{'fraction_non_alphanumeric': 0.042071197411003236, 'fraction_numerical': 0.011326860841423949, 'mean_word_length': 4.615524402907581, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Some general bounds for 1-D scattering

One-dimensional scattering problems are of wide physical interest and are encountered in many diverse applications. In this article I establish some very general bounds for reflection and transmission coefficients for one-dimensional potential scattering. Equivalently, these results may be phrased as general bounds on the Bogolubov coefficients, or statements about the transfer matrix. A similar analysis can be provided for the parametric change of frequency of a harmonic oscillator. A number of specific examples are discussed---in particular I provide a general proof that sharp step function potentials always scatter more effectively than the corresponding smoothed potentials. The analysis also serves to collect together and unify what would otherwise appear to be quite unrelated results.


I. INTRODUCTION
One-dimensional scattering problems occur in a wide variety of physical contexts. In acoustics one might be interested in the propagation of sound waves down a long pipe, while in electromagnetism one might be interested in the physics of wave-guides. In quantum physics the canonical examples are barrier penetration and reflection, while in classical physics an equivalent problem is the analysis of parametric resonances. All of these physical problems can be analyzed in the same mathematical framework, though for definiteness I shall present the discussion in terms of the Schrodinger equation, commenting on alternative formulations as appropriate.
For one-dimensional scattering problems there is a large catalog of specific potentials for which exact analytic results are known. There are also well-developed numerical techniques for estimating the scattering properties. In this paper I wish to take a different tack: I shall develop a number of very general and rather simple bounds on the reflection and transmission probabilities (equivalently, these bounds can be presented in terms of the Bogolubov coefficients, or in terms of statements about the transfer matrix). These bounds, because they are so general, are powerful aids in the qualitative understanding of one-dimensional scattering. Furthermore, this analysis provides a unifying theme that serves to connect together seemingly quite disparate results obtained in individual special cases.

A. Shabat-Zakharov systems
Consider the one-dimensional time-independent Schrodinger equation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] −h If the potential asymptotes to a constant, then in each of the two asymptotic regions there are two independent solutions to the Schrodinger equation Here the ±i distinguishes right-moving modes e +ikx from left-moving modes e −ikx , while the ±∞ specifies which of the asymptotic regions we are in. Furthermore To even begin to set up a scattering problem the minimum requirements are that the potential asymptote to some constant, and this assumption will be made henceforth. The so-called Jost solutions [16] are exact solutions J ± (x) of the Schrodinger equation that satisfy and Here α and β are the (right-moving) Bogolubov coefficients, which are related to the (right-moving) reflection and transmission amplitudes by These conventions correspond to an incoming flux of right-moving particles (incident from the left) being partially transmitted and partially scattered. The leftmoving Bogolubov coefficients are just the complex conjugates of the right-moving coefficients, however it should be borne in mind that the phases of β and β * are physically meaningless in that they can be arbitrarily changed simply by moving the origin of coordinates. The phases of α and α * on the other hand do contain real physical information.
In this article I will derive some very general bounds on |α| and |β|, which also lead to general bounds on the reflection and transmission probabilities The key idea is to re-write the second-order Schrodinger equation as a particular type of Shabat-Zakharov [17] system: a particular set of two coupled first-order differential equations for which bounds can be easily established. A similar representation of the Schrodinger equation is briefly discussed by Peirls [18] and related representations are well-known, often being used used without giving an explicit reference (see e.g. [19]). However an exhaustive search has not uncovered prior use of the particular representation of this article, nor the idea of using the representation to place bounds on one-dimensional scattering. I start by introducing an arbitrary auxiliary function ϕ(x) which may be either real or complex, though I do demand that ϕ ′ (x) = 0, and then defining This representation effectively seeks to use quantities resembling the "phase integral" wavefunctions as a basis for the true wavefunction [20]. This representation is of course highly redundant, since one complex number ψ(x) has been traded for two complex numbers a(x) and b(x) plus an essentially arbitrary auxiliary function ϕ(x). In order for this representation to be most useful it is best to arrange things so that a(x) and b(x) asymptote to constants at spatial infinity, which we shall soon see implies that we should pick the auxiliary function to satisfy To trim down the number of degrees of freedom it is useful to impose a "gauge condition" Subject to this gauge condition, I now re-write the Schrodinger equation in terms of two coupled first-order differential equations for these position-dependent Bogolubov coefficients. To do this note that (The last two relations use the "gauge condition".) Now insert these formulae into the Schrodinger equation written in the form to deduce It is easy to verify that this first-order system is compatible with the "gauge condition" (13), and that by iterating the system twice (subject to this gauge condition) one recovers exactly the original Schrodinger equation. These equations hold for arbitrary ϕ, real or complex, and when written in matrix form, exhibit a deep connection with the transfer matrix formalism [21].

B. Bounds
To obtain our bounds on the Bogolubov coefficients we start by restricting attention to the case that ϕ(x) is a real function of x. (Since ϕ is an essentially arbitrary auxiliary function this is not a particularly restrictive condition). Under this assumption the probability current is Now at x ∼ +∞ the wavefunction is purely right-moving and normalized to 1, because we are considering onedimensional Jost solutions [16]. Then for all x we have a conserved quantity It is this result that makes it useful to interpret a(x) and b(x) as position-dependent Bogolubov coefficients relative to the auxiliary function ϕ(x). Now use the fact that and use equation (20) to obtain That is The right hand side can now be bounded from above, by systematically using Re(A B) ≤ |A| |B|. This leads to It is essential that ϕ be real to have | exp(−2iϕ)| = 1 which is the other key ingredient above. Now define the non-negative quantity and use the conservation law (23) to write Integrate this inequality Taking limits as That is Which automatically implies Since this result holds for all real choices of the auxiliary function ϕ(x), (subject only to ϕ ′ = 0 and ϕ ′ → k ±∞ as x → ±∞), it encodes an enormously wide class of bounds on the Bogolubov coefficients. When translated to reflection and transmission coefficients the equivalent statements are and I shall soon turn this general result into more specific theorems.

C. Transfer matrix representation
The system of equations (20)- (21) can also be written in matrix form. It is convenient to define Then d dx This has the formal solution in terms of a generalized position-dependent "transfer matrix" [21] E(x f , x i ) = P exp where the symbol P denotes "path ordering". In particular, if we take x i → −∞ and x f → +∞ we obtain a formal but exact expression for the Bogolubov coefficients The matrix E is not unitary, though it does have determinant 1. It is in fact an element of the group SU (1, 1). Taking then (σ z ) 2 = +I, and defining E † = (E * ) T , it is easy to see This is the analog of the invariance of the Minkowski metric for Lorentz transformations in SO (3,1). Similarly, if we define the "complex structure" J by then J 2 = −I and

III. SPECIAL CASE 1
Suppose now that the potential satisfies V +∞ = V −∞ . Also, choose the phase function ϕ(x) to be ϕ = k ∞ x. We also require k ∞ = 0, that is E > V ±∞ . This is the special case discussed in a different context by Peirls [18]. Then the evolution equations simplify tremendously, and These bounds are exact non-perturbative results, however for high energies it may be convenient to use the slightly less restrictive (but analytically much more tractable) bounds and This version of the bounds also holds for all energies, but is not very restrictive for low energy. The transfer matrices can be analyzed by checking that the evolution equations simplify to This can be written in matrix form as This version of the Shabat-Zakharov system [17] has a formal solution in terms of the transfer matrix The formal but exact expression for the Bogolubov coefficients is now Furthermore, the form of the system (50)-(51) suggests that it might be useful to define Then This representation simplifies some of the results, for instance This can be used as the basis of an approximation scheme forβ. Suppose that for all This may be immediately integrated to yield This is immediately recognizable as the (first) Born approximation. If we instead work in terms of the original definition β This is one form of the distorted Born wave approximation.
In short, this type of analysis collects together a large number of results that otherwise appear quite unrelated. By taking further specific cases of these bounds and related results it is possible to reproduce many analytically known results, such as for delta function potentials, double delta function potentials, square wells, and sech 2 potentials, as discussed later in this article. (See Section VI.)

IV. SPECIAL CASE 2
Suppose now we take k(x) = ϕ ′ (x). This means that we are choosing our auxiliary function so that we use the WKB approximation for the true wavefunction as a "basis" for calculating the Bogolubov coefficients. This choice is perfectly capable of handling the case V +∞ = V −∞ but because of the assumed reality of ϕ is limited to considering scattering over the potential barrier. (This is the special case implicit in a different context in [19]). The evolution equations again simplify tremendously, to yield This form of the evolution equations can be related to the qualitative discussion of scattering over a potential barrier presented by Migdal [22,23]. For this choice of auxiliary function and the bounds become and The relevant transfer matrix is now The Bogolubov coefficients are now This type of analysis collects together and unifies several analytically known results for scattering over the barrier, such as for asymmetric square wells and Poschl-Teller potentials. (See Section VI.) After a few general comments, I shall turn to specializing this still rather general result to more specific cases.
A. Reflection above the barrier The system (63)-(64) can also be used as the basis of an approximation scheme for β. Suppose that for all x we have |b(x)| ≪ 1, so that |a(x)| ≈ 1. Then This may be immediately integrated to yield Or the equivalent This result serves to clarify the otherwise quite mysterious discussion of "reflection above the barrier" given by Migdal [22,23]. Even though the WKB wavefunctions are buried in the representation of the wavefunction underlying the analysis leading to this approximation, the validity of this result for |β| does not require validity of the WKB approximation.
If the shifted potential, V − V ∞ , is "small" we can recover the Born approximation in the usual manner. In that case

B. Under the barrier?
What goes wrong when we try to extend this analysis into the classically forbidden region? Analytically continuing the system (63)-(64) is trivial, replace and write to obtain Thus we are violating our previous condition that ϕ be real, though we still require ϕ ′ = 0. This is a perfectly good Shabat-Zakharov system that works in the forbidden region. But you cannot now use this to derive bounds on the transmission coefficient. The fly in the ointment resides in the fact that the formula for the probability current is modified, and that in the forbidden region the probability current is For a properly normalized flux in the allowed region (|a| 2 − |b| 2 = 1), we have in the forbidden region 2 Im {ab * exp(+2iϕ tp )} = 1.
While this does imply 2|a||b| > 1, the inequality is unfortunately in the wrong direction to be useful for placing bounds on the transmission coefficient.

C. Special Case 2-a
Suppose now that V (x) is continuous and monotonic increasing or decreasing, varying from and the transmission and reflection probabilities satisfy and These bounds are immediately recognizable as the exact analytic results for a step-function potential [1,7,8], and the result asserts that for arbitrary smooth monotonic potentials the step function provides upper and lower bounds on the exact result. If we are interested in physical situations such as a time-dependent refractive index [25,26], or particle production due to the expansion of the universe [27], this technique shows that sudden changes in refractive index or size of the universe provide a strict upper bound on particle production.

D. Special Case 2-b
Suppose now that V (x) has a single unique extremum (either a peak or a valley), and provided that E ≥ max{V ∞ , V extremum , V +∞ } so that there is no classical turning point, then k(x) moves monotonically from k −∞ to k extremum and then back to k +∞ . Under these circumstances This implies Which yields To be more specific, if in addition V (−∞) = 0 = V (+∞), so that k −∞ = k +∞ , then we have and Translated into statements about the transmission and reflection probabilities this becomes and Equivalently and the present result (when it is applicable) leads to tighter bounds on the transmission and reflection coefficients. Numerous generalizations of these formulae are possible. For example, at the cost of a little extra notation, we also already have enough information to provide a bound on an asymmetric barrier or asymmetric well, as long as it has only a single extremum (maximum or minimum) we apply the previous equations to derive and Translated into statements about the transmission and reflection probabilities this becomes and and This can be compared, for example, with known analytic results for the asymmetric square well, see equation (144) in Section VI.

E. Special Case 2-c
Suppose now that V (x) has a number of extrema, (both peaks and valleys). I allow V (+∞) = V (−∞), but demand that for all extrema E ≥ max{V −∞ , V +∞ , V i extremum } so that there is no classical turning point.
For definiteness, suppose the ordering is: Defining we see This bounds the Bogolubov coefficients as That is and .
Then the transmission and reflection probabilities satisfy and In these formulae, peaks and valleys can be interchanged in the obvious way, and by letting the initial or final peak sink down to V ±∞ as appropriate we obtain bounds for sequences such as: −∞ → valley → peak ... valley → peak → +∞, or: −∞ → peak → valley ... peak → valley → +∞. In the case of one or zero extrema these formulae reduce to the previously given results. [Equations (96)-(97).] Further modifications of these formulae are still possible, the cost is that more specific assumptions are needed to derive more specific results.

V. PARAMETRIC OSCILLATIONS
Though the discussion so far has been presented in terms of the spatial properties of the time-independent Schrodinger equation, the mathematical structure of parametrically excited oscillations is identical, needing only a few minor translations to be brought into the current form. For a parametrically excited oscillator we have Just map t → x, ω(t) → k(x), and φ → ψ. In the general analysis of equations (28)-(35) the quantity ϑ should be replaced by The analysis then parallels that of the Schrodinger equation. Some key results are given below.
For definiteness, suppose the ordering is: −∞ → peak → valley ... valley → peak → +∞. Define The Bogolubov coefficients satisfy and In these formulae, peaks and valleys can be interchanged in the obvious way, and by letting the initial or final peak sink down to ω ±∞ as appropriate we obtain bounds for sequences such as: −∞ → valley → peak ... valley → peak → +∞, or: −∞ → peak → valley ... peak → valley → +∞. In the case of one or zero extrema these formulae reduce to the previously given results.
Again, further specializations of these formulae are still possible. As always there is a trade-off between the strength of the result and its generality.

VI. COMPARISON WITH KNOWN ANALYTIC RESULTS
For comparison purposes, in this section I collect several known analytic results and show how they relate to the general results presented in this article.

A. Delta-function potential
For a delta function potential the transmission coefficient is known to be [2,3] This satisfies the bound (46), and also (48), and for E → ∞ asymptotically approaches the bound, thus showing that the bound cannot be improved in the high-energy regime unless additional hypotheses are made. Though these bounds were derived assuming wellbehaved functions the statements (46) and (48) continue to make good sense even for delta-function potentials. Thus any smooth set of well-behaved functions tending to a delta-function limit may be used to establish (46) and (48) even for potentials containing delta-function contributions.

B. Double-delta-function potential
For the double delta function the transmission coefficient is [11] T = 1 It is an easy exercise to check that this satisfies the bounds (46) and (48).

C. Square barrier
Tunneling over a square barrier is an elementary problem which however is not always discussed in the textbooks. (Tunneling under a square barrier is much more popular.) The exact transmission coefficient is (See Landau-Lifshitz [1] or Schiff [9].) If we re-write this as then it is clear that the bound (46) is satisfied. It is also possible to verify that this satisfies the general lower bound (66) that I have presented above, and in fact oscillates between this lower bound and the upper T ≤ 1 unitarity limit. For certain values of the barrier width [k extremum L = (2n + 1)π/2] the square well saturates this bound thus showing that this bound cannot be improved unless additional hypotheses are made.

D. Tanh potential
For a smoothed step function of the form the reflection coefficient is known analytically to be [1] This certainly satisfies the general bounds (81)-(82) enunciated above, and as L → 0 approaches and saturates the bound.

E. Sech potential
For a sech 2 potential of the form the transmission coefficient is known analytically to be [1] provided 8mV e L 2 <h 2 . This satisfies the general bounds, both (34) and (81), enunciated above. (Though proving this is tedious.) Start by noting that for this sech potential and use the inequality (x > 0) Then Provided that the extremum is a peak, V peak > 0 we can use the bound 8mV peak L 2 <h 2 to deduce This is the particularization of (34) to the present case.
If V e < 0 we need a different analysis.

F. Asymmetric square-well potential
For the asymmetric square well we define k i ≡ 2m(E − V i )/h. The transmission coefficient is [15] T = 4k 1 k 2 2 k 3 (k 1 + k 3 ) 2 k 2 2 + [k 2 1 k 2 3 + k 2 2 (k 2 2 − k 2 1 − k 2 3 )] sin 2 (k 2 L) . Then Similarly to the case for the symmetric square well, the transmission probability for the asymmetric square well oscillates between the bound (96) and the unitarity limit T = 1. For certain values of the width of the well [k 2 L = (2n+1)π/2] the transmission coefficient saturates the bound thus showing that this bound cannot be improved unless additional hypotheses are made. Because V −∞ = V +∞ the bound (34) is not applicable, at least not without modification from its original form.

VII. DISCUSSION
The various special cases discussed above are merely specific examples of the general results (32)-(33) and (34)-(35) illustrating the power of the technique. There are many other variations on the bounds presented above that can be derived for specific choices of ϕ(x) and specific restrictions on the scattering potential V (x).
The most general form of the bounds are given in equations (32)-(33) and (34)-(35). Because of the large amount of freedom in choosing the function ϕ these bounds encode even more specific cases beyond those discussed in this article, and have the potential for leading to new interesting specific cases. The special cases I discussed in this article were chosen for directness and simplicity.
For instance, Special Case 1, as presented in equations (46)-(47) and (48)-(49), has the advantage that it applies to both scattering over the barrier and under the barrier. On the other hand, Special Case 2, as presented in equations (66)-(67) and their specializations, applies only to scattering over the barrier but has the advantage of being much more selective in how much information is needed concerning the scattering potential.
In summary, the bounds presented in this article are useful in establishing qualitative analytic properties of one-dimensional scattering, and as such are complementary to both explicit numerical investigations and the guidance extracted from exact analytic solutions.