MULTIPLE HOMOCLINIC SOLUTIONS FOR A ONE-DIMENSIONAL SCHR¨ODINGER EQUATION

. In this paper we study the problem of the existence of homoclinic solutions to a Schr¨odinger equation of the form where is a stepwise potential. The technique of proof is based on a topological method, relying on the properties of the transformation of continuous planar paths (the S.A.P. method), together with the application of the classical Conley-Wa˙zewski’s method.


1.
Introduction. Let us consider the equation where V : R → R is a L ∞ -function; we assume that there exist 0 < V 1 < V 2 and t 0 < t 1 < . . . < t 2K < t 2K+1 such that V (t) = V 1 , ∀ t ∈ [t 2j , t 2j+1 ], j = 0, . . . , K, V (t) = V 2 , ∀ t ∈ [t 2j+1 , t 2j+2 ], j = 0, . . . , K − 1. ( The equation 1 is the very classical one-dimensional Schrödinger equation, which comes from the study of stationary waves of a nonlinear Schrödinger equation in R N ; there is a huge literature on this equation and several different results have been proved (see for instance, among the others, [1], [5], [8], [20]). Very recently, some results on the lines of our approach have been proved in [24]; indeed, in the quoted paper the authors study the existence of chaotic dynamics for the equation 1, using the S.A.P. method. We will be concerned with the existence of solutions of 1 which are homoclinic to the equilibrium point (0, 0) (in the phase-plane) and have some prescribed behavior -in terms of the number of zeros of the derivative -in the intervals [t i , t i+1 ], i = 0, . . . , 2K. Many papers can be found in the literature on this topic; we only mention [10,15,17] and the very recent [2,6,9]. In particular, the results in [2,6] are exactly in the same spirit of the one of the present paper, but they are concerned with a different form of equations. For related results on this direction see also [3].
We will able to prove the following result: Theorem 1.1. There exists S * > 0 and for any integer M ≥ 1, there exists T * = T * (M ) > 0, such that if there exist M 2K−1 geometrically distinct globally defined positive solutions u of 1 such that lim t→±∞ (u(t), u (t)) = (0, 0).
We point out that the same result can be proved also for a more general equation of the form where f (x) = xh(x), for every x ∈ R, and h ∈ C 1 (R; R) is such that h(0) = 0, h (x) > 0 for every x > 0 and lim x→+∞ h(x) = +∞ (cfr. [24]). The proof of Theorem 1.1 is based on the combination of the Conley-Ważewski's method (see [4,23]) and of the so-called S.A.P. method (see [16,17,19]); the technique is a slight variant of the ones already used in [2,6]. More precisely, we first show the existence of the stable and unstable manifolds of the equilibrium point (0, 0); then we are able to connect these two sets by suitable orbits of 4 (coming from the application of the S.A.P. method), solving a kind of generalized Sturm-Liouville problem for 1.1. The crucial point in the present situation is that we need an approximation procedure in order to replace the stable and unstable manifolds (which are in general merely continua) by images of continuous paths. We observe that the problem of connecting continuous paths can be found already in the paper [22]; more recently, it has been considered in [7,12,13,15,21].
The plan of the paper is the following. In Section 2, we prove the existence of stable and unstable manifolds to the equilibrium (0, 0); in Section 3 we study the dynamics of the equation on bounded intervals, while in Section 4 we prove our result.
Throughout the paper by a path or curve γ we mean a continuous map γ : [0, 1] → R 2 . We will make a systematic abuse of notation by using the same symbol γ to stand for the curve and for its image γ([0, 1]).
2. Continua of asymptotic solutions. In this section we prove the existence of stable and unstable manifolds to the equilibria of the nonautonomous planar system with V : R → R a locally integrable function. We use the notation z(t; p, t 0 ) for the (unique) solution to 4 satisfying the condition z(t 0 ) = p. Moreover, we define let us observe that E µ is the energy associated to the autonomous system which are drown in figure 1. We are able to prove the following result: Theorem 2.1. The following statements hold true.
Proof. We observe that Statement 2 is a consequence of Statement 1 and the symmetry enjoyed by the vector field in 4 with respect to the axis y = 0. Concerning the proof of Statement 1, let us fix > 0 such that We first prove that any solution z(t) = (x(t), y(t)) which remains in T for every t ≤ t 0 has to satisfy lim Indeed, for such a solution we have that, for t ≤ t 0 , and, hence, there exists If L > 0 then there exists t 1 ≤ t 0 such that As a consequence, since (x(t), x (t)) ∈ T for every t ≤ t 1 , this implies that This contradicts the fact that L is finite. Finally, since z(t) ∈ T for every t ≤ t 0 , the fact that x(t) → 0 for t → −∞ implies that x (t) → 0 for t → −∞. Now we write system 4 as an autonomous system in R 3 : and let π(·; s 0 , P ) be the unique solution to 6 starting from P ∈ R 3 at s = s 0 . We set where γ − and γ + are the portions of the boundary of T lying on E V1− = 0 and E V −2+ = 0, respectively. Let us study the behavior of the vector field associated with system 6 at any point P = (x 1 , y 1 , t 1 ) ∈ ∂W . First, if (x 1 , y 1 ) = (0, 0), it is clear that π(s; t 1 , P ) = (0, 0, s) for all s. Next, if P ∈ U ∪ V , the vector field points strictly inwards W ; therefore, π(s; t 1 , P ) / ∈ W for all s in a left neighborhood of t 1 . Finally, if either and t 1 = 0 then π(s; t 1 , P ) ∈ W for all s in a left neighborhood of t 1 , since in those points the vector field of 6 points strictly outwards W .
Following the terminology introduced in [4], a point P ∈ ∂W is called an entry point in W for the flow π if π(s; 0, P ) ∈ W for all s ∈ [− , 0) and some > 0; moreover it is called a strict entry point if in addition π(s; 0, ) ∈ • W for all s ∈ (0, ] and some > 0. The analysis carried above shows that all the entry points in W are strict and that their set is U ∪ V . Now, we consider the set D ⊂ W given by and the map Φ : D → ∂W such that Φ(P ) is the last entry point in W before P for the solution of 6 starting from the point P , namely Φ(P ) = π(s * ; t 1 , P ) where s * = sup{s < t 1 : π(s; t 1 , P ) ∈ D}.
Using the fact that all entry points in W are strict, it follows from the results in [4] that Φ is continuous on D; moreover, by the previous discussion, Φ(D) = U ∪ V is not connected and U and V are its connected components.
3. Dynamics on bounded intervals. In this section, we fix two real positive constants V 1 < V 2 and we study the dynamics associated to the autonomous system which will be referred to as (V i ). For T ∈ R and i = 1, 2, we define the map Ψ T i as the Poincaré map associated with system (V i ) from t = 0 to t = T , i.e.
where (x(·; x 0 , y 0 ), y(·; x 0 , y 0 )) is the unique solution to (V i ) satisfying the initial condition (x(0), y(0)) = (x 0 , y 0 ). Observe that (Ψ T i ) −1 = Ψ −T i . It is well known that the orbits of (V i ) with negative energy E Vi are periodic and turn around the equilibrium ( √ V i , 0) in clockwise sense. Hence it is useful to introduce a pair of angular coordinates θ i , for i = 1, 2, which are centered at ( and are measured in clockwise sense starting from the half line √ V i , +∞ × {0}. In particular we denote by θ(t; (x 0 , y 0 )) the angular coordinate of the solution of (V i ) starting at time t = 0 from the point ( Let us now consider the points ( √ 2V 1 , ± 2V 1 (V 2 − V 1 )) which are two vertices of the triangular regions T ± lying where the homoclinic orbit of (V 2 ) crosses the vertical line x = √ 2V 1 . They also lie on the same orbit of (V 1 ), namely the one with energy E V1 (x, y) = V 1 (V 2 −V 1 ), which crosses the positive x-axis at the abscissâ . We observe that each solution of (V 1 ) that starts from a point in T − crosses the positive x-axis at a point lying between ( √ 2V 1 , 0) and (x, 0), and the same holds for the solutions of (V 1 ) arriving at a point in T + .
We set with c 1 , c 2 fixed in the following way: We observe that condition 1. guarantees that A 2 is filled up by periodic orbits of system (V 2 ), that surround the segment [ √ 2V 1 ,x] × {0} by condition 2. We denote by ∂ e A 2 and ∂ i A 2 the outer (E V2 = c 2 ) and inner (E V2 = c 1 ) boundary of A 2 , respectively. In particular, ∂ i A 2 and ∂ e A 2 cross the segment (0, √ V 2 ) × {0} on the x-axis at the point (x i , 0) and (x e , 0), respectively, which both lie inside the region bounded by the homoclinic orbit of (V 1 ) by condition 2, i.e. they satisfy is non-empty and it has exactly two connected components. We denote by R 1 the connected component contained in {(x, y) ∈ R 2 : y > 0} and by R 2 the other one (see Figure 2). Moreover, we name explicitly some of the "sides" of R 1 and R 2 : where the superscript l, r stands for "left" and "right", respectively. Setting R − 1 = R l 1 ∪ R r 1 and R − 2 = R l 2 ∪ R r 2 , we obtain that each of the pairs is an oriented rectangle, according to [16,18].
Oriented rectangles are crucial in the following definition (see again [16,18]).
We say that Ψ stretches R 1 to R 2 along the paths with crossing number M ≥ 1 and write Ψ : As an easy consequence of the definition, we have that the stretching property has a good behavior with respect to compositions of maps.
The proof is essentially contained in [24,Sect. 3]. We only highlight some key points of the argument for Ψ T2 2 : R 1 −→ M R 2 which will be useful in the sequel. We observe that Ψ T2 2 moves the points of R 1 along the periodic orbits of system (V 2 ) which fill up the annulus A 2 and that the period τ i of the orbit on ∂ i A 2 is smaller than the period τ e of the orbit on ∂ e A 2 . Hence, if we choose T 2 such that any path γ : 2 into a path winding at least M + 1 times around ( √ V 2 , 0) in clockwise sense. In particular Ψ T2 2 (γ) crosses R 2 at least M times across the components of the boundary in R − 2 . As a consequence it is possible to select M sub-paths γ 1 , . . . , γ M of γ such that Ψ T2 2 (γ i ) is contained in R 2 and crosses both components of R − 2 for i = 1, . . . , M . More precisely, we observe that and, therefore, as s ranges in [0, 1], the angular function s → θ 2 (T 2 ; γ(s)) spans all the M pairwise disjoint intervals [2π(n * +n−1), 2π(n * +n)−π], where n * = T 2 /τ e and n = 1, . . . , M . Indeed, the choice 9 of T 2 implies that and, thus: In particular, if we define it is possible to find points s n 1 , s n 2 ∈ [0, 1], with n = 1, . . . , M and 0 ≤ s 1 This shows that the following stretching properties hold: (H 1 n , Ψ T2 2 ) : R 1 −→ R 2 for all n = 1, . . . , M. We remark that each solution of (V 2 ) starting at time t = 0 from any point z ∈ H 1 n ends in R 2 at t = T 2 after exactly 2(n * + n) − 1 crossings with the x-axis.
In a very similar way, one can show also that, if we denote by σ i the period of the orbit ∂ i A i of (V 1 ) and choose T 1 ≥ M σ i , then also the following stretching properties are true: and with analogous considerations about the nodal behavior of the solution of (V 1 ) starting from points of H 2 n . Actually in this case the argument is slightly simpler thanks to the fact that the external boundary ∂ e A 1 of A 1 is the homoclinic orbit of (V 1 ) and not a periodic one.
In conclusion, the lemma is proved with the choice

4.
Proof of the result. We fix M ≥ 1 and {(n j )} j=1,...,2K−1 such that n j ∈ {1, . . . , M } for all j = 1, . . . , 2K − 1. We then consider T * = T * (M ) as given in Proposition 2 and a weight function V as in 2-3, with S * given below. We first apply Proposition 2.1 on the intervals (−∞, t 0 ] and [t 2K , +∞) in order to find the continua Γ ±∞ . We then select P ± = ( √ 2V 1 , y ± ) ∈ Γ ±∞ and let where, for every µ > 0, F µ is the primitive of such that F µ (0) = 0, and x ± > 0 are such that as before. In other words, S * is the maximum between the times needed to run from P − to (x − , 0) and from (x + , 0) to P + along the orbits of (V 1 ). 1 In order to prove Theorem 1.1 we will show that the following Sturm-Liouvilletype boundary value problem: has at least a solution whose oscillatory behavior in each interval (t j , t j+1 ) is described by the fixed number n j ∈ {1, . . . , M }, for each j = 1, . . . , 2K − 1, in a way that will be clearer along the proof and that, however, is closely related to the definition of the sets H i n in Proposition 2. In fact, any solution of 11 is necessarily 1 The constant S * thus defined is not very explicit since it depends on the knowledge of points of the continua Γ ±∞ . A more explicit, albeit rougher, constant S * could be obtained in the following way. Each solution of (V 1 ), starting from a point ( √ 2V 1 , y 0 ) on the "right side" , hits the positive x-axis at the abscissax(y 0 ) ∈ ( √ 2V 1 , √ 2V 2 ) which can be explicitly evaluated together with the time needed to reach the point (x(y 0 ), 0). By letting we obtain a larger bound that doesn't depend on the explicit knowledge of Γ ±∞ . a global solution of 1 and satisfies (x(±∞), x (±∞)) = (0, 0) by the very definition of Γ ±∞ . Since our shooting argument involves mainly paths of initial points and the sets Γ ±∞ are just continua, we first replace the two continua in the boundary conditions with (the image of) two suitable paths that lie in an -neighborhood of Γ ±∞ , as in the following claim. For each > 0 we look for solutions of the approximating boundary value problems with suitable oscillatory behavior. At the end of the argument we will let → 0. Our shooting argument starts by following the forward evolution of the points on γ − from t = t 0 to t = t 1 and the backward evolution of the points on γ + from t = t 2K+1 to t = t 2K , along the flow generated by the differential equations in 12.
Claim 2. There exist 0 < s ± < s ± < 1 such that Proof. As before, we give the details only for the time interval [t 0 , t 1 ]. Let us consider z(·; γ − (1), t 0 ) = (x(·; γ − (1), t 0 ), y(·; γ − (1), t 0 )) and let s 0 = inf{s ∈ (0, 1] | ∃ t ∈ (t 0 , t 1 ] : y(t; γ − (t 0 ), t 0 ) = 0}, which is well defined by the construction of γ − and the fact that t 1 − t 0 ≥ S * . Indeed, the condition t 1 − t 0 ≥ S * implies that z(·; γ − (1), t 0 ) = z(·; P − , t 0 ) crosses the positive x-axis at least once in (t 0 , t 1 ] (see 10). Moreover, the crossing point lies in the interior of the inner boundary ∂ i A 2 of the annulus A 2 by the very construction of A 2 (see 8). By the way, we observe that 0 < s 0 ≤ 1. Hence, Ψ t1−t0 ] is a continuous path in T − which connects (0, 0) with the point Ψ t1−t0 1 (γ − (s 0 )) on the x-axis. Therefore, its support crosses A 2 in the upper half plane; this allows to define (γ − (s)) ∈ ∂ i A 2 } and conclude the proof of the claim. Now we follow the forward evolution of the solutions starting at t = t 0 from γ − up to the time t = t 2K . The main idea here is to select a sub-path of γ − that gives rise to solutions with the desired oscillatory properties in the intermediate intervals and that is transformed at time t = t 2K into a path whose position inside the annulus A 2 is in some sense good.
By Claim 2, the path s → z(t 1 ; γ − (s), t 0 ) crosses the annulus A 2 in the upper half plane as s ranges in [s − , s − ]. Since t 2 − t 1 ≥ T * (M ) it is possible to argue as in the proof of Proposition 2 and show that the map Ψ t2−t1 2 transforms the curve s → z(t 1 ; γ − (s), t 0 ) into a path that winds in A 2 and crosses R 2 at least M times as s ranges in [s − , s − ]. In fact, there exists a non-trivial subinterval [s 2 , s 2 ] ⊂ [s − , s − ] such that z(t 2 ; γ − (s), t 0 ) ∈ R 2 for all s ∈ [s 2 , s 2 ], z(t 2 ; γ − (s 2 ), t 0 ) ∈ R l 2 and z(t 2 ; γ − (s 2 ), t 0 ) ∈ R r 2 . More precisely the interval [s 2 , s 2 ] can be chosen among M different ones, according to the number of oscillations that the solution performs in the interval [t 1 , t 2 ]. Based on the choice of n 1 ∈ {1, . . . , M } that we made at the beginning of the proof, the interval [s 2 , s 2 ] can be characterized by the following property: where n 1 * = (t 2 − t 1 )/τ e . We omit the details since they are analogous to those employed in the proof of Proposition 2. We only remark that all the solutions z(·; γ − (s), t 0 ), for all s ∈ [s 2 , s 2 ] cross the positive x-axis exactly 2(n 1 * + n 1 ) − 1 times in the interval (t 1 , t 2 ). Now, about the dynamics in the intermediate interval [t 2 , t 2K−1 ], we observe that the map p → z(t 2K ; p, t 2 ) is the alternate composition, for suitable values of j, of the maps Ψ t2j+2−t2j+1 2 and Ψ t2j+1−t2j 1 for which the stretching properties of Proposition 2 with crossing number M hold thanks to assumption 3. However, the compact sets H i n with respect to which the S.A.P. property holds are different for each of those maps since they depend explicitly on the length of the time intervals t j+1 − t j . To make things slightly more complicated, in the case i = 1 (i.e. j odd) one has also to be careful with the minimum number of turns n * which also depends on the lengths t j+1 −t j . In order to take into account all these facts, we modify our notation in the following way: the superscript j in H j and H j n is allowed to range in 2, . . . , 2K − 2 and, from now on, it refers to the stretching property in the interval [t j , t j+1 ] for the appropriate map, depending on the value that V has in that interval according to 2. More precisely, by Proposition 2 we then have