aa r X i v : . [ m a t h . SP ] J un Spectral properties of soft quantum waveguides
Pavel Exner
Nuclear Physics Institute, Czech Academy of Sciences, Hlavn´ı 130,25068 ˇReˇz near Prague, Czech RepublicDoppler Institute, Czech Technical University, Bˇrehov´a 7, 11519 Prague,Czech RepublicE-mail: [email protected]
Abstract.
We consider a soft quantum waveguide described by a two-dimensionalSchr¨odinger operators with an attractive potential in the form of a channel of a fixedprofile built along an infinite smooth curve which is not straight but it is asymptoticallystraight in a suitable sense. Using Birman-Schwinger principle we show that thediscrete spectrum of such an operator is nonempty if the potential well defining thechannel profile is deep and narrow enough. Some related problems are also mentioned.PACS numbers: 03.65.Ge, 03.65Db
Keywords : Quantum waveguides, Schr¨odinger operators, regular potentials, Birman-Schwinger principle, discrete spectrum
Submitted to:
J. Phys. A: Math. Gen.
1. Introduction
Properties of motion confined to regions infinitely extended in some direction, and ofa constant ‘size’ in the other, attracted over the years a lot of attention. The originalmotivation came from quantum mechanics where such systems appeared naturally asmodels of electrons in semiconductor wires, atoms in hollow optical fibers, and otherwaveguide-type systems [EK15], but related effects were also investigated, theoreticallyand experimentally, for instance in electromagnetism [LCM99] or acoustics [DP98]. Themost common framework for such a waveguide analysis is based on Schr¨odinger operatorsin tubular regions with appropriate conditions at their boundary; for a recent survey ofthe existing results we refer to the monograph [EK15].An alternative approach, for which names like ‘leaky quantum wires’ or similar isused, is motivated by the fact that in actual quantum systems the confinement is rarelyperfect, and tunneling between different parts of the structure is possible. It works withsingular Schr¨odinger operators which can be formally written as − ∆ − αδ ( x − Γ) with oft quantum waveguides α >
0, where the interaction support Γ is a curve, a graph, or more generally, a complexof lower dimensionality [EK15, Chap. 10]. In this situation the particle can be foundeverywhere, but the states with the energy support in the negative part of the spectrumare localized in the vicinity of Γ.The two model classes share some properties. A notable one among them is theexistence of curvature-induced bound states . If such a tube, or the set Γ supportingthe attractive interaction, is not straight but it is straight outside a bounded region,or it is at least asymptotically straight in a suitable sense, then the system exhibitslocalized states associated with discrete eigenvalues below the continuum, their numberand location being determined by the geometry of the system.One can note that both the described system types contain a degree of idealization.We have mentioned already that the tube walls may not be impenetrable. On the otherhand, real guides we try to model are often thin but never infinitely thin. A morerealistic way to describe this sort of physical constraints would be to use an attractivepotential, but a regular one in the form of a ditch, or channel, of a finite width. This isthe topic we are going to address in this paper. The first question that naturally arisesis whether in such a situation again ‘bending can cause binding’, that is, whether theHamiltonian with a bent potential channel would have isolated eigenvalues.We are going to discuss this problem in the simplest two-dimensional setting wherethe potential channel has a finite width. There are naturally many extensions tothe question posed in this way, and we will mention them briefly in the concludingremarks. The tool we choose is the Birman-Schwinger method which has the advantagethat it allows us employ the ‘straightening trick’ known from the theory of hard-wallwaveguides. This is not the only possibility, of course, one can think, for instance, ofa variational method, but that could constitute the contents of another paper. Ourmain result, Theorem 7.1, is a sufficient condition for the existence of curvature-inducedbound states which shows, in particular, that such states exist provided the potentialwell determining the channel profile is deep and narrow enough.Before proceeding, let us briefly describe the contents of the paper. The problem isstated in the next section where the assumption we use are formulated. In Section 3 wecombine the bracketing technique and Weyl criterion in order to determine the essentialspectrum of the Schr¨odinger operators in question; it is vital to show that it is preservedunder the geometric perturbations we are considering. Before coming to our main topic,we mention in Section 4 two simple asymptotic results establishing the discrete spectrumexistence based on the known facts about the hard-wall waveguides and leaky wires. Themain part of the paper are Sections 5–8 where we apply Birman-Schwinger method toour case, prove the main result, and discuss its consequences. We finish the paper with asurvey of open questions related to the present problem, noting with pleasure a result ona similar problem in the three-dimensional setting that appeared very recently [EKP20]. oft quantum waveguides
2. Statement of the problem
Let Γ be an infinite and smooth planar curve without self-intersections, naturallyparametrized by its arc length s , that is, the graph of a function Γ : R → R ;with the abuse of notation we employ the same symbol for the map Γ and for itsrange. Writing Γ in the Cartesian coordinates we introduce the signed curvature γ : γ ( s ) = ( ˙Γ ¨Γ − ˙Γ ¨Γ )( s ) where the dot conventionally denotes the derivative withrespect to s . The curve is supposed to satisfy the following assumptions:(a) Γ is C -smooth so, in particular, γ ( s ) makes sense,(b) γ is either of compact support, supp γ ⊂ [ − s , s ] for some s >
0, or Γ is C -smoothand γ ( s ) together with its first and second derivatives tend to zero as | s | → ∞ ,(c) | Γ( s ) − Γ( s ′ ) | → ∞ holds as | s − s ′ | → ∞ .The last assumption excludes U-shaped curves and their various modifications.The knowledge of γ allows us to reconstruct the curve uniquely, up to Euclideantransformations: putting β ( s , s ) := R s s γ ( s ) d s , we haveΓ( s ) = (cid:16) x + Z ss cos β ( s , s ) d s , x − Z ss sin β ( s , s ) d s (cid:17) (2.1)for some s ∈ R and x = ( x , x ) ∈ R . Next we define the strip Ω a in the plane, ofhalfwidth a >
0, built over Γ asΩ a := { x ∈ R : dist( x, Γ) < a } , in particular, Ω a := R × ( − a, a ) corresponds to a straight line for which we use thesymbol Γ . We assume that(d) a k γ k ∞ < a can be uniquely parametrized by the arc length andthe distance from Γ as follows, x ( s, u ) = (cid:0) Γ ( s ) − u ˙Γ ( s ) , Γ ( s ) + u ˙Γ ( s ) (cid:1) , (2.2)which constitute a natural locally orthogonal system of coordinates on Ω a , N ( s ) =( − ˙Γ ( s ) , ˙Γ ( s )) being the unit normal vector to Γ at the point s .The main object of our interest are Schr¨odinger operators with an attractivepotential supported in the strip Ω a . To introduce it we consider(e) a nonzero V ≥ L ∞ ( R ) with supp V ⊂ [ − a, a ]and define ˜ V : Ω a → R + , ˜ V ( x ( s, u )) = V ( u ) , (2.3a) H Γ ,V = − ∆ − ˜ V ( x ); (2.3b) oft quantum waveguides D ( − ∆) = H ( R ). It is also useful tointroduce the comparison operator on L ( R ), h V = − ∂ x − V ( x ) (2.4)with the domain H ( R ) which has in accordance with (e) a nonempty and finite discretespectrum such that ǫ := inf σ disc ( h V ) = inf σ ( h V ) ∈ (cid:0) − k V k ∞ , (cid:1) . (2.5)Moreover, we know that ǫ is a simple eigenvalue and the associated eigenfunction φ ∈ H ( R ) can be chosen strictly positive; we will use the same symbol for this functionand its restriction to the interval ( − a, a ). The relation (2.5) helps us to find the spectrumof H Γ ,V in the situation when the generating curve is a straight line because in that casethe variables separate and we have σ ( H Γ ,V ) = σ ess ( H Γ ,V ) = [ ǫ , ∞ ) . (2.6)The question we address in this paper is about the spectrum of the operator H Γ ,V inthe situation where Γ satisfies the assumptions (a)–(e) and is not straight . An exampleof particular interest is the soft flat-bottom waveguide referring to the function V J, ( u ) = V χ J ( u ) , V > , (2.7)where χ J is the indicator function of an interval J = [ − a , a ] ⊂ [ − a .a ].
3. The essential spectrum
If the potential ditch is straight outside a compact, or at least asymptotically straightin the sense of (b), the essential spectrum is preserved.
Proposition 3.1.
Under assumptions (a) – (e) we have σ ess ( H Γ ,V ) = [ ǫ , ∞ ) .Proof. If Γ is straight outside a compact we can divide the plane into four regions. Thefirst two of them is a pair of disjoint halfstrips Σ ± = { x ( s, u ) : ± s > s , | u | < u } , therest consists of a compact set Σ c containing supp V \ (Σ + ∪ Σ − ) and its complement to R \ (Σ + ∪ Σ − ). We estimate H Γ ,V from below by imposing additional Neumann conditionat the boundaries between the four regions. The compact one does not contribute tothe essential spectrum, hence we have to inspect the spectral thresholds of the otherthree. The operator part referring to the last named one is positive, while the operatorscorresponding to the two halfstrips have separated variables, and consequently, theiressential spectra start at ǫ ( u ) := inf σ ( h N V ( u )), where h N V ( u ) is the operator (2.4)restricted to the interval ( − u , u ) with Neumann boundary conditions. Furthermore,in analogy with [DH93] one can check that ǫ ( u ) → ǫ as u → ∞ , in fact exponentiallyfast, because V ( x ) = 0 outside ( − a, a ) and φ is exponentially decreasing there. Noting oft quantum waveguides u can be chosen arbitrarilylarge by picking s large enough, we can conclude thatinf σ ess ( H Γ ,V ) = ǫ . (3.1)If Γ satisfies the other part of assumption (b) we replace the halfstrips Σ ± by familiesof pairwise disjoint regions Σ ( j ) ± := { x ( s, u ) : ± s ∈ ( s j − , s j ) , u ∈ ( − u j , u j ) } , j ∈ N ,determined by increasing sequences { s j } ∞ j =1 and { u j } ∞ j =1 of positive numbers, and imposeagain Neumann conditions at their boundaries. Using the parametrization (2.2) we canpass from the corresponding Neumann restrictions of h V to unitarily equivalent operators H ( j ) ± on L (Σ ( j )0 , ± ), where Σ ( j )0 , + := ( s j − , s j ) × ( − u j , u j ) and Σ ( j )0 , − is defined analogously,in the way described in (5.3) below. According to [EK15, Sec. 1.1] they are of the form H ( j ) ± = h N V ( u ) ⊗ ( − ∂ s ) N + V γ ( s, u ) ,V γ ( s, u ) := − γ ( s ) uγ ( s )) + u ¨ γ ( s )2(1 + uγ ( s )) − u ˙ γ ( s ) (1 + uγ ( s )) . Since γ ( s ) → | s | → ∞ , in view of assumptions (c) and (d) the rectangles Σ ( j )0 , ± canbe made arbitrarily wide in the u variable by choosing large enough s j − . The spectrumof each of the operators H ( j ) ± is discrete, of course, it is the accumulation point of theirprincipal eigenvalues which determines the threshold of σ ess ( H Γ ,V ). Since the groundstate of ( − ∂ s ) N is zero and the eigenvalues of H ( j ) ± differ from those of h N V ( u ) ⊗ ( − ∂ s ) N at most by k V γ ↾ Σ ( j )0 , ± k ∞ which tends to zero as j → ∞ , we arrive at (3.1) again.To prove that there are no spectral gaps above ǫ , we use Weyl criterion. Toconstruct a suitable sequence, we consider functions v, w ∈ C ∞ ( R ) with the supportsin [ − ,
1] such that their norms are k v k = k w k = 1 and v ( s ) = w ( s ) = 1 holds in thevicinity of zero, and put ψ ( s, u ) = √ µν v ( µ ( s − s )) w ( νu ) φ ( u ) e iks (3.3)for µ, ν >
0, a fixed k ∈ R , and some s . The conclusions we are going to make do notdepend on s , so we can put s = 0. Using the fact that φ is the eigenfunction of h V corresponding to the eigenvalue ǫ we find that( − ∆ − V − ǫ − k ) ψ ( s, u ) = √ µν (cid:2)(cid:0) − µ v ′′ ( µs ) − ikµv ′ ( µs ) (cid:1) w ( νu ) φ ( u )+ v ( µs ) (cid:0) − ν w ( νu ) φ ( u ) − νw ′ ( νu ) φ ′ ( u ) (cid:1)(cid:3) e iks The right-hand side is a sum of four terms, f + f + f + f . Let us estimate theirnorms. After a simple change of variables we get k f k = Z − Z − µ v ′′ ( ξ ) w ( η ) φ (cid:0) uν (cid:1) d ξ d η = µ k w ′′ k φ (0) (cid:0) O ( ν ) (cid:1) so that k f k = O ( µ ), in a similar way we find k f k = O ( µ ), k f k = O ( ν ), and k f k = O ( ν ), and therefore k ( − ∆ − V − ǫ − k ) ψ k → µ, ν →
0. To prove that oft quantum waveguides ǫ belongs to σ ess ( H Γ ,V ) it is thus sufficient to find a family ofincreasing regions threaded by the curve that support functions ψ j of the form (3.3).If Γ is straight outside a compact we use the regions Σ ( j )0 , + and put s = ( s j − + s j )in (3.3). In contrast to the first part of the proof, in addition to u j → ∞ it is alsoimportant to have s j − s j − → ∞ as j → ∞ , which is possible due to assumption (c).Then it is enough to choose µ j = 2 c ( s j − s j − ) − and ν j = cu − j for some c ∈ (0 , ( j )0 , + bythe ‘bent rectangles’ Σ ( j )+ , assumptions (c) and (d) again allow us to choose a disjointfamily of them expanding in both the longitudinal and transversal directions. Usingonce more the straightening transformation we find that the norms of the correspondingWeyl approximants, k ( − ∆ − ˜ V − ǫ − k ) ψ j k , differ from the above estimate at mostby k V γ ↾ Σ ( j )0 , ± k ∞ , and since this quantity vanishes as we follow the curve to infinity, theproof is complete.
4. Asymptotic results
The other types of quantum waveguides, the hard-wall ones [EK15] and the leaky wires,described by singular Schr¨odinger operators, are not only much better understood thanoperators of the type (2.3b), but they represent in a sense extreme cases of such systems.This makes it possible to prove some sufficient conditions for the existence of discretespectrum. One comes from the approximation result proven in [EI01], and in greatergenerality in [BEHL17]. To state it, we consider the family of potentials V ε : V ε ( u ) = ε V (cid:0) uε (cid:1) obtained by scaling of a given V satisfying assumption (e). Proposition 4.1.
Consider a non-straight C -smooth curve Γ : R → R such that | Γ( s ) − Γ( s ′ ) | < c | s − s ′ | holds for some c ∈ (0 , . If the support of its signed curvature γ is noncompact, assume, in addition to (b) , that γ ( s ) = O ( | s | − β ) with some β > as | s | → ∞ . Then σ disc ( H Γ ,V ε ) = ∅ holds for all ε small enough.Proof. By the results of [EI01, BEHL17] the operators H Γ ,V ε converge as ε → H Γ ,α formally writtenas − ∆ − αδ ( x − Γ) with α := R a − a V ( u ) d u , which is the unique self-adjoint operatorassociated with the quadratic form q Γ ,α [ ψ ] = k∇ ψ k L ( R ) − α k ψ k L (Γ) , closed and below bounded on H ( R ). Consequently, the spectrum of H Γ ,V ε convergesin the set sense to that of H Γ ,α as ε →
0. In particular, it is not difficult to checkdirectly that the essential spectrum threshold ǫ ( ε ) of H Γ ,V ε converges to the the essentialspectrum threshold − α of H Γ ,α .The assumptions (a), (b), and (e) are satisfied, the inequality | Γ( s ) − Γ( s ′ ) | < c | s − s ′ | implies further the validity of (c). Next we note that the support of V ε is contained oft quantum waveguides − εa, εa ], hence (d) is valid for sufficiently small values of ε . At the same time,the additional decay requirement imposed on γ means by Remark 5.6 of [EI01] thatthe assumptions of Theorem 5.2 of the said paper are satisfied, and consequently, theoperator H Γ ,α has at least one isolated eigenvalue below the the essential spectrumthreshold. From the convergence result indicated above, it then follows that the sameis true for H Γ ,V ε with ε small enough.The most common quantum waveguide model works with the particle confined toa tubular region with hard walls; if no other forces are involved, the Hamiltonian is(a multiple of) the appropriate Dirichlet Laplacian. In the two-dimensional situationwhere the region is a strip of a fixed width in the plane, the known results can againbe used for comparison with our present problem. Let us recall the heuristic statementthat the Dirichlet condition corresponds to ‘infinitely high potential wall’, which can begiven a mathematically rigorous meaning, cf. [DK05, Sec. 4.2.3] or [Si05, Sec. 21]. Proposition 4.2.
Suppose that Γ is not straight and assumptions (a) – (d) are satisfied,then the operator H Γ ,V J, referring to the potential (2.7) has nonempty discrete spectrumfor all V large enough.Proof. { H Γ ,V J, : V ≥ } is clearly a holomorphic family of type (A) in the sense ofKato, and moreover, it is monotonous with respect to V . The same is true for operators G Γ ,V J, := H Γ ,V J, + V which are positive and form an increasing family, G Γ ,V J, ≥ G Γ ,V ′ J, for V > V ′ . Thus their eigenvalues λ j ( V ) are continuous functions of V , increasingby the minimax principle, and as such that they have limits as V → ∞ ; the samealso applies to the threshold of their essential spectrum. The limiting values refer to thecorresponding spectral quantities of the (negative) Dirichlet Laplacian of Ω J , the supportof the function (2.7). Under the hypotheses made, the assumptions of Theorem 1.1 andits corollary in [EK15] are fulfilled, hence the said limiting operator has a nonemptydiscrete spectrum below the continuum which starts at π | J | − , | J | = a + a . It followsthat for all sufficiently large V we have σ disc (cid:0) G Γ ,V J, (cid:1) = ∅ , and the same is true, ofcourse, for the shifted operators H Γ ,V J, . Remark 4.1.
We note that the assumptions under which the used spectral propertiesof the limiting operators have been derived are in no way optimal. This concerns, forinstance, the existence of a wedge separating the curve ends in Proposition 4.1 or theflat bottom of the potential channel in Proposition 4.2. There is, no doubt, a room formathematical activity here.
5. Birman-Schwinger analysis
Given a function V and z ∈ C \ R + we consider the operator K Γ ,V ( z ) := ˜ V / ( − ∆ − z ) − ˜ V / (5.1) oft quantum waveguides V given by (2.3a); we are particularly interested in the negative values of thespectral parameter, z = − κ with κ >
0. In view of assumption (e) it is a boundedoperator, positive for z = − κ , which we can regard as a map L (Ω a ) → L (Ω a ). ByBirman-Schwinger principle this operator can be used to determine the discrete spectrumof H Γ ,V . Proposition 5.1. z ∈ σ disc ( H Γ ,V ) holds if and only if ∈ σ disc ( K Γ ,V ( z )) . The function κ K Γ ,V ( − κ ) is continuous and decreasing in (0 , ∞ ) , tending to zero in the normtopology, that is, k K Γ ,V ( − κ ) k → holds as κ → ∞ .Proof. The first claim is a particular case of a more general and commonly known result,see, e.g., [BGRS97]. The continuity follows from the functional calculus and we havedd κ ( ψ, ˜ V / ( − ∆ + κ ) − ˜ V / ψ ) = − κ ( ψ, ˜ V / ( − ∆ + κ ) − ˜ V / ψ ) < ψ ∈ L (Ω a ) with ˜ V / ψ = 0 which proves the monotonicity. Finally, the simpleestimate k K Γ ,V ( − κ ) k ≤ κ − k V k ∞ concludes the proof.Note also that if g is an eigenfunction of the operator (5.1) with eigenvalue one,the corresponding eigenfunction of H Γ ,V is given by φ ( x ) = Z supp ˜ V G κ ( x, x ′ ) ˜ V ( x ′ ) / g ( x ′ ) d x ′ , (5.2)where G κ is the integral kernel of ( − ∆ + κ ) − .Using the knowledge of the Laplacian resolvent we can write the action of K Γ ,V ( z )explicitly, in particular, for z = − κ with κ > K Γ ,V ( x, x ′ ; − κ ) = 12 π ˜ V / ( x ) K ( κ | x − x ′ | ) ˜ V / ( x ′ ) , where K is the Macdonald function, mapping L (Ω a ) to itself.In order to treat the geometry of Γ as a perturbation it is useful to pass tothe ‘straightened’ strip in the way used in treatment of the ‘hard-wall’ waveguides[EK15, Sec. 1.1]. The first step is the natural coordinate change, passing from theCartesian coordinates in the plane to s, u , which amounts to a unitary map L (Ω a ) → L (Ω a , (1 + uγ ( s )) / d s d u ). To get rid of the Jacobian, we use the unitary operator L (Ω a ) → L (Ω a ) , ( U ψ )( s, u ) = (1 + uγ ( s )) / ψ ( x ( s, u )) . (5.3)Using it we pass from the Birman-Schwinger operator K Γ ,V ( − κ ) to the unitarily equi-valent one, R κ Γ ,V := U K Γ ,V ( − κ ) U − , which is an integral operator on L (Ω a ) with thekernel R κ Γ ,V ( s, u ; s ′ , u ′ ) = 12 π W ( s, u ) / K ( κ | x − x ′ | ) W ( s ′ , u ′ ) / , (5.4)where x = x ( s, u ), x ′ = x ( s ′ , u ′ ), and the modified potential is W ( s, u ) := (1 + uγ ( s )) V ( u ) . oft quantum waveguides
6. The straight case
The Birman-Schwinger operator corresponding to the straight potential ditch has thekernel R κ Γ ,V ( s, u ; s ′ , u ′ ) = 12 π V ( u ) / K ( κ | x − x ′ | ) V ( u ′ ) / , (6.1)where | x − x ′ | = (cid:2) ( s − s ′ ) + ( u − u ′ ) (cid:3) / . To find its spectrum we notice that, withrespect to the longitudinal variable s , it acts as a convolution. The unitarily equivalentoperator obtained by means of the Fourier-Plancherel operator F on L ( R ), has thusthe form of a direct integral,( F ⊗ I ) R κ Γ ,V ( F ⊗ I ) − = Z ⊕ R R κ Γ ,V ( p ) d p, (6.2)where the fibers are integral operator on L ( − a, a ) with the kernels R κ Γ ,V ( u, u ′ ; p ) = V ( u ) / e − √ κ + p | u − u ′ | p κ + p V ( u ′ ) / , (6.3)because by [GR07, 6.726.4] and [AS72, 10.2.17] we have12 π Z R K (cid:0) κ p ξ + | u − u ′ | (cid:1) e ipξ d ξ = e − √ κ + p | u − u ′ | p κ + p . However, in view of (6.3) R κ Γ ,V ( p ) is nothing but the Birman-Schwinger operatorassociated with (2.4) referring to the spectral parameter z = − ( κ + p ). By assumption, ǫ is the smallest eigenvalue of h V , and consequently, by Proposition 5.1 in combinationwith (6.2), the number − κ = ǫ + p belongs to the spectrum of H Γ ,V for any p ∈ R ,in accordance with (2.6).At the same time, the operator with kernel (6.1) satisfiessup σ ( R κ Γ ,V ) = 1 , where κ = √− ǫ , because, was the left-hand side larger than one, by Proposition 5.1there would exist a ˜ κ > κ such that 1 ∈ σ ( R ˜ κ Γ ,V ), and consequently, we would have − ˜ κ ∈ σ ( H Γ ,V ), however, this contradicts relation (2.6).We also note that one can relate the eigenfunction φ of h V to the eigenfunction g of R κ Γ ,V (0) corresponding to the unit eigenvalue. On the one hand, we have g ( u ) = V / ( u ) φ ( u ) , (6.4)on the other hand, φ can be expressed in the way analogous to (5.2) and, mutatismutandis , one can write the generalized eigenfunction associated with the bottom of σ ( H Γ ,V ) as f ( s, u ) = φ ( u ) = Z a − a e − κ | u − u ′ | κ V ( u ′ ) / g ( u ′ ) d u ′ . oft quantum waveguides
7. Existence of bound states
Our aim is now to use the Birman-Schwinger method to derive a condition which wouldensure the existence of curvature-induced bound states, that is, a discrete spectrum ofoperator H Γ ,V . Theorem 7.1.
Let assumptions (a) – (e) be valid and set C κ Γ ,V ( s, u ; s ′ , u ′ ) = 12 π φ ( u ) (cid:2) ((1 + uγ ( s )) K ( κ | x ( s, u ) − x ( s ′ , u ′ ) | ) (1 + u ′ γ ( s ′ )) − K ( κ | x ( s, u ) − x ( s ′ , u ′ ) | ) (cid:3) φ ( u ′ ) for all ( s, u ) , ( s ′ , u ′ ) ∈ Ω a , then we have σ disc ( H Γ ,V ) = ∅ provided Z R d s d s ′ Z a − a Z a − a d u d u ′ C κ Γ ,V ( s, u ; s ′ , u ′ ) > . (7.1) holds for κ = √− ǫ .Proof. The main idea is to treat the geometry of the system, translated by (5.3) into thecoefficients of the operator, as a perturbation of the straight case. By Proposition 3.1the essential spectrum is preserved, hence in order to demonstrate that the spectrum of H Γ ,V has at least one eigenvalue below ǫ , it is in view of Proposition 5.1 sufficient tofind a function ψ η ∈ L (Ω a ) such that( ψ, R κ Γ ,V ψ ) − k ψ k > . (7.2)As usual a natural way to construct such a trial function is to combine the generalizedeigenfunction, associated with the edge of the spectrum, with a mollifier which makesit an element of the Hilbert space. Let us first inspect the effect of the mollifier in thestraight case, Γ = Γ . Lemma 7.1.
Let ψ η ∈ L (Ω a ) be of the form ψ η ( s, u ) = h η ( s ) g ( u ) , where h η ( s ) = h ( ηs ) with a function h ∈ C ∞ ( R ) such that h ( s ) = 1 in the vicinity of s = 0 . Then ( ψ η , R κ Γ ,V ψ η ) − k ψ η k = O ( η ) holds as η → .Proof. Since g is by assumption the eigenfuction of R κ Γ ,V (0), we can rewrite the secondterm on the left-hand side as −k h η k ( g , R κ Γ ,V (0) g ). Using relations (6.1)–(6.3) we castthe expression to be estimated into the form Z a − a Z a − a g ( u ) V ( u ) / (cid:20) Z R | ˆ h η ( p ) | e − √ κ + p | u − u ′ | p κ + p d p − k h η k e − κ | u − u ′ | κ (cid:21) × V ( u ′ ) / g ( u ′ ) d u d u ′ , oft quantum waveguides g and V are bounded, it is enough to check that Z R | ˆ h η ( p ) | e − √ κ + p | u − u ′ | p κ + p d p − k h η k e − κ | u − u ′ | κ = O ( η )holds as η →
0. We have ˆ h η ( p ) = η ˆ h (cid:0) pη (cid:1) , hence, changing the integration variable andusing the mean value theorem, we can rewrite the first term on the left-hand side of thisexpression as 1 η Z R | ˆ h ( ζ ) | e − √ κ + η ζ | u − u ′ | p κ + η ζ d ζ == 1 η (cid:16) e − κ | u − u ′ | κ + O ( η ) (cid:17) , and using further the relation k h η k = η k h k we arrive at the result.Now we can return to the proof of Theorem 7.1. Consider the difference of theBirman-Schwinger operators D κ Γ ,V := R κ Γ ,V − R κ Γ ,V (7.3)which is by (5.4) and (6.1) an integral operator on L (Ω a ) with the kernel D κ Γ ,V ( s, u ; s ′ , u ′ ) = 12 π (cid:16) W ( s, u ) / K ( κ | x ( s, u ) − x ( s ′ , u ′ ) | ) W ( s ′ , u ′ ) / − V ( u ) / K ( κ | x ( s, u ) − x ( s ′ , u ′ ) | ) V ( u ′ ) / (cid:17) In view of (7.2), (7.3), and Lemmma 7.1 we would have sup σ ( R κ Γ ,V ) > η → ( ψ η , D κ Γ ,V ψ η ) > , and using the choice of the function h η , in combination with the dominant convergencetheorem, we find that this is equivalent to Z R d s d s ′ Z a − a Z a − a d u d u ′ g ( u ) D κ Γ ,V ( s, u ; s ′ , u ′ ) g ( u ′ ) > , however, in view of (6.4) the integrated function is nothing but C κ Γ ,V , hence the lastinequality yields the condition (7.1).In contrast to the results of Section 4, the sufficient condition (7.1) is of aquantitative nature. Indeed, given Γ and V , we are able to find κ and the respectiveintegral can be evaluated. We have already mentioned that the argument of theMacdonald function in the straight case is | x ( s, u ) − x ( s ′ , u ′ ) | = (cid:2) ( s − s ′ ) + ( u − u ′ ) (cid:3) / , oft quantum waveguides s ′ = s and (2.2) that the squareddistance is given by the formula | x ( s, u ) − x ( s ′ , u ′ ) | = | Γ( s ) − Γ( s ′ ) | + u + u ′ − uu ′ cos β ( s, s ′ ) + 2( u cos β ( s, s ′ ) − u ′ ) Z ss ′ sin β ( ξ, s ′ ) d ξ, where the first term on the right-hand side equals | Γ( s ) − Γ( s ′ ) | = Z ss ′ Z ss ′ cos β ( ξ, ξ ′ ) d ξ d ξ ′ .
8. Another existence result
While we know the explicit form of the integral in (7.1), its evaluation may becomplicated. Even without it, however, we can use Theorem 7.1 to make conclusionsabout the existence of a discrete spectrum. As an example, let us show a result whichmay be regarded as a generalization of Proposition 4.1:
Proposition 8.1.
Let V ǫ be the family of potentials V satisfying assumption (e) andsuch that inf σ ( h V ) ≤ ǫ . Then to any ǫ > there exists an a = a ( ǫ ) such that σ disc ( H Γ ,V ) = ∅ holds for all V ∈ V ǫ with supp V ⊂ [ − a , a ] .Proof. To begin with, it is sufficient to consider potentials V such that inf σ ( h V ) = ǫ .Indeed, the family of operators { h λV : λ > } is monotonous and by Proposition 3.1they have the same essential spectrum, hence if the claim is valid for h λV , the same istrue for all h λ ′ V with λ ′ > λ . With the properties of the functions involved in mind,the integration in (7.1) can be performed in any order; we rewrite thus the expressionin question as 12 π Z a − a Z a − a φ ( u ) F ( u, u ′ ) φ ( u ) d u d u ′ , (8.1)where F ( u, u ′ ) := Z R (cid:2) ((1 + uγ ( s )) K ( κ | x ( s, u ) − x ( s ′ , u ′ ) | ) (1 + u ′ γ ( s ′ )) − K ( κ | x ( s, u ) − x ( s ′ , u ′ ) | ) (cid:3) d s d s ′ . The function F ( · , · ) is well defined as long as assumption (d) is satisfied and continuous.Furthermore, we have F (0 ,
0) = Z R (cid:2) K ( κ | Γ( s ) − Γ( s ′ ) | ) − K ( κ | s − s ′ | ) (cid:3) d s d s ′ > . (8.2)To see that this is the case, recall that Γ is parametrized by the arc length so that | Γ( s ) − Γ( s ′ ) | ≤ | s − s ′ | , and because the curve is not straight by assumption, thereis an open set on which the inequality is sharp; this yields the inequality (8.2) since K ( · ) is decreasing in (0 , ∞ ). The continuity implies the existence of a neighborhood( − a , a ) × ( − a , a ) of (0 ,
0) on which F ( u, u ′ ) is positive, and that in view of (8.1) incombination with the positivity of φ concludes the proof. oft quantum waveguides
9. Concluding remarks
In contrast to the cases of hard-wall waveguides and leaky wires, where any bendinggives rise – under appropriate asymptotic straightness requirement – to a nonemptydiscrete spectrum, the present analysis does not yield such a universal result. Thequestion whether all bent soft waveguides, in particular the wide and shallow ones, dobind, remains open, and it is possible that the Birman-Schwinger technique is not anoptimal tool to address it. True, it produced the universal result for leaky wires [EI01]but it might have been a lucky coincidence. Recall also that there are other geometricperturbations for which the question about weakly coupled bound states is more subtle,see [EV97] and notes to Sec. 6.4. in [EK15].Apart from this question, the above discussion brings to mind numerousmodifications and extensions of the present problem for which the soft waveguideprogram might be an appropriate name:(i) Potential channels in which the generating curve is not smooth , as well as related polygonal potential channels . The simplest example is a flat-bottom channel in theform of a broken strip. One expects that it will have a discrete spectrum for anychannel depth provided the angle is sharp enough, while the answer in the generalcase is not clear. Note that for a cross-shaped potential channel, related to thequestion (iv) below, such a discrete spectrum exists independently of the ditchdepth as it was shown by a variational method in [EKP18] ‡ . Unfortunately, thisfact alone does not allow us to make conclusions about a single broken channel.(ii) Tubular potential channels in three dimensions . In view of [EK15, Thms 1.3.and 10.3] one can expect the validity of asymptotic results analogous to those ofSection 4. If the channel profile lacks the rotational symmetry with respect toits axis Γ, one can also expect additional effects coming from the channel torsion which could give rise to an effective repulsive interaction in analogy with the resultsin [EKK08] and [EK15, Sec. 1.7](iii) Local perturbations of potential channels, coming from variation either of theirdepth or the width. The question about the existence of a discrete spectrumis easy to answer if such a perturbation is sign-definite, the mentioned result of[EV97] suggests that in general it may be harder.(iv) Potential channels of a more complicated geometry, in first place branched ones where the ‘axis’ Γ is a metric graph. If the ‘ends’ of such a channel system havethe same profile being (asymptotically) straight, the essential spectrum thresholdis determined by that of a single semi-infinite channel; the existence question canbe sometimes answered using the minimax principle, provided there is a part ofthe potential in the form of a suitable bent or polygonal channel. Of course, tohave the problem well defined one must specify the potential in the vicinity of the ‡ The aim of that paper was to investigate two interacting particles on a halfline. Let us add thatit is not for the first time when a link was made between a two-body system and a two-dimensionalwaveguide, cf. the caricature model of meson confinement in [LLM86]. oft quantum waveguides − V at the pointsthe distance of which from Γ is less than a and zero otherwise.(v) In addition to the discrete spectrum existence, one is interested in the number ofeigenvalues and their other properties in dependence on the system geometry. Ofparticular interest are the weakly bound states corresponding to mild geometricperturbations. One such question is whether the gap between the ground stateand the continuum in slightly bent potential channels would be proportional tothe fourth power of the bending angle as it is the case for hard-wall tubes [EK15,Thm. 6.3] and leaky wires [EKo15].(vi) Various spectral optimization problems come to mind. For instance, if the potentialchannel is finite and Γ is a loop of a fixed length, one can ask about the shape whichmakes the principal eigenvalue maximal; in analogy with [EK15, Prop. 3.2.1 andThm. 10.6] we conjecture that a sharp maximum is reached by a circular shape.(vii) The discrete spectrum is not the only interesting feature of such system, anotherone concerns the scattering in a bent or locally perturbed potential channel; thefirst task here is to establish the existence and completeness of wave operators asit is done for hard-wall tubes and leaky wires, cf. [EK15, Sec. 2.1, Thm. 10.4] and[Di20, EKo19]. Furthermore, if the profile of the channel is narrow and sufficientlydeep so that h V has more than one bound state, one can expect the presence of resonances near the corresponding higher thresholds in analogy with [Ne97] and[EK15, Thm. 2.6].(viii) Another extension to three dimensions concerns potential layers , that is potentialsof a fixed transverse profile built over an infinite surface Σ in R , different from aplane but asymptotically flat in a suitable sense. Here we know that the discretespectrum of the hard-wall layer is nonempty if the total Gauss curvature of Σ isnon-positive [EK15, Thm. 4.1], in the opposite case we have partial results only[EK15, Sec. 4.2]. In the spirit of Proposition 4.2 we can establish the discretespectrum existence for potential layers with the profile deep enough, while in theregime different from the asymptotic one, the question is open.(ix) A substantial difference between the tubes and layers is that in the lattercase the spectrum depends on the global geometry of the interaction support.A nice illustration is provided by conical hard-wall layers [ET10] and leakysurfaces [BEL14] which both have an infinite discrete spectrum accumulatinglogarithmically at the continuum threshold, even in the situations when thegenerating surfaces lacks a rotational symmetry [OP18]. We noted in theintroduction a recent result [EKP20] showing that the spectrum of a cylindricalpotential layer § has the same behavior. One can conjecture that also potentiallayers with a different geometry might behave as their hard-wall counterparts, § The channel profile in this paper is supposed to have mirror symmetry, while the other assumptionsare less restrictive than here, in particular, the support of V may not be compact and the potentialneed not be sign-definite. oft quantum waveguides periodic waveguides one is interested primarily in the absolute continuity ofthe spectrum and the existence of spectral gaps. As for the latter, having againProposition 4.2 in mind, one can conjecture that the gaps would exist for profilesdeep an narrow enough. The (global) absolute continuity is in the hard-wall caseproved in the two-dimensional situation [SW02], however, even for periodic leakywires it remains an open problem.(xii) All the above listed problems concerned the one-particle situation. A completelynew area opens when we consider a system of many particles interacting mutually,for instance, due the charges they carry, confined in a soft waveguide. In the hard-wall case, for instance, a condition is known [EK15, Thm. 3.7] under which morethan one particle cannot be bound. In the soft case one can expect that the samewould happen for a large enough particle charge, but the specific form of such acondition remains to be found.This list is by no means complete but we prefer to stop here with the hope that weconvinced the reader that this area offers a large number of interesting questions. Acknowledgements
The author is grateful to the referees for useful comments. The research was supportedin part by the European Union within the project CZ.02.1.01/0.0/0.0/16 019/0000778.
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