum Mechanics, (Israel Program for Scientific Translations, U.S.Dept. of Commerce, Clearinghouse for Federal Scientific and Technical Informa-tion, Springfield, 1969)[5] G.N. Gibson, G. Dunne, K.J. Bergquist, Tunneling Ionization Rates from Arbi-trary Potential Wells, Phys. Rev. Lett. 81 (1998) 2663-2666[6] E. Torrontegui, J. Muñoz, Yue Ban, J.G. Muga, Explanation and Observability ofDiffraction in Time, arXiv:1011.4278, 2010[7] A. del Campo, J.G. Muga, Dynamics of a Tonks-Girardeau gas released from ahard-wall trap, Europhys. Lett. 74 (2006) 965-971[8] M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto, Y. Iye, Observation of theFano-Kondo Antiresonance in a Quantum Wire with a Side-Coupled QuantumDot, Phys. Rev. Lett. 95 (2005) 066801 [4 pages][9] A. Goldberg, H.M. Schey, J.L. Schwartz, Computer-generated Motion Pictures ofOne-dimensional Quantum-mechanical Transmission and Reflection Phenomena,Am. J. Phys. 35 (1967) 177-186[10] H. Ohanian, C.G. Ginsburg, Antibound ‘States’ and Resonances, Am. J. Phys. 42(1974) 310-315[11] N. Hatano, K. Sasada, H. Nakamura, T. Petrosky, Some Properties of the ResonantState in Quantum Mechanics and Its Computation, Prog. Theor. Phys. 119 (2008)187-222[12] N. Moiseyev, Quantum Theory of Resonances: Calculating Energies, Widths andCross Sections by Complex Scaling, Phys. Rep. 302 (1998) 211-2932213] M. Kawasaki, T. Maehara, M. Yonezawa, Mutual Transformation AmongBound, Virtual and Resonance States in One-dimensional Rectangular Potentials,preprint, 2008 [arXiv:quant-ph/0810.3368v1][14] A.F.J. Siegert, On the Derivation of the Dispersion Formula for Nuclear Reactions,Phys. Rev. 56 (1939) 750-752[15] R.E. Peierls, Complex Eigenvalues in Scattering Theory, Proc. R. Soc. London A256 (1959) 16-36[16] N. Hokkyo, A Remark on the Norm of the Unstable State: A Role of Adjoint WaveFunctions in Non-Self-Adjoint Quantum Systems, Prog. Theor. Phys. 33 (1965)1116-1128;G. García-Calderón, R. Peierls, Resonant States and their Uses, Nucl. Phys. A265(1976) 443-460;A. Bohm, M. Gadella, G.B. Mainland, Gamow Vectors and Decaying States, Am.J. Phys. 57 (1989) 1103-1108;W. van Dijk, Y. Nogami, Novel Expression for the Wave Function of a DecayingQuantum System, Phys. Rev. Lett. 83 (1999) 2867-2871[17] H. M. Nussenzveig, The poles of the S-matrix of a Rectangular Potential Well orBarrier, Nucl. Phys. 11 (1959) 499-521[18] R. Zavin, N. Moiseyev, One-dimensional Symmetric Rectangular Well: fromBound to Resonance via Self-orthogonal Virtual State, J. Phys. A: Math. Gen.37 (2004) 4619-4628[19] G. Rawitscher, C. Merow, M. Nguyen, I. Simbotin, Resonances and QuantumScattering for the Morse Potential as a Barrier, Am. J. Phys. 70 (2002) 935-944[20] L.D. Faddeev, Properties of the S-matrix of the One-dimensional SchrodingerEquation, Trudy Mat. Inst. Steklov. 73 (1964) 314-336 (in Russian)[21] T. Regge, Nuovo Cimento, Construction ot Potentials from Resonance Parameters,9 (1958), 491-503[22] R. de la Madrid, M. Gadella, A Pedestrian Introduction to Gamow Vectors, Am.J. Phys. 70 (2002) 626-638[23] R. de la Madrid, G. García-Calderón, J.G. Muga, Resonance Expansions in Quan-tum Mechanics, Czech. J. Phys. 55 (2005) 1141-1150[24] L.D. Faddeyev, The Inverse Problem in the Quantum Theory of Scattering, J.Math. Phys. 4 (1963) 72-104[25] S.H. Tang, M. Zworski, Potential Scattering on the Real Line, available athttp://math.berkeley.edu/ ∼∼