THE EVANS FUNCTION AND THE WEYL-TITCHMARSH FUNCTIONfaculty.missouri.edu/~latushkiny/preprints/EWTf.pdf · connections of the Weyl-Titchmarsh functions and spectral measures. The paper

THE EVANS FUNCTION ANDTHE WEYL-TITCHMARSH FUNCTION

YURI LATUSHKIN AND ALIM SUKHTAYEV

Abstract. We describe relations between the Evans function, a modern tool

in the study of stability of traveling waves and other patterns for PDEs, andthe classical Weyl-Titchmarsh function for singular Sturm-Liouville differential

expressions and for matrix Hamiltonian systems. Also, we discuss a related

issue of approximating eigenvalue problems on the whole line by that on finitesegments.

1. Introduction

The primary goal of this semi-expository paper is to establish connections be-tween the classical Weyl-Titchmarsh function for Hamiltonian ordinary differentialequations, and the Evans function, a Wronskian-type determinant designed to de-tect point spectrum of ordinary differential operators arising in the study of stabilityof traveling waves and other patterns for partial differential equations. As a byprod-uct, we give an elementary explanation of the relations of the point spectrum andthe Evans function for problems on the full line to that on finite segments.

In addition to the classical sources [9, 10] of the Weyl-Titchmarsh theory, we referto recent work [7, 19, 20, 21, 27] containing illuminating surveys, and, especially, to[14]. There is of course big literature on approximating the spectra of differentialoperators on the line by the spectra of respective operators on finite segments, see,for instance, [4, 5, 22, 31, 33, 35, 39] and the literature cited therein. Finally, werefer to [3, 12, 13, 15, 25, 26, 30, 32, 36, 37] for the general discussion of the Evansfunction some relevant aspects of which will be briefly reviewed next.

Given a partial differential equation in one space dimension, say, a reaction dif-fusion equation on (−∞,+∞), one is interested in studying stability of such specialsolutions as traveling waves. Passing to the moving coordinates and linearizing thepartial differential equation about the wave, one considers an eigenvalue problem,Hu = zu, for the linearized operator H, an ordinary differential operator of someorder. The instability of the wave is related to the presence of unstable eigenvalues,and thus the problem of locating the isolated eigenvalues is of importance. Rewrit-ing the high order differential equation Hu = zu as a first order matrix differentialequation, one is looking for the values of the spectral parameter z for which thelatter equation has solutions exponentially decaying at both plus and minus infin-ity. These values are the zeros of the Evans function, defined as the determinantof the matrix whose columns are the solutions of the matrix differential equations

Date: March 8, 2011.Key words and phrases. Schrodinger equation, Hamiltonian systems, Jost function, traveling

waves, eigenvalues.Partially supported by the US National Science Foundation Grant DMS-0754705 and by the

Research Council and the Research Board of the University of Missouri.1

2 Y. LATUSHKIN AND A. SUKHTAYEV

that are exponentially decaying at +∞ and at −∞. For instance, in the particularcase when H is the Schrodinger operator with a summable real valued potential onthe line, the Evans function is known to be equal, see [15, 26], to the Jost function,that is, to the rescaled Wronskian of the exponentially decaying at +∞ and −∞Jost solutions of the Schrodinger equation, see, e.g., [6, Chapter XVII].

In summary, the main idea of constructing the Evans function is to match so-lutions with appropriate asymptotics at +∞ and at −∞. This idea works formany important situations, most notably, for quite general non-selfadjoint prob-lems; however, the matching solutions are assumed to be exponentially decaying atthe respective singular ends which are always assumed to be +∞ and −∞. Al-though the Weyl-Titchmarsh function is constructed only for selfadjoint problems,it is designed to produce solutions that are merely square summable at the re-spective singular ends (which, in addition, are not necessarily infinite). One of themany nice features of the Weyl-Titchmarsh function is that it is constructed byapproximation, that is, using respective objects defined on regular (in particular,finite) segments. This is very much in concert with the approach of approximat-ing the Evans function by respective objects on finite segments frequently used bynumerical analysts working in stability analysis of traveling waves.

A typical result of the current paper, see, e.g., Corollary 2.18, is that the Evansfunction (which is matching solutions at two singular ends) is, essentially, equalto the difference of the two Weyl-Titchmarsh functions (corresponding to each ofthe singular ends). We do not really know of any previous literature relating theEvans and the Weyl-Titchmarsh functions. The relation between the Jost and theWeyl-Titchmarsh functions, however, should look familiar to experts in the Weyl-Titchmarsh theory although we were not able to pinpoint in the literature the exactfacts that we want. Our methods are totally elementary, e.g., we even do not useconnections of the Weyl-Titchmarsh functions and spectral measures.

The paper is organized as follows. In Section 2 we study general Sturm-Liouvilledifferential operators on (a, b) ⊂ R where each end can be singular (finite or infinite,either in the limit point or in the limit circle case). After a brief review of theWeyl-Titchmarsh theory, we construct the solutions of the Sturm-Liouville equationwhose Wronskian is equal to zero exactly at the points of the discrete spectrum ofthe differential operator. This could be viewed as constructing an analogue of theEvans function in our quite general situation. In Theorem 2.12 we offer formulasrelating this Wronskian and the Weyl-Titchmarsh function for the values of thespectral parameter z outside of the essential spectrum. (Formulas of this type areof course well known for non-real values of z; here, we just carefully recorded whathappens on the real line, in particular, at the points of the discrete spectrum).We then use the Weyl-Titchmarsh functions to describe the discrete spectrum,see Corollary 2.14. At the end of Section 2 we specialize to the situation of theSchrodinger operator on the line with a summable potential when the situationbecomes more transparent.

In Section 3, still working on the model case of the Schrodinger equation, wediscuss how to approximate the Jost (or the Evans) function, J , on the line byrestricting the problem to finite intervals [−L,L] and imposing some boundaryconditions at ±L. Of course, this topic belongs to a huge and well-studied area,but our contribution here is a formula, see Theorem 3.3, relating the asymptoticbehavior for large L of the Jost function JL on the finite segments to that of the

THE EVANS FUNCTION AND THE WEYL-TITCHMARSH FUNCTION 3

product of J and some quantities induced by the boundary conditions. We alsooffer an operator-theoretic interpretation of these quantities, which appears to behelpful in explaining some numerically important results in [4, 33].

In Section 4 we study the first order matrix self-adjoint Hamiltonian differentialsystems on (a, b) ⊂ R for which the matrix valued Weyl-Titchmarsh theory isavailable, see [7, 8, 23, 27, 28, 34]. Considering either the limit point or the limitcircle case at either end, we describe square summable at each end solutions andmatch them by means of a matrix valued Wronskian. Our main result, Theorem4.8, gives a formula relating the Evans function for the first order system and theWronskians of the matrix valued Weyl-Titchmarsh functions. We conclude with anumber of examples for which our results are applicable.

Notations. We denote by[α β

]a row-vector or a rectangular block-matrix,

so that[α β

]> is a column vector, where > is transposition. We denote byei = [δij ]nj=1, i = 1, . . . , n, the standard unit column (n × 1) vectors in Cn; thesame symbols ei, i = 1, . . . , 2n, are used to denote the standard unit vectors inC2n. For an operator T , we denote by σ(T ) the spectrum, by σd(T ) the discretespectrum (the set of isolated eigenvalues of finite algebraic multiplicity), and byσess(T ) = σ(T ) \ σd(T ) the essential spectrum. Given a measurable and almost-everywhere positive function p, for f, g and pf ′, pg′ absolutely continuous, we de-note by Wt(f, g) = f(t)pg′(t) − pf ′(t)g(t) the value of the Wronskian W (f, g) ofthe functions f, g at the point t (as a rule, we prefer to write pf ′(t) instead ofp(t)f ′(t)). Also, we will use notations Wa(f, g) = limc→aWc(f, g) with a < c < 0and Wb(f, g) = limd→bWd(f, g) with 0 < d < b. Given a measurable and almost-everywhere positive function r, we denote by L2(a, b) the Hilbert space with thescalar product 〈f, g〉L2(a,b) =

∫ baf(t) · g(t)r(t) dt; here and below bar stands for

complex conjugation.Acknowledgments. The authors are indebted to Fritz Gesztesy for many valu-

able discussions. The first author thanks Lai-Sang Young for the opportunity tovisit the Courant Institute of Mathematical Sciences in Fall Semester 2010, whena part of this paper was written.

2. Singular Sturm-Liouville differential expressions

Let us consider the general Sturm-Liouville differential expression τ ,

τy(t) =1r(t)

(− (py′)′(t) + V (t)

), t ∈ (a, b), −∞ ≤ a < b ≤ ∞, (2.1)

and impose the following assumptions.

Hypothesis 2.1. Assume that p, r, V are real-valued measurable functions on (a, b)such that p(t), r(t) > 0 almost everywhere in (a, b), and 1/p, r, V belong to L1

loc(a, b),that is, belong to L1(c, d) for any a < c < d < b.

First, we briefly review some basics of the Weyl-Titschmarsh theory [9, 10].Without loss of generality we will assume that 0 ∈ (a, b). The point b is calledregular if b is finite and assumptions in Hypothesis 2.1 hold on (a, b], and singularotherwise; analogously for a. Following [9, Chapter IX], we fix an ω ∈ [0, π), andlet θ(· , z), φ(· , z) denote the solutions (a, b) of the differential equation τy = zy,


z ∈ C, satisfyingθ(0, z) = sinω, φ(0, z) = cosω,

pθ′(0, z) = − cosω, pφ′(0, z) = sinω.(2.2)

Then clearly θ(· , z), φ(· , z) are linearly independent solutions, and θ, θ′, φ, φ′ areentire functions of z and continuous in t, z. Moreover, since W0(θ, φ) = 1 one hasWt(θ, φ) = 1 for all t. These solutions are real for real z.

Every solution χ of τy = zy except φ is, up to a constant multiple, of the form

χ(·, z) = θ(·, z) +mφ(·, z), z ∈ C, (2.3)

for some m = m(z) which will depend on z ∈ C. Consider now a real boundarycondition at some point d, 0 < d < b,

cos η y(d) + sin η py′(d) = 0, 0 ≤ η < π. (2.4)

We want to find m so that the solution χ in (2.3) would satisfy (2.4). Clearly, mmust satisfy:

m = − cot η θ(d, z) + pθ′(d, z)cot η φ(d, z) + pφ′(d, z)

. (2.5)

As z, d, η vary, m becomes a function of these arguments, m = m(z, d, η), andsince θ, θ′, φ, φ′ are entire in z it follows that m(·, d, η) is meromorphic in z andreal for real z. For fixed (z, d), when η changes from 0 to π, the complex numbersm(z, d, η) form a circle, denoted by Cd, whose equation in the m-plane, by a directverification, is given by the formula

Wd(χ, χ) = 0, where χ(·, z) = θ(·, z) +mφ(·, z), m ∈ C. (2.6)

Theorem 2.2. ([9, Theorem 9.2.2]) If z ∈ C \ R and θ, φ are the linearly inde-pendent solutions of τy = zy satisfying (2.2), then the solution χ(·, z) = θ(·, z) +m(z)φ(·, z) satisfies the real boundary condition (2.4) if and only if m lies on thecircle Cd in the complex m-plane whose equation is given by (2.6)

As d → b, either Cd → Cb, a limit circle, or Cd → mb, a limit point. If z ∈ Cthen all solutions of the differential equation τy = zy belong to L2(0, b) in the limitcircle case (lcc), and if z ∈ C\R then exactly one solution (up to a constant multiple)belongs to L2(0, b) in the limit point case (lpc). Moreover, in the limit-circle case,a point m is on the circle Cb if and only if Wb(χ, χ) = 0 with χ = θ +mφ.

Analogous assertions hold for a and c, a < c < 0, as c→ a.

In what follows, we denote by ma (respectively, mb) a point on the limit circleCa (respectively, Cb) if τ is in lcc at a (respectively, at b), or the limit point ma (re-spectively, mb) if τ is in lpc at a (respectively, at b). The functions ma, mb,ma,mb

are called the Weyl-Titchmarsh functions.

Theorem 2.3. ([9, Theorem 9.2.3]) The Weyl-Titchmarsh functions are analyticfunctions of z for Im(z) > 0 and Im(z) < 0. Moreover, Im(mb) > 0 and Im(ma) <0 for Im(z) > 0, and if mb or ma has zeros or poles on the real axis then they areall simple.

Next, we define differential operators associated with τ , see [29, Chapter 6], abrief review in [39], and a systematic exposition in [38]. The maximal operatorHmax in L2(a, b) associated with τ is defined by

Hmaxf = τf,

f ∈ domHmax ={g ∈ L2(a, b)

∣∣ g, pg′ ∈ ACloc(a, b); τg ∈ L2(a, b)}.

(2.7)


The minimal operator Hmin in L2(a, b) associated with τ is defined as the closureof the operator H ′min given as follows:

H ′minf = τf,

f ∈ domH ′min ={g ∈ domHmax

∣∣ g has compact support in (a, b)}.

(2.8)

Theorem 2.4. ([29, Theorem 6.1], [39, Theorem 3.9]) Assume Hypothesis 2.1.Then the adjoint of Hmin defined in L2(a, b) is the maximal operator Hmax:

H∗minf = Hmaxf = τf,

f ∈ domH∗min = {g ∈ L2(a, b)|g, pg′ ∈ ACloc(a, b); τg ∈ L2(a, b)}.(2.9)

We will impose some boundary conditions in order to define self-adjoint exten-sions H of the operator Hmin in L2(a, b), see [29, Section 6.3], [39, Section 4.5], and[38]. For this purpose, we define smooth functions ρa and ρb such that

ρa ≡ 1 near a, ρa ≡ 0 near b, (2.10)

ρb ≡ 1 near b, ρb ≡ 0 near a. (2.11)

Next, we fix a z ∈ C\R, then fix any ma(z) ∈ Ca and mb(z) ∈ Cb (in particular,ma(z) = ma(z) if τ is in lpc at a and mb(z) = mb(z) if τ is in lpc at b), and defineua(·, z) ∈ L2(a, b) and ub(·, z) ∈ L2(a, b) by

ua(t, z) = ρa(t)(θ(t, z) + ma(z)φ(t, z)),

ub(t, z) = ρb(t)(θ(t, z) + mb(z)φ(t, z)).(2.12)

Let us define in L2(a, b) the differential operator H1 byH1f = τf,

f ∈ domH1 = domH ′min + {ua(·, z)}+ {ub(·, z)},(2.13)

where ua(·, z) and ub(·, z) are given by (2.12). In other words, the domain of H1

consists of compactly supported in (a, b) functions from domHmax together withua(·, z), ub(·, z), and all linear combinations of finitely many such functions.

Theorem 2.5. ([29, Theorem. 6.5.], [38, Theorem 5.8]) Assume Hypothesis 2.1.Let H1 be as in (2.13). Then H1 is an essential self-adjoint extension of the minimaloperator Hmin given in (2.8), (2.9). Let H be the closure of H1. Then

Hf = τf,

f ∈ domH = {g ∈ L2(a, b)|g, pg′ ∈ ACloc(a, b), τg ∈ L2(a, b),

Wa(g, ua(z)) = 0,Wb(g, ub(z)) = 0}.(2.14)

We will also need the operators Ha in L2(a, 0) and Hb in L2(0, b) defined by

Haf = τf, (2.15a)

f ∈ domHa = {g ∈ L2(a, 0)|g, pg′ ∈ ACloc(a, 0], τg ∈ L2(a, 0), (2.15b)

Wa(g, ua(z)) = 0,W0(g, φ) = sinωg(0)− cosωpg′(0) = 0}, (2.15c)

Hbf = τf, (2.16a)

f ∈ domHb = {g ∈ L2(0, b)|g, pg′ ∈ ACloc[0, b), τg ∈ L2(0, b), (2.16b)

W0(g, φ) = sinωg(0)− cosωpg′(0) = 0,Wb(g, ub(z)) = 0}. (2.16c)


As shown in [29, page 25] or [27, Section VII.7], the boundary conditions at a andb in (2.14), (2.15c), (2.16c) are z-independent.

Remark 2.6. If τ is in limit point case either at a or at b then the correspondingboundary condition at a or b in (2.14), (2.15c), (2.16c) may be omitted (that is, if τis in lpc at a (respectively, at b) then the condition Wa(g, u(z)a) = 0 (respectively,Wb(g, u(z)b) = 0) is automatically satisfied for any g ∈ domHmax, see e.g. [38],[39, Theorem 5.7], [29, Theorem 6.5 (iv)]).

Since (see [10, Theorem XIII.7.4])

σess(H) = σess(Ha) ∪ σess(Hb), (2.17)

the following lemma holds (see, e.g. [19, (A.25)]).

Lemma 2.7. Assume Hypothesis 2.1. Then the Weyl-Titchmarsh functions aremeromorphic in z ∈ C\σess(H).

Proof. Let ma be the Weyl-Titchmarsh function (that is, a point on the circle Caif τ is in lcc at a, or the point ma if τ is in lpc at a). Consider the operator Ha

defined in L2(a, 0) by (2.15). Green’s function of Ha for z ∈ C \ σ(Ha) is given bythe formula

G(t, s, z) =

{φ(t, z)χa(s, z), t ≤ s,φ(s, z)χa(t, z), s ≤ t,

(2.18)

where χa(t, z) = θ(t, z) + ma(z)φ(t, z) (see [9, Chapter IX, (4.10)]). In particular,

G(t, t, z) = φ(t, z)(θ(t, z) + ma(z)φ(t, z)

), t ≤ 0.

Since Green’s function is meromorphic in z away from the essential spectrum of Ha

for any fixed t, s (cf. [10, Section XIII.5, Theorem XIII.5.18, Corollary XIII.5.30]),ma is meromorphic for z ∈ C\σess(Ha) and, therefore, in C\σess(H) due to (2.17).Analogously, one can prove that mb is meromorphic in C\σess(H). �

Remark 2.8. If τ is in limit point case at b, if m is any point on Cd and d→ b, thenm→ mb, where mb is the limit point, and the relation m→ mb holds independentlyof the choice of η in the boundary condition (2.4). In particular, it holds when η = 0,and thus the limit point is given by the formula

mb(z) = − limd→b

θ(d, z)φ(d, z)

for z ∈ C \ R. (2.19)

Similarly, if τ is in limit point case at a then

ma(z) = − limc→a

θ(c, z)φ(c, z)

for z ∈ C \ R. (2.20)

The purpose of the next proposition is to establish the existence and uniquenessof the solutions fa(· , z) and fb(· , z) of the differential equation τy = zy that aresquare summable near a and b, respectively, and satisfy the respective boundaryconditions in (2.15c) and (2.16c), if any. For z ∈ C \ R this is contained in [10,Theorem XIII.2.32] or [29, Theorem 6.2]; we emphasize that z in Proposition 2.9below can be real.

Proposition 2.9. Assume Hypothesis 2.1. If z ∈ C \ σess(Ha) then the followingassertions hold:


(i) If τ is in lpc at a then there is a unique (up to a constant multiple) solutionfa(· , z) of τy = zy that is square integrable near a;

(ii) If τ is in lcc at a then there is a unique (up to a constant multiple) solu-tion fa(· , z) of τy = zy that is square integrable near a and satisfy the boundarycondition at a in (2.15c).

Moreover, in either case, if z ∈ C \ σ(Ha) then (up to a constant multiple)fa(· , z) is equal to χa(· , z) = θ(· , z) + ma(z)φ(· , z), where ma(z) is used in (2.12)and (2.15c) to define the operator Ha if τ is in lcc at a, and ma(z) = ma(z) if τ isin lpc at a. Lastly, if z ∈ σd(Ha) then fa(· , z) is the nonzero eigenfunction of theoperator Ha − z.

Analogous assertions hold for the point b.

Proof. First, assuming z ∈ C\σ(Ha), we let µ denote the number of linearly in-dependent solutions of τy = zy that are square integrable near a and satisfy theboundary condition at a in (2.15c) provided τ is in lcc at a, and that are justsquare integrable near a provided τ is in lpc at a. Also, we let ν denote the numberof linearly independent solutions that are square integrable near 0 and satisfy theboundary condition at 0 in (2.15c). Then µ + ν = 2 by [10, Theorem XIII.3.11]or [38, Theorem 7.1]. Since 0 is a regular point, ν = 1. Thus µ = 1 proving(i) and (ii) for z ∈ C\σ(Ha). Furthermore, if τ is in lpc at a then the solutionχa(· , z) = θ(· , z) + ma(z)φ(· , z) is square integrable near a, and if τ is in lcc at athen the solution χa(· , z) = θ(· , z) + ma(z)φ(· , z) is square integrable near a andsatisfies the boundary condition at a in (2.15c). Indeed, for nonreal z this followsfrom the construction of the Weyl-Titchmarsh function in Theorem 2.2, cf. [10,Theorem XIII.2.32] or [29, Theorem 6.2]. For z ∈ R \ σ(Ha) this follows from thefirst line in formula (2.18) for Green’s function, and the fact that Green’s functionis square integrable and G(·, s, z) (for a fixed s) satisfies the boundary conditionsdefining Ha, see, e.g., [10, Lemma XIII.3.4, Lemma XIII.3.7].

Next, we assume that z ∈ σd(Ha), the discrete spectrum of Ha; in particular,z ∈ R. If τ is in lpc at a, then the nonzero eigenfunction of Ha corresponding to thegiven z is the required solution fa(· , z) since the number of linearly independentsolutions that are square integrable near a is at most one. It remains to considerthe case when z ∈ σd(Ha) and τ is in lcc at a. We know that the eigenfunctioncorresponding to the given z is a solution of τy = zy that is square integrable neara and satisfy the boundary condition at a in (2.15c). Seeking a contradiction, let usassume that there are two linearly independent solutions, f (1)

a (· , z) and f(2)a (· , z),

of τy = zy that are square integrable near a and satisfy the boundary condition ata in (2.15c). According to [38, Theorem 5.8(v)], the domain of Ha can be rewrittenin the following way:

domHa ={g ∈ L2(a, 0)

∣∣ g, pg′ ∈ ACloc(a, 0], τg ∈ L2(a, 0),

Wa(v, g) = W0(w, g) = 0},

(2.21)

where v and w are nontrivial real solutions of τy = zy. Indeed, as required in [38,Theorem 5.8(iv)], τ is in lcc at both a and 0, and the operator Ha is defined usingseparated boundary conditions (2.15c). Let us introduce the functions f (1)

a and f (2)a

as follows:

f (j)a (t, z) =

{f

(j)a (t, z) for t close to a,

0 for t close to 0,(2.22)


and such that f (j)a (· , z) ∈ domHa,max (the latter inclusion is possible as described in

[38, page 50]). Then f(j)a ∈ domHa. Consequently, f (1)

a (· , z) and f(2)a (· , z) should

satisfy the boundary condition at a in (2.21). Therefore, f (1)a (· , z) and f

(2)a (· , z)

should satisfy the boundary condition at a in (2.21) as well. Since f (1)a (· , z) and v

are solutions of τy = zy, they are linearly dependent. Analogously, f (2)a (· , z) and

v are linearly dependent. Thus, f (1)a (· , z) and f

(2)a (· , z) are linearly dependent, a

contradiction. �

Remark 2.10. If z ∈ C \ σess(Ha) and fa is the solution described in Proposition2.9 then both Wronskians W

(θ, fa(· , z)

)and W

(fa(· , z), φ

)are finite, and cannot

be equal to 0 simultaneously as otherwise fa(0, z0) = pf ′a(0, z0) = 0. Analogousfacts hold for the point b.

Our next objective is to relate the Weyl-Titschmarsh m-function to the Wron-skian determinant of the solutions θ, φ, fa and fb. As the following lemma shows,this information can be used to characterize the discrete spectrum of the operatorsHa, Hb and H.

Lemma 2.11. Assume Hypothesis 2.1. If z0 ∈ C \ σess(Ha) then the followingassertions are equivalent:

(i) W(fa(· , z0), φ(· , z0)

)= 0;

(ii) z0 ∈ σd(Ha);(iii) z0 is a pole of ma(·),

and if the equivalent assertions hold then fa(· , z0) is an eigenfunction of Ha. Anal-ogous facts hold for the point b. Finally, if z0 ∈ C \ σess(H) then the followingassertions are equivalent:

(iv) W(fa(· , z0), fb(· , z0)

)= 0;

(v) z0 ∈ σd(H).

Proof. (i)⇔ (ii): By Proposition 2.9, fa is the unique solution of τy = zy squaresummable at a (when τ is in lpc at a) that satisfies the boundary condition at a in(2.15c) (when τ is in lcc at a). Then (i) holds if and only if fa is proportional to φif and only if fa satisfies both boundary conditions in (2.15c) (when τ is in lcc ata) if and only if fa is an eigenfunction of Ha.

(ii)⇔ (iii): Indeed, (ii) holds if and only if there exists at least one point (t, s)for which Green’s function G(t, s, ·) of Ha has a pole at z0 as a function of z, andthis is the case if and only if z0 is a pole of ma(·) due to (2.18).

(iv)⇔ (v): Indeed, (iv) holds if and only if fa is proportional to fb if and onlyif fa satisfies Proposition 2.9 at both a and b if and only if fa is an eigenfunctionof H. �

We refer to [19, (A.36)] and [14, (5.8),(5.10)] for versions of the next theoremwhen z ∈ C \ R. Our main point, however, is that z can be real in (2.23)-(2.25).

Theorem 2.12. Assume Hypothesis 2.1 and suppose that θ and φ satisfy (2.2) forsome ω ∈ [0, π). Let fa(· , z) and fb(· , z) denote the square integrable near a andb solutions of τy = zy described in Proposition 2.9. Then the following formulashold:

ma(z) =W0

(θ(· , z), fa(· , z)

)W0

(fa(· , z), φ(· , z)

) , z ∈ C \ σess(Ha), (2.23)


mb(z) =W0

(θ(· , z), fb(· , z)

)W0

(fb(· , z), φ(· , z)

) , z ∈ C \ σess(Hb), (2.24)

mb(z)−ma(z) =W0

(fa(· , z), fb(· , z)

)W0

(fa(· , z), φ(· , z)

)W0

(fb(· , z), φ(· , z)

) , z ∈ C \ σess(H).

(2.25)

Proof. Let us prove (2.24), the proof of (2.23) is analogous. By Proposition 2.9,fb(· , z) and χb(· , z) = θ(· , z)+mb(z)φ(· , z) have to be proportional whenever mb(z)is finite, that is, for z ∈ C\σ(Hb). Hence, there exists a t-independent constantcb(z) such that fb(t, z) = cb(z)

(θ(t, z)+mb(z)φ(t, z)

). Differentiating, letting t = 0,

and using (2.2) yields the system of equations for mb(z) and cb(z),

cb(z) sinω + cb(z)mb(z) cosω = fb(0, z),

−cb(z) cosω + cb(z)mb(z) sinω = pf ′b(0, z),

whose solution is given by

mb(z) =fb(0, z) cosω + pf ′b(0, z) sinωfb(0, z) sinω − pf ′b(0, z) cosω

(2.26)

=W0

(θ(· , z), fb(· , z)

)W0

(fb(· , z), φ(· , z)

) ,thus proving (2.24) for z ∈ C \ σ(Hb). Further, if z ∈ σd(Hb) then mb has a poleat z by Lemma 2.11. Since fb(·, z) is the eigenfunction of Hb by Proposition 2.9,W0(fb(·, z), φ(· , z)) = 0 by (2.16c). Also, W0(θ(· , z), fb(·, z)) 6= 0 by Remark 2.10.This extends (2.24) for z ∈ C \ σess(Hb). Using (2.26), its analogue for a, and(2.17), a short calculation yields (2.25). �

Remark 2.13. Assume Hypothesis 2.1 and let z ∈ C\(σ(Ha)∪σ(Hb)). If we choosefa(·, z) = χa(·, z) and fb(·, z) = χb(·, z) in Theorem 2.12, then (2.25) becomes

mb(z)−ma(z) = W(χa(· , z), χb(· , z)

), z ∈ C\(σ(Ha) ∪ σ(Hb)). (2.27)

This formula could be found in [19, (A. 36)] for a = −∞, b = ∞ and z ∈ C \ R.For ω = π/2 formulas (2.23), (2.24), (2.25) become:

ma(z) =pf ′a(0, z)fa(0, z)

, mb(z) =pf ′b(0, z)fb(0, z)

,mb(z)−ma(z) =W(fa(· , z), fb(· , z)

)fa(0, z)fb(0, z)

,

(2.28)and, in particular, mb(z) = pχ′b(0, z), ma(z) = pχ′a(0, z). Choosing fa = χa, fb =χb shows that the Wronskians in (2.23), (2.24), (2.25) are not necessarily analyticin z ∈ C \ σess(Ha), z ∈ C \ σess(Hb), z ∈ C \ σess(H).

As shown in Lemma 2.11, the Wronskian W(fa(· , z0), fb(· , z0)

)plays the role

of the Evans function. Usually, the Evans function is defined by means of expo-nentially decaying solutions, see e.g. [3, 15, 32, 36] and the literature therein. Animportant point of our analysis is that in the more general than in [32] settings of thecurrent paper the Evans function is defined using square integrable solutions. Also,combining Lemma 2.11 and the next corollary, one can use the Weyl-Titchmarshfunction to detect the discrete spectrum of H.


Corollary 2.14. If the assumptions in Theorem 2.12 are satisfied, then for anyz0 ∈ C \ σess(H) the Wronskian W

(fa(· , z0), fb(· , z0)

)= 0 if and only if the fol-

lowing alternative holds: Either both ma(z0) and mb(z0) have poles at z0, or theyboth have finite values at z0 and ma(z0) = mb(z0).

Proof. Since ma is meromorphic, (2.23) implies that z0 is a pole of ma if and only ifW(fa(· , z0), φ

)= 0. Analogous assertion holds for W

(fb(· , z0), φ

). Formula (2.25)

yields:

W(fa(· , z0), fb(· , z0)

)=(mb(z0)−ma(z0)

)W(fa(· , z0), φ

)W(fb(· , z0), φ

)(2.29)

= mb(z0)W(fb(· , z0), φ

)·W(fa(· , z0), φ

)−ma(z0)W

(fa(· , z0), φ

)·W(fb(· , z0), φ

).

= W(θ, fb(· , z0)

)·W(fa(· , z0), φ

)(2.30)

−W(θ, fa(· , z0)

)·W(fb(· , z0), φ

). (2.31)

Thus, if both W(fa(· , z0), φ

)and W

(fb(· , z0), φ

)are not zero then (2.29) im-

plies that W(fa(· , z0), fb(· , z0)

)= 0 if and only if mb(z0) = ma(z0). If one

of the expressions W(fa(· , z0), φ

)or W

(fb(· , z0), φ

)is not zero and another is

zero, then one of the expressions in (2.30), (2.31) is zero and another is not; thus,W(fa(· , z0), fb(· , z0)

)is not zero. Finally, if W

(fa(· , z0), φ

)= W

(fb(· , z0), φ

)= 0

then both (2.30), (2.31) are zero, and thus W(fa(· , z0), fb(· , z0)

)= 0. �

Let us now consider the Schrodinger equation on the line:

−y′′(t) + V (t)y(t) = k2y(t), −∞ < t <∞. (2.32)

We denote the spectral parameter by k so that z = k2 ∈ C, and assume throughoutthat Im(k) ≥ 0. The Schrodinger equation (2.32) is equivalent to the first ordersystem

Y ′(t) =(A(k) +R(t)

)Y (t), A(k) =

[0 1−k2 0

], R(t) =

[0 0

V (t) 0

], t ∈ R, (2.33)

where Y (t) =[y(t) y′(t)

]> is the column-vector. As soon as the values y(t0)and y′(t0) at any point t0 are given, one can find the corresponding solution y of(2.32) on R such that y, y′ are locally absolutely continuous functions by solvingthe corresponding Cauchy problem for (2.33) .

Hypothesis 2.15. Assume that a = −∞, b = +∞, p(t) = r(t) = 1, t ∈ R, andthe potential in (2.32) is real valued and satisfies V ∈ L1(R).

As it is well known, see, e.g. [6], under Hypothesis 2.15 all solutions of theSchrodinger equation (2.32) can be obtained as linear combinations of the solutionsf±(t, k) of (2.32) satisfying the asymptotic boundary conditions

limt→±∞

e∓iktf±(t, k) = 1, Im(k) > 0. (2.34)

These solutions are called the Jost solutions; they are defined as solutions of theVolterra integral equations

f±(t, k) = e±ikt −∫ ±∞t

sin(k(t− s))k

V (s)f±(s, k)ds,

Im(k) > 0, t ∈ R.(2.35)


The Jost function, J = J (k), is defined by

J (k) =1

2ikW(f−(· , k), f+(· , k)

), Im(k) > 0. (2.36)

The isolated zeros of the Jost function are the discrete eigenvalues of the self-adjointoperator H associated with (2.32), see, e.g. [6, Chapter XVII].

Assuming Hypothesis 2.15 and Im(k) > 0, the first order matrix system (2.33)has a solution Y+(· , k) exponentially decaying on R+ and a solution Y−(· , k) expo-nentially decaying on R−; each of them is defined up to a constant multiple. TheEvans function D(k) is then defined as D(k) = det

[Y+(0, k) Y−(0, k)

], see, e.g.,

[32]. As shown in [15], the Jost function, J(k), for (2.32) is equal to the appropri-ately chosen Evans function, D(k), for (2.33). We will now show how to calculatethe Evans and the Jost functions via the Weyl-Titchmarsh functions.

Lemma 2.16. [9, Problem 9.4] Assume Hypothesis 2.15. Then the equation τy =k2y is in the limit point case at both −∞ and ∞.

Proof. As known from [9, Problem 9.4], for k > 0 the first equation in (2.35) has abounded solution on the positive semi-axis and, hence, f+(t, k)−eikt → 0 as t→∞.Therefore, f+ is not square integrable at +∞ and thus according to Theorem 2.2the equation τy = k2y is in the limit point case at +∞. Analogously one can showthat the equation τy = k2y is in the limit point case at −∞. �

Lemma 2.17. Assume Hypothesis 2.15. Then the Weyl-Titchmarsh functionsm−∞(·) and m∞(·) are meromorphic in C\[0,∞).

Proof. This follows from Lemma 2.7 and the fact that σess(H) = [0,∞), see, e.g.,[38, Theorem 15.3]. �

Corollary 2.18. Assume Hypothesis 2.15 and let ω = π/2 in (2.2). Then z0 ∈σd(H) if and only if either m∞(z0) = m−∞(z0) or m−∞ and m∞ both have asimple pole at z0. Moreover, the following formula holds:

D(k) = J (k) =1

2ik(f−(0, k)f+(0, k)

)(m∞(k2)−m−∞(k2)

), Im(k) > 0. (2.37)

Proof. The Jost solutions f− and f+ are the solutions fa, a = −∞, and fb, b = +∞,described in Proposition 2.9. Thus, the required assertions follow from Lemma2.11, Theorem 2.12, Corollary 2.14, (2.36), and equality D(k) = J (k) proved in[15, Theorems 5.4(i), 9.4]. �

3. The reduced Jost function

In this section we derive an asymptotic formula relating the Jost function (2.36)for the Schrodinger equation (2.32) on the line, and the reduced Jost function forthe Schrodinger equation on a large but finite segment. We assume throughout thatHypothesis 2.15 holds, and choose k with Im(k) > 0 so that eikt → 0 as t → ∞.We recall that the Jost solutions f±(· , k) from (2.34), (2.35) are the solutions ofthe Schrodinger equation (2.32) that are asymptotic to the plane waves e±ikt whichare solutions of the Schrodinger equation with zero potential. The asymptoticproperties of the Jost solutions are summarized in the following elementary lemma(for a related material see, e.g. [6, Sec. XVII.1.2]).


Lemma 3.1. Assume Hypothesis 2.15. Then, with Im(k) > 0, the following as-ymptotic relations hold:

limt→±∞

f±(t, k)e∓ikt = 1, limt→±∞

1±ik

f ′±(t, k)e∓ikt = 1, (3.1)

limt→±∞

f∓(t, k)e±ikt = J (k), limt→±∞

1∓ik

f ′∓(t, k)e±ikt = J (k). (3.2)

Proof. Formulas (3.1) follow from (2.35). To show (3.2), we begin by recallingthe following well-known property of the asymptotically constant coefficient ODEsystem (2.33) (see, e.g., [9, Problem III.29] or [11, Chapter 1]): There exist solutionsY±(· , k) and Y±(· , k) of (2.33) such that

limt→±∞

Y±(t, k)e∓ikt = [1 ± ik]>, limt→±∞

Y±(t, k)e±ikt = [1 ∓ ik]>, (3.3)

where ±ik are the eigenvalues and [1 ± ik]> are the corresponding eigenvectors ofthe matrix A(k) defined in (2.33). By (3.1), we have Y±(· , k) = [f±(· , k) f ′±(· , k)]>

with the Jost solutions f±(· , k) of (2.32). Denoting by f±(· , k) the solutions of(2.32) such that Y±(· , k) = [f±(· , k) f ′±(· , k)]>, we derive from the second equationin (3.3) the following asymptotic formulas:

limt→±∞

f±(t, k)e±ikt = 1, limt→±∞

1∓ik

f ′±(t, k)e±ikt = 1. (3.4)

By letting t→ ±∞, we note that (3.1), (3.4) also imply

W(f±(· , k), f±(· , k)

)= ∓2ik, Im(k) > 0. (3.5)

Therefore, f+(· , k) and f+(· , k) are linearly independent solutions of (2.3), andthus there are some constants c1, c2 such that

f−(t, k) = c1f+(t, k) + c2f+(t, k) for all t ∈ R. (3.6)

Computing the WronskianW(f−(· , k), f+(· , k)

)in (3.6) and using (3.5), we see that

c2 = J (k). Multiplying (3.6) by eikt, letting t→ +∞, and using (3.1), (3.4) yieldsthe first formula in (3.2), that is, limt→+∞ f−(t, k)e+ikt = J (k). Differentiating(3.6), in the same manner one obtains the formula in (3.2) for the derivative off−(· , k). Expressing f+(· , k) as a linear combination

f+(t, k) = c1f−(t, k) + c2f−(t, k) for all t ∈ R, (3.7)

computing the Wronskian W(f−(· , k), f+(· , k)

)in (3.7) and using (3.5), we see

that c2 = J (k). Multiplying (3.7) by e−ikt, letting t→ −∞, and using (3.1), (3.4)yields the formula limt→−∞ f+(t, k)e−ikt = J (k) in (3.2). Differentiating (3.7), inthe same manner one obtains the formula in (3.2) for the derivative of f+(· , k). �

Let us introduce the operator Hf = −f ′′ + V (t)f on L2(R) with the domain

domH ={f ∈ L2(R)

∣∣∣f, f ′ ∈ ACloc(R),−f ′′ + V f ∈ L2(R)}, (3.8)

and remark that k2 ∈ σd(H) if and only if k ∈ iR \ {0} and J (k) = 0, see, e.g., [6,Sec. XVII.1.3]. Indeed, if J (k) = 0 then f+(· , k) is proportional to f−(· , k) andthus is an eigenfunction for H as it is exponentially decaying at both +∞ and −∞.Conversely, expressing an eigenfunction f ∈ L2(R) of H as a linear combination off+(· , k) and f−(· , k), we see that the Jost solutions must be proportional yieldingJ (k) = 0.


We will now construct a reduced Jost function, JL(k), that corresponds to thedifferential operator on the segment [−L,L] with (large) positive L whose discretespectrum approximates the spectrum of H. Fix an L > 0 and two angles, ω− andω+, in [0, π). We will consider the following boundary conditions at the points −Land +L,

f(−L) cosω− + f ′(−L) sinω− = 0, (3.9)

f(+L) cosω+ + f ′(+L) sinω+ = 0, (3.10)

and define the operator HLf = −f ′′ + V (t)f on L2(−L,L) with the domain

domHL ={f ∈ L2(−L,L)

∣∣∣f, f ′ ∈ ACloc[−L,L],−f ′′ + V f ∈ L2(−L,L),

and both conditions (3.9), (3.10) hold}.

(3.11)

Solving the corresponding Cauchy problems for (2.33), we let g+(· , k) and g−(· , k)denote the locally absolutely continuous together with their derivatives solutions ofthe Schrodinger equation (2.32) on R satisfying the following conditions at ±L,

g±(±L, k) = eikL sinω±, g′±(±L, k) = −eikL cosω±, (3.12)

and define the reduced Jost function, JL = JL(k), as follows:

JL(k) =1

2ikW(g−(· , k), g+(· , k)

), Im(k) > 0. (3.13)

Clearly, g+(· , k) satisfies (3.10) while g−(· , k) satisfies (3.9). We claim that k2 ∈σd(HL) if and only if JL(k) = 0. Indeed, if JL(k) = 0 then g+(· , k) is propor-tional to g−(· , k) and thus is an eigenfunction for HL as it satisfies both conditions(3.9) and (3.10). Conversely, if g is an eigenfunction for HL then it satisfies bothconditions (3.9) and (3.10). Evaluating the Wronskian at t = +L, we notice thatW (g, g+(· , k)) = 0 and hence g is proportional to g+(· , k). Evaluating the Wron-skian at t = −L, we notice that W (g, g−(· , k)) = 0, and hence g is proportional tog−(· , k), proving the claim.

In addition, we will consider two more operators with zero potential, H+L,0 and

H−L,0, such that H±L,0f = −f ′′. The operator H+L,0 is defined in L2(−∞,+L) with

the domain

domH+L,0 =

{f ∈ L2(−∞, L)

∣∣∣f, f ′ ∈ ACloc(−∞, L],−f ′′ ∈ L2(−∞, L),

and condition (3.10) holds}.

The operator H−L,0 is defined in L2(−L,+∞) with the domain

domH−L,0 ={f ∈ L2(−L,+∞)

∣∣∣f, f ′ ∈ ACloc[−L,+∞),−f ′′ ∈ L2(−L,+∞),

and condition (3.9) holds}.

Furthermore, we let h+(· , k) and h−(· , k) denote the solutions of the differentialequation −h′′ = k2h on R which are locally absolutely continuous together withtheir derivatives and satisfy the following conditions at ±L,

h±(±L, k) = eikL sinω±, h′±(±L, k) = −eikL cosω±, (3.14)


and define the functions, J±L,0 = J±L,0(k), as follows:

J +L,0(k) =

12ik

W(e−ikt, h+(· , k)

),

J−L,0(k) =1

2ikW(h−(· , k), eikt

), Im(k) > 0.

(3.15)

Computing the Wronskian for J±L,0(k) at ±L yields

2ikJ +L,0(k) = −det

[sinω+ 1− cosω+ −ik

]= − cosω+ + ik sinω+, (3.16)

2ikJ−L,0(k) = det[

sinω− 1− cosω− ik

]= cosω− + ik sinω−; (3.17)

in particular,J±L,0(k) does not depend on L. (3.18)

Clearly, h+(· , k) satisfies (3.10) and h−(· , k) satisfies (3.9) while eikt ∈ L2(−L,+∞)and e−ikt ∈ L2(−∞,+L) since Im(k) > 0.

Remark 3.2. We claim that k2 ∈ σd(H±L,0) if and only if J±L,0(k) = 0. Indeed,if, say, J +

L,0(k) = 0 then h+(· , k) is proportional to e−ikt and thus is an eigen-function for H+

L,0 as it satisfies conditions (3.10) and h+(· , k) ∈ L2(−∞,+L).Conversely, if h is an eigenfunction for H+

L,0 then it satisfies conditions (3.10)and h ∈ L2(−∞,+L). Evaluating the Wronskian at t = +L, we notice thatW (h, h+(· , k)) = 0 and hence h is proportional to h+(· , k). Expressing h ∈L2(−∞,+L) as a linear combination of the plane waves e−ikt and eikt, we concludethat h is proportional to e−ikt, thus proving the claim for H+

L,0. The argument forH−L,0 is analogous. 3

We are ready to present the main result of this section (recall (3.18)).

Theorem 3.3. Assume that V ∈ L1(R; R). Then, for the Jost functions J (k),JL(k), J±L,0(k) defined in (2.36), (3.13), (3.15), we have:

limL→∞

JL(k) = J (k)J−L,0(k)J +L,0(k), Im(k) > 0. (3.19)

Proof. Varying k a little, if needed, with no loss of generality we may and willassume throughout the proof that k2 is not an isolated eigenvalue of H, that is, thatJ (k) 6= 0. Then the Jost solutions f+(· , k) and f−(· , k) are linearly independentand hence there are exist constants c−1 , c

−2 and c+1 , c

+2 such that for all t ∈ R one

has the following representations:

g−(t, k) = c−1 f+(t, k) + c−2 f−(t, k),

g+(t, k) = c+1 f+(t, k) + c+2 f−(t, k).(3.20)

Computing the Wronskians, we find:

c−1 =W(f−(· , k), g−(· , k)

)W(f−(· , k), f+(· , k)

) , c−2 =W(g−(· , k), f+(· , k)

)W(f−(· , k), f+(· , k)

) ,c+1 =

W(f−(· , k), g+(· , k)

)W(f−(· , k), f+(· , k)

) , c+2 =W(g+(· , k), f+(· , k)

)W(f−(· , k), f+(· , k)

) . (3.21)


A computation using (3.20), (3.21) reveals:

W(g−(· , k),g+(· , k)

)= W

(c−1 f+(· , k) + c−2 f−(· , k), c+1 f+(· , k) + c+2 f−(· , k)

)=(− c−1 c

+2 + c−2 c

+1

)W(f−(· , k), f+(· , k)

)= W1 +W2, (3.22)

where we temporarily introduced the following notations:

W1 = −W(f−(· , k), g−(· , k)

)W(g+(· , k), f+(· , k)

)/W(f−(· , k), f+(· , k)

), (3.23)

W2 = W(g−(· , k), f+(· , k)

)W(f−(· , k), g+(· , k)

)/W(f−(· , k), f+(· , k)

). (3.24)

Assertion (3.19) now follows from (3.12) and Lemma 3.1. Indeed, the expression

W(f−(· , k), g−(· , k)

)= f−(−L, k)g′−(−L, k)− f ′−(−L, k)g−(−L, k)

= −f−(−L, k)eikL cosω− − f ′−(−L, k)eikL sinω− (3.25)

= −(f−(−L, k)eik(−L)

)e2ikL cosω−

− (−ik)( 1−ik

f ′−(−L, k)eik(−L))

e2ikL sinω−

tends to zero as L→∞ due to (3.1) and Im(k) > 0. This and a similar argumentfor W

(g+(· , k), f+(· , k)

)yields W1 → 0 as L→∞, and hence it suffices to handle

the term W2 in (3.22). If L→∞ then, using (3.2), the expression

W(g−(· , k), f+(· , k)

)= g−(−L, k)f ′+(−L, k)− g′−(−L, k)f+(−L, k)

= eikL sinω−f ′+(−L, k) + eikL cosω−f+(−L, k)

=( 1ik

e−ik(−L)f ′+(−L, k))

(ik) sinω− +(

e−ik(−L)f+(−L, k))

cosω−

tends toJ (k)(ik) sinω− + J (k) cosω− = 2ikJ (k)J−L,0(k),

where in the last equality we used (3.17). Also, the expression

W(f−(· , k), g+(· , k)

)= f−(L, k)g′+(L, k)− f ′−(L, k)g+(L, k)

= −f−(L, k)eikL cosω+ − f ′−(L, k)eikL sinω+

= −(f−(L, k)eikL

)cosω+ +

( 1−ik

f ′−(L, k)eikL)

(ik) sinω+

tends to−J (k) cosω+ + J (k)(ik) sinω+ = 2ikJ (k)J +

L,0(k),where in the last equality we used (3.16). Combining this with (3.22) yields

limL→∞

JL(k) = limL→∞

12ik

W (g−(· , k), g+(· , k)) = limL→∞

12ik

W2

= limL→∞

12ik

W (g−(· , k), f+(· , k))W (f−(· , k), g+(· , k))2ikJ (k)

= J (k)J−L,0(k)J +L,0(k),

as required. �

We conclude this section with a short discussion of the relation between Theorem3.3 and certain results in [33] and [4]. In particular, as a part of a general theoryrelating the spectra of first order differential operators on the line and on finitesegments, [33, Theorem 3] relates the multiplicity of the eigenvalues of a general firstorder differential operator TL in the space L2((−L,L); Cd) of d-dimensional vector


valued functions to that of the operator T in L2(R; Cd), and to the multiplicity ofzeros of certain functions, D±(z), of the spectral parameter z, determined by theboundary conditions at ±L used to define TL. The discussion in [33] involves theEvans functions D∞(z), defined for T , and DL(z), defined for TL, such that thezeros of the Evans functions are the eigenvalues of the respective operators. Oneof the major conclusions in [33, Theorem 3], see also the preceding analysis in [4]and a more recent paper [31], is the following eigenvalue multiplicity result: Undercertain natural assumptions, for L sufficiently large, the algebraic multiplicity ofan isolated eigenvalue of TL (that is, the order of a zero of DL) is equal to thesum of the orders of zeros of the functions D∞, D−, and D+ in a small vicinity ofthe eigenvalue. We will now furnish the definitions of the functions D∞(z), DL(z),D±(z) of the spectral parameter z = k2 for the particular case of the first ordersystem (2.33) corresponding to the Schrodinger equation (2.32) and relate them tothe Jost functions J (k), JL(k), J±L,0(k) studied earlier in this section.

First, we recall that the Evans function D∞(z) is the determinant of the (2× 2)matrix whose first column is the initial data at t = 0 of the solution of (2.33) thatexponentially decays to zero as t → +∞ and whose second column is the initialdata at t = 0 of the solution of (2.33) that exponentially decays to zero as t→ −∞.It is known that if these solutions are appropriately chosen then D∞(z) = J (k),z = k2, see [15, Section 9] and Corollary 2.18.

Next, we define DL(z) as follows. Let us fix two arbitrary unit vectors, denotedby [sinω− − cosω−]> and [sinω+ − cosω+]>, and let Q± denote the subspacespanned by the vector [sinω± − cosω±]>. The subspaces Q− and Q+ determinethe boundary conditions at −L and +L respectively in the sense that a solutionf of the Schrodinger equation (2.32) satisfies the boundary conditions (3.9), (3.10)if and only if the corresponding solution Y (t) = [f(t) f ′(t)]> of the first ordersystem (2.33) satisfies the boundary conditions Y (±L) ∈ Q±. Following [33], giventwo subspaces F and E of Cd with dimE + dimF = d, we denote by E ∧ E thedeterminant of the (d× d) matrix whose columns are the bases vectors of E and Fput consequently. We let ϕ(t, s; z) denote the propagator of the system (2.33). Withthis notations, we define DL(z) = ϕ(0,−L; z)Q−∧ϕ(0, L; z)Q+ (see, e.g., [33, (4.5)]where notation Dsep was used instead of DL). Of course, DL(z) depends, up to aconstant factor, on the choice of the bases vectors in Q±. In particular, if g±(· , k)is the solutions of (2.32) that satisfy (3.12) then [g±(±L, k) g′±(±L, k)]> ∈ Q± andthus DL(z) = JL(k) by (3.13).

Finally, we define D±(z). We recall that ±ik are the eigenvalues of the martixA(k) from (2.33) with the corresponding eigenvectors [1 ± ik]>. Let us denoteby Es−(z) the linear subspace spanned by the vector [1 + ik]>, and by Eu+(z) thelinear subspace spanned by the vector [1 −ik]>. Then Es−(z) is the set of the initialdata of the solutions of the constant coefficient differential equation Y ′ = A(k)Ythat exponentially grow to infinity as t → −∞ and Eu+(z) is the set of the initialdata of the solutions of the constant coefficient differential equation Y ′ = A(k)Ythat exponentially grow to infinity as t → +∞. Using subspaces Q± introducedin the previous paragraph, following [33, (4.5)], we define D−(z) = Q− ∧ Es−(z)and D+(z) = Q+ ∧ Eu+(z). By (3.16), (3.17) we conclude that, up to a constantfactor, D±(z) is equal to J±L,0(k). Remark 3.2 provides an operator theoretical


interpretation of the quantities D±(z). Theorem 3.3 yields

limL→∞

DL(z) = D∞(z)D−(z)D+(z), z ∈ C \ [0,∞), (3.26)

which is consistent with the eigenvalue multiplicity results from [4, 33] mentionedabove.

4. The Weyl-Titschmarsh M-function and Hamiltonian systems

Following [8, 23, 27, 28], we will briefly review the basics of the Weyl-Titschmarshtheory for the matrix valued Hamiltonian systems. We consider two singular end-points problem for the (2n× 2n) Hamiltonian system

JY ′(t) =(zA(t) +B(t)

)Y (t), t ∈ (a, b), −∞ ≤ a < b ≤ ∞, (4.1)

where Y (t) is a (2n×1)-vector, z is a complex parameter, and J =[

0 −In×nIn×n 0

].

The coefficients A,B satisfy the following assumptions.

Hypothesis 4.1. (i) A(t) and B(t) are (2n×2n) Hermitian matrices for t ∈ (a, b)whose entries are complex valued measurable and locally integrable funtions on (a, b);

(ii) A(t) ≥ 0 a.e. in the sense of quadratic forms and satisfies Atkinson’s defi-niteness condition, i.e., if Y is a nontrivial solution of (4.1) then∫ d

c

Y (t)∗A(t)Y (t)dt > 0 for any c, d such that a < c < d < b.

A solution of (4.1) is said to be A-square integrable if∫ baY ∗(t)A(t)Y (t) dt <∞,

and we denote this by Y ∈ L2A(a, b). As already mentioned in (4.1), we allow

the endpoints a and b to be finite or infinite, and, with no loss of generality, weassume that 0 ∈ (a, b). We will now discuss boundary conditions at c, 0, d for anyc, d satisfying a < c < 0 < d < b. Let us fix three (n × 2n) matrices,

[α1 α2

],[

γ1 γ2

]and

[β1 β2

], where α1, α2, γ1, γ2, β1, β2 are (n × n) matrices satisfying

the conditions

rank[α1 α2

]= rank

[γ1 γ2

]= rank

[β1 β2

]= n, (4.2)

α1α∗2 − α2α

∗1 = 0n×n, γ1γ

∗2 − γ2γ

∗1 = 0n×n, β1β

∗2 − β2β

∗1 = 0n×n . (4.3)

If (4.2), (4.3) hold then, cf. [27, page 108], with no loss of generality we may andwill assume that

α1α∗1 + α2α

∗2 = I, γ1γ

∗1 + γ2γ

∗2 = I, β1β

∗1 + β2β

∗2 = I. (4.4)

Let Y(t, z) be the fundamental matrix solution of the differential equation (4.1)normalized such that

Y(0, z) =[γ∗1 −γ∗2γ∗2 γ∗1

]. (4.5)

Then Y(t, z) satisfies the following conditions at t = 0:[γ1 γ2

]Y(0, z) =

[In×n 0n×n

],

Y∗(0, z)Y(0, z) = I2n×2n = Y(0, z)Y∗(0, z).(4.6)

We decompose Y(· , z) into 2n×n blocks θ, φ, where θ and φ are further decomposedinto n× n blocks as follows:

Y(t, z) =[θ(t, z) φ(t, z)

]=[θ1(t, z) φ1(t, z)θ2(t, z) φ2(t, z)

]. (4.7)


We note that θ, φ are entire functions in z and are continuos in t, z. Also, welet φ1(t, z), . . . , φn(t, z) denote the columns of the (2n× n) matrix φ(t, z) and, forfuture use, derive from (4.5) the following identities:

φ(0, z) = Jθ(0, z), θ(0, z) = −Jφ(0, z),

φ∗(0, z) = −θ∗(0, z)J, θ∗(0, z) = φ∗(0, z)J.(4.8)

We impose a regular, self-adjoint boundary condition at c and d for solutions of(4.1), [

α1 α2

]Y (c) = 0n×1,

[β1 β2

]Y (d) = 0n×1, (4.9)

For z ∈ C \R, one attempts to satisfy the boundary condition (4.9) at d for thesolution χd(t, z) = θ(t, z)+φ(t, z)Md(z) with some (n×n) matrix Md(z). Insertingχd into the equation

[β1 β2

]Y (d) = 0 shows that

Md(z) = −( [β1 β2

]φ(d, z)

)−1( [β1 β2

]θ(d, z)

). (4.10)

A direct calculation shows that Md(z) satisfies the following circle equations:

±χd(d, z)∗(J/i)χd(d, z) = 0, where χd(d, z) = θ(d, z) + φ(d, z)Md(z). (4.11)

It can be shown, see [27, Section VII.3], that, as d approaches b, Md(z) approachesone of the matrices of the form

Mb(z) = Cb(z) +Rb(z)Ub(z)Rb(z), (4.12)

where we define

Cb(z) = − limd→b

(2 Im(z)

∫ d

0

φ∗Aφdt)−1(

2 Im(z)∫ d

0

φ∗Aθdt− iIn×n),

Rb(z) = limd→b

(2| Im(z)|

∫ d

0

φ∗Aφdt)−1/2

, z ∈ C \ R,(4.13)

so that Rb(z) = Rb(z), and where Ub(z) is a unitary matrix. It can be furthershown, see [27, Section VII.4], that if χb(t, z) = θ(t, z) + φ(t, z)Mb(z), then∫ b

0

χ∗b(t, z)A(t)χb(t, z)dt ≤(Mb(z)−M∗b (z)

)/(2i Im(z)), (4.14)

and thus by Atkinson’s condition in Hypothesis 4.1 (ii) one has

Im(Mb(z))/ Im(z) =(Mb(z)−M∗b (z)

)/(2i Im(z)) > 0, z ∈ C \ R. (4.15)

Similarly, one attempts to satisfy boundary conditions (4.9) at c for the solutionχc(t, z) = θ(t, z) + φ(t, z)Mc(z) with some n × n matrix Mc(z). Inserting χc intothe equation

[α1 α2

]Y (c) = 0 shows that

Mc(z) = −( [α1 α2

]φ(c, z)

)−1( [α1 α2

]θ(c, z)

). (4.16)

The circle equations, satisfied by Mc(z), are as follows:

∓χc(c, z)∗(J/i)χc(c, z) = 0, where χc(c, z) = θ(c, z) + φ(c, z)Mc(z). (4.17)

As c approaches a, Mc(z) approaches

Ma(z) = Ca(z) +Ra(z)Ua(z)Ra(z), (4.18)


where

Ca(z) = − limc→a

(2 Im(z)

∫ 0

c

φ∗Aφdt)−1(

2 Im(z)∫ 0

c

φ∗Aθdt+ iI),

Ra(z) = limc→a

(2| Im(z)|

∫ 0

c

φ∗Aφdt)−1/2

, z ∈ C \ R,(4.19)

so that Ra(z) = Ra(z), and where Ua(z) is a unitary matrix. If χa(t, z) = θ(t, z) +φ(t, z)Ma(z), then∫ 0

a

χ∗a(t)A(t)χa(t)dt ≤(M∗a (z)−Ma(z)

)/(2i Im(z)), (4.20)

and by Atkinson’s condition in Hypothesis 4.1 (ii)

Im(Ma(z))/(Im(z)) =(M∗a (z)−Ma(z)

)/(2i Im(z)) < 0, z ∈ C \ R. (4.21)

The matrix valued functions Ma(z) and Mb(z) are called the Weyl-TitchmarshM -functions.

We introduce the following spaces:

N(b, z) ={Y ∈ L2

A(0, b)∣∣JY ′(t) = (zA(t) +B(t))Y (t) a.e. on (0, b)

}, (4.22)

N(a, z) ={Y ∈ L2

A(a, 0)∣∣JY ′(t) = (zA(t) +B(t))Y (t) a.e. on (a, 0)

}. (4.23)

The Hamiltonian system (4.1) is said to be (see [27, Page 88], [23, Page 274], [28])in the limit point case at b (respectively, at a) whenever

dimC(N(b, z)) = n (respectively, dimC(N(a, z)) = n) for all z ∈ C\R, (4.24)

and in the limit circle case at b (respectively, at a) whenever

dimC(N(b, z)) = 2n (respectively, dimC(N(a, z)) = 2n) for all z ∈ C. (4.25)

Let us define the maximal operator Tmax in L2A(a, b) associated with (4.1):

Tmaxf =F,

f ∈ domTmax ={g ∈ L2

A(a, b)∣∣g ∈ ACloc((a, b); C2n) and there exists

F ∈ L2A(a, b) such that Jg′(t)−B(t)g(t) = A(t)F (t) for a.e. t ∈ (a, b)

}.

The minimal operator Tmin in L2A(a, b) associated with (4.1) is defined as the closure

of the operator T ′min given as follows:

T ′minf = F,

f ∈ domT ′min ={g ∈ domTmax

∣∣∣ g has compact support in (a, b)}.

(4.26)

We recall from [34, Theorem 4.1] that the deficiency indices of Tmin are given by

dim ker(Tmax − zIL2

A

)= dimC

(N(a, z)

)+ dimC

(N(b, z)

)− 2n, z ∈ C \ R. (4.27)

Also, we recall from [27, Section VII.8] Green’s formula associated with (4.1): IfF1, F2 ∈ domTmax then∫ b

a

(F ∗2 (JF ′1 −BF1)− (JF ′2 −BF2)∗F1

)dt

= F ∗2 JF1

∣∣∣ba

:= F ∗2 (b)JF1(b)− F ∗2 (a)JF1(a).

(4.28)


The following assumption holds provided that the entries of A, B are real valued.It is also holds for all examples discussed at the end of this section.

Hypothesis 4.2. In addition to Hypothesis 4.1 we assume that

dimC(N(a, z)

)= dimC

(N(a, z)

),

dimC(N(b, z)

)= dimC

(N(b, z)

), for all z ∈ C \ R.

(4.29)

Next, we fix some z ∈ C\R, and then fix any Ma(z) and Mb(z) in (4.18), (4.12).We introduce the operator T on L2

A(a, b) associated with (4.1) as follows:

Tf =F,

f ∈ domT ={g ∈ domTmax

∣∣∣Ba(g) := limc→a

χa(c, z)∗Jg(c) = 0, (4.30)

Bb(g) := limd→b

χb(d, z)∗Jg(d) = 0}. (4.31)

Theorem 4.3. [27, Theorem VIII.3.4] Assume Hypothesis 4.2. The operator Tdefined in (4.30) is self-adjoint.

Also, we introduce the self-adjoint operators Ta in L2A(a, 0) and Tb in L2

A(0, b)(see e.g., [27, Theorem VII.6.4]) by

Taf =F, (4.32a)

f ∈ domTa ={g ∈ domTmax

∣∣∣Ba(g) := limc→a

χa(c, z)∗Jg(c) = 0, (4.32b)[γ1 γ2

]g(0) = 0}. (4.32c)

Tbf =F, (4.33a)

f ∈ domTb ={g ∈ domTmax

∣∣∣ [γ1 γ2

]g(0) = 0, (4.33b)

Bb(g) := limd→b

χb(d, z)∗Jg(d) = 0}. (4.33c)

As shown in [27, Section VII.7], the domain of the operators, defined via the bound-ary conditions at a and b in (4.30), (4.31), (4.32b), (4.33c), is z-independent.

Remark 4.4. Using Green’s formula (4.28), the deficiency index theorems in [38,Sections 4–6] can be proved for the closed symmetric operator Tmin, cf. [34]. Also,as in Remark 2.6, if (4.1) is in lpc at a (respectively, at b) then the boundaryconditions in (4.30) and (4.32b) (respectively, in (4.31) and (4.33c)) can be omittedas they hold automatically for all g ∈ domTmax, see, e.g., [27, Section VI.10].

Since (see [38, Theorem 11.5])

σess(T ) = σess(Ta) ∪ σess(Tb), (4.34)

the following lemma holds.

Lemma 4.5. Assume Hypothesis 4.2. Then the Weyl-Titchmarsh M -functions Ma

and Mb are meromorphic in C\σess(T ).

Proof. Let Mb be the matrix-valued Weyl-Titchmarsh function. Consider the op-erator Tb defined in L2

A(0, b) by (4.33). Green’s function of Tb for z ∈ C \ σ(Tb) isgiven by the formula

G(t, s, z) =

{χb(t, z)φ∗(s, z), s < t,

φ(t, z)χ∗b(s, z), t < s,(4.35)


where χb(s, z) = θ(s, z) + φ(s, z)Mb(z) (see [27, Theorem VII.6.3]). Denoting thecolumns of φ by φj , for a < c < d < b and j = 1, . . . , n we define

φj(t, z) =

{φj(t, z), t ∈ (c, d),0, otherwise,

(4.36)

and introduce the matrix valued function

G(z) =∫ d

c

φ∗(t, z)A(t)φ(t, z)dt. (4.37)

Clearly, G(·) is analytic for z ∈ C. Due to Atkinson’s condition in Hypothesis4.1 (ii), 〈·, ·〉L2

A(c,d) is an inner product in the vector space of the solutions of (4.1).Therefore, G(z) is the Gram matrix of the linearly independent solutions φ1, . . . , φn.Consequently, G(z) is non-singular for any z ∈ C.

For z ∈ ρ(Tb) the resolvent operator (Tb − z)−1 satisfies:

〈(Tb − z)−1φj(·, z), φi(·, z)〉L2A(a,b) =

∫ b

a

φ∗i (t, z)A(t)∫ b

a

G(t, s, z)A(s)φj(s, z)dsdt

=∫ b

a

φ∗i (t, z)A(t)(φ(t, z)

∫ b

t

(θ∗(s, z) +M∗(z)φ∗(s, z)

)A(s)φj(s, z)ds

+(θ(t, z) + φ(t, z)M(z)

) ∫ t

a

φ∗(s, z)A(s)φj(s, z)ds)dt. (4.38)

Let R(z) denotes the (n× n) matrix whose entries Rij(z) are defined as follows:

Rij(z) = 〈(Tb − z)−1φj(·, z), φi(·, z)〉L2A(a,b)

−∫ b

a

φ∗i (t, z)A(t)(φ(t, z)

∫ b

t

θ∗(s, z)A(s)φj(s, z)ds

+ θ(t, z)∫ t

a

φ∗(s, z)A(s)φj(s, z)ds)dt.

(4.39)

Then (4.38) yields

Rij(z) =(∫ b

a

φ∗i (t, z)A(t)φ(t, z)dt)Mb(z)

(∫ b

a

φ∗(s, z)A(s)φj(s, z)ds)

=(

e∗i

∫ d

c

φ∗(t, z)A(t)φ(t, z)dt)Mb(z)

(∫ d

c

φ∗(s, z)A(s)φ(s, z)ds ej),

where ei are the standard unit vectors in Cn. In other words,

R(z) = G(z)Mb(z)G(z), (4.40)

where G(z) is defined in (4.37). It is clear from (4.40) that Mb(·) is meromorphicfor z ∈ C\σess(Tb) and, due to (4.34), for C\σess(T ) since R(z) is meromorphicand G(z) is analytic and nonsingular. Analogously, one can prove that Ma ismeromorphic for z ∈ C\σess(T ). �

Next, we prove an analog of Proposition 2.9. We stress again that z can be realin Proposition 4.6.

Proposition 4.6. Assume Hypothesis 4.1. If z ∈ C \ σess(Ta) then the followingassertions hold:


(i) If the Hamiltonian system (4.1) is in lpc at a then there exist exactly nlinearly independent solutions of (4.1) that are A-square integrable near a (that is,belonging to L2

A(a, 0));(ii) If the Hamiltonian system (4.1) is in lcc at a then there exist exactly n lin-

early independent solutions of (4.1) that are A-square integrable near a and satisfythe boundary condition at a in (4.32b).

Moreover, assuming that (4.1) is either in lpc or lcc at a, let z ∈ C \ σ(Ta)and let Fa(· , z) denote the (2n × n) matrix valued function whose columns arethe solutions described in part (i) or (ii). Then Fa(· , z) = χa(· , z)C(z), whereχa(· , z) = θ(· , z) + Ma(z)φ(· , z), C(z) is some constant matrix, and Ma(z) is theWeyl-Titchmarsh M -function used in (4.32b) to define the operator Ta. Analogousassertions hold for the point b.

Proof. We let µ denote the number of linearly independent solutions of (4.1) thatare A-square integrable near a and satisfy the boundary conditions at a in (4.32b)provided (4.1) is in lcc at a, and that are just A-square integrable near a provided(4.1) is in lpc at a. Also, we let ν denote the number of linearly independentsolutions that are A-square integrable near 0 and satisfy the boundary conditions at0 in (4.32c). Assuming z ∈ C\σ(Ta), we have µ+ν = 2n by [38, Theorem 7.1]. Since0 is a regular point, ν = n due to (4.32c) and (4.2). Thus µ = n proving (i) and (ii)for z ∈ C\σ(Ta). Furthermore, if (4.1) is in lpc at a then the columns of χa(· , z) =θ(· , z)+φ(· , z)Ma(z) are the solutions of (4.1) that are A-square integrable near a,and if (4.1) is in lcc at a then the columns of χa(· , z) = θ(· , z) + φ(· , z)Ma(z) arethe solutions of (4.1) that are A-square integrable near a and satisfy the boundaryconditions at a in (4.32c). Indeed, this holds because χa(· , z) enters the formulafor Green’s function, cf. (4.35), and Green’s function is composed of the A-squareintegrable near a solutions that satisfy the boundary conditions when appropriate(see, e.g., [38, Theorem 7.6]). This proves the representation Fa(· , z) = χa(· , z)C(z)for z ∈ C\σ(Ta).

Next, we assume that z ∈ σd(Ta), the discrete spectrum of Ta, and stress that zis real. We claim that µ ≥ n provided (4.1) is either in lpc or lcc at a. Starting theproof of the claim, let Fa(· , z) denote the matrix valued function whose columnsare the solutions Y (j)

a (· , z) of (4.1) with the properties described in assertions (i)or (ii). Seeking a contradiction, let us suppose that µ < n. Since µ is the rankof Fa(· , z), by multiplying Fa(· , z) from the right by a constant (n × n) matrix,we may and will assume that the first µ columns of Fa(· , z) are linearly indepen-dent and the remaining columns are zero. Using µ = rankFa(· , z) again, let ussuppose that the linearly independent rows of Fa(· , z) are located in the rows withthe numbers k1, . . . , kµ, where k1 ≥ · · · ≥ kµ. Let us consider the (n × 2n) ma-trix γ =

[ek1 . . . ekµ 02n×1 . . . 02n×1

]>, where ei are the standard unit columnvectors in C2n. If j = 1, . . . , µ, then the j-th row of the matrix γJ is equal to−e>kj+n provided kj ≤ n and is equal to e>kj−n provided kj > n, and the remainingrows are zero. It follows that γJγ∗ = 0, and thus γ =

[γ1 γ2

]satisfies (4.3). Since

the first µ rows of the matrix γFa(· , z) are linearly independent, each of the firstµ columns of this matrix is not zero. In other words, all columns of the product[γ1 γ2

]·[Y

(1)a (· , z) . . . Y

(µ)a (· , z)

]are nonzero. We denote by Ta the oper-

ator defined as in (4.32) but with γ1, γ2 replaced by γ1, γ2, and remark that thesolutions Y (1)

a (· , z), . . . , Y (µ)a (· , z) of (4.1) do not satisfy the boundary condition

THE EVANS FUNCTION AND THE WEYL-TITCHMARSH FUNCTION 23[γ1 γ2

]g(0) = 0. Thus, ker(Ta − z) = {0} and so z /∈ σd(Ta). Since Ta is yet

another self-adjoint extension of the operator Tmin, one has σess(Ta) = σess(Ta),and thus z /∈ σess(Ta) since z /∈ σess(Ta) by the assumption in the proposition.Therefore, z ∈ C \ σ(Ta). As we have seen in the previous paragraph, the latterinclusion implies µ = n, a contradiction which proves the claim.

It remains to prove that µ ≤ n. Let us first consider the case when (4.1) is inlpc at a (still assuming z ∈ σd(Ta)). In this case the proof is analogous to theproof of [38, Theorem 4.8]. Indeed, since Y (j)

a (· , z) are solutions of (4.1) and z isreal, we have Y (j)∗

a (t, z)JY (j)a (t, z) = const for each i, j = 1, . . . , µ. By Remark 4.4,

these constants are in fact equal to zero since Y (j)a ∈ domTmax and the boundary

condition (4.30) and (4.32b) at a are satisfied automatically for all functions fromdomTmax because (4.1) is in lpc at a. So, Y (i)∗

a (t, z)JY (j)a (t, z) = 0 for all i, j =

1, . . . , µ and t ∈ (a, 0). Fix any c ∈ (a, 0) and define the linear functionals on C2n byρi(e) = Y

(i)∗a (c, z)Je, i = 1, . . . , µ. They are linearly independent since Y (i)∗

a (· , z)are linearly independent. Then ∩j−1

i=1 ker ρi * ker ρj for each j = 1, . . . , µ. This andρi(Y

(j)a (c, z)) = 0 for all i, j = 1, . . . , µ imply µ ≤ dim∩µi=1 ker ρi ≤ 2n − µ, thus

proving µ ≤ n.It remains to prove that µ ≤ n in the case when (4.1) is in lcc at a (still assuming

z ∈ σd(Ta)). Let us assume that µ > n. Let Y (j)a , j = 1, . . . , µ, be the linearly

independent solutions of (4.1) that are A-square integrable near a and satisfy theboundary conditions in (4.32b) at a. Next, we introduce the functions Y (j)

a , j =1, . . . , µ, as follows:

Y (j)a (t, z) =

{Y

(j)a (t, z) for t close to a,

0 for t close to 0,(4.41)

and such that Y (j)a (· , z) ∈ domTa,max (the latter inclusion is possible as described

in [38, page 50]). Then Y(j)a ∈ domTa. Moreover, Y (j)

a are linearly independentmodulo domTa,min (see, e.g., [38, Theorem 5.4(a)]). We also know that ν = n. LetY

(j)0 , j = 1, . . . , n, be the linearly independent solutions of (4.1) that are A-square

integrable near 0 and satisfy the boundary conditions in (4.32c) at 0. Now, weconstruct the functions Y (j)

0 , j = 1, . . . , n, as follows:

Y(j)0 (t, z) =

{Y

(j)0 (t, z) for t close to 0,

0 for t close to a,(4.42)

and such that Y (j)a ∈ domTa. Moreover, Y (j)

0 are linearly independent modulodomTa,min. Thus, there are µ+n linearly independent modulo domTa,min elementsfrom domTa. This contradicts the fact that

dim(

domD(Ta)/ domD(Ta,min))

= 2n, (4.43)

which holds because the deficiency index of Ta,min is 2n since (4.1) is in lcc at a. �

Next, we describe the connections between the Weyl-Titschmarsh M -functions,the isolated eigenvalues of the respective differential operators, and the determi-nants of matrices composed of the solutions θ, φ, Fa and Fb; here and below Fa(· , z)and Fb(· , z) are the matrix valued functions whose columns are the solutions of (4.1)with the properties described in assertions (i) or (ii) of Proposition 4.6.


Lemma 4.7. Assume Hypothesis 4.1. If z0 ∈ C \ σess(Ta) and (4.1) is either inlpc or lcc at a then the following assertions are equivalent:

(i) det[Fa(t, z0) φ(t, z0)

]= 0 for some/all t ∈ (a, 0);

(ii) z0 ∈ σd(Ta);(iii) z0 is a pole of Ma(·).

Analogous facts hold for the point b. Finally, if z0 ∈ C \σess(T ) and (4.1) is eitherin lpc or lcc at either a or b then the following assertions are equivalent:

(iv) det[Fa(t, z0) Fb(t, z0)

]= 0 for some/all t ∈ (a, b);

(v) z0 ∈ σd(T ).

Proof. (i) ⇔ (ii): By Proposition 4.6, the columns of Fa(· , z0) span the space ofsolutions of (4.1) that are A-square integrable at a (when (4.1) is in lpc at a) andthat satisfy the boundary conditions at a in (2.15c) (when (4.1) is in lcc at a).Assertion (i) holds if and only if the columns of Fa(· , z0) and φ(· , z0) are linearlydependent. Since rankFa(· , z0) = rankφ(· , z0) = n, this holds if and only if there isa solution of (4.1) which is equal to a linear combination of the columns of Fa(· , z0)and is equal to a linear combination of the columns of φ(· , z0). Thus, this solutionsatisfies both boundary conditions in (4.32b), (4.32c), yielding equivalence to (ii).

(ii) ⇔ (iii): By (4.40), z0 is a pole of Ma(·) if and only if z0 is a pole of theresolvent (Ta − z)−1. Since z0 ∈ C \ σess(Ta), the latter is equivalent to (ii).

(iv)⇔ (v): The proof is similar to the proof of (i)⇔ (ii). �

Lemma 4.7 shows that the Evans function, D(z), for the Hamiltonian system(4.1) can be naturally defined in terms of Fa and Fb as follows:

D(z) = det[Fa(0, z) Fb(0, z)

], z ∈ C \ σess(T ), (4.44)

so that D(z0) = 0 if and only if z0 ∈ σd(T ). We will now express the Evans functionin terms of the Weyl-Titchmarsh M -functions. For this, we introduce an (n × n)matrix Wronskian, W(F,G), by denoting

W(F,G) = F ∗1G2 − F ∗2G1 = −F ∗JG (4.45)

for any 2n× n-matrices F =[F1 F2

]> and G =[G1 G2

]>, and remark that

W∗(F,G) =(− F ∗JG

)∗ = −W(G,F ). (4.46)

Also, we recall that Y(· , z) is the fundamental matrix solution of (4.1) satisfying(4.5), (4.6).

Theorem 4.8. Assume Hypothesis 4.28 and that (4.1) is either in lpc or lcc ateither a or b. Let Fa(· , z) and Fb(· , z) be the matrix valued functions whose columnsare the A-square integrable solutions of (4.1) described in assertions (i) or (ii) ofProposition 4.6. Then the following formulas hold:

Ma(z) =W(θ(0 , z), Fa(0 , z)

)W∗(Fa(0 , z), φ(0 , z)

)−1, z ∈ C \ σess(Ta), (4.47)

Mb(z) =W(θ(0 , z), Fb(0 , z)

)W∗(Fb(0 , z), φ(0 , z)

)−1, z ∈ C \ σess(Tb), (4.48)

Mb(z)−M∗a (z) =(W(Fa(0 , z), φ(0 , z)

))−1

W(Fa(0 , z), Fb(0 , z)

)×(W∗(Fb(0 , z), φ(0 , z)

))−1

, z ∈ C \ σess(T ). (4.49)


Moreover, the Evans function D(z) for (4.1) and the Weyl-Titchmarsh M -functionsare related as follows:

D(z) = detY(0, z) detW∗(Fa(0 , z), φ(0 , z)

)detW∗

(Fb(0 , z), φ(0 , z)

)× det

(Mb(z)−Ma(z)

). (4.50)

Proof. Let us prove (4.48), the proof of (4.47) is analogous. By Proposition 4.6,Fb(· , z) = χb(· , z)Cb(z) for z ∈ C\σ(Tb). Letting t = 0 and using (4.5) yield thesystem of equations for Mb(z) and Cb(z),(

γ∗1 − γ∗2Mb(z))Cb(z) = Fb1(0, z),

(γ∗2 + γ∗1Mb(z)

)Cb(z) = Fb2(0, z), (4.51)

where we subdivide Fb(0 , z) =[F>b1(0 , z) F>b2(0 , z)

]> into two (n× n) blocks. Inother words, using (4.6), (4.7), (4.8), (4.45),

Fb(0, z) = Y(0, z)[

Cb(z)Mb(z)Cb(z)

],[

Cb(z)Mb(z)Cb(z)

]= Y∗(0, z)Fb(0, z) =

[θ∗(0, z)Fb(0, z)φ∗(0, z)Fb(0, z)

]=[φ∗(0, z)JFb(0, z)−θ∗(0, z)JFb(0, z)

]=[−W

(φ(0, z), Fb(0, z)

)W(θ(0, z), Fb(0, z)

) ] . (4.52)

By Lemma 4.7, z ∈ σd(Tb) if and only if Mb has a pole at z or, equivalently,det[Fb(0 , z) φ(0 , z)

]= 0. Since, cf. (4.6), (4.8),

det(Y∗(0, z)

[Fb(0 , z) φ(0 , z)

] )= det

([θ∗(0, z)φ∗(0, z)

]·[Fb(0 , z) φ(0 , z)

] )= det

[θ∗(0, z)Fb(0, z) 0φ∗(0, z)Fb(0, z) In×n

]= det

[−W

(φ(0, z), Fb(0, z)

)0

φ∗(0, z)Fb(0, z) In×n

],

the matrix W∗(Fb(0 , z), φ(0 , z)

)is singular if and only if z ∈ σd(T ). Now (4.52),

(4.46) imply

Mb(z) =(γ1Fb2(0 , z)− γ2Fb1(0 , z)

)(γ1Fb1(0 , z) + γ2Fb2(0 , z)

)−1 (4.53)

=W(θ(0 , z), Fb(0 , z)

)(W∗(Fb(0 , z), φ(0 , z)

))−1,

Cb(z) = γ1Fb1(0 , z) + γ2Fb2(0 , z) =W∗(Fb(0 , z), φ(0 , z)

)(4.54)

= −W(φ(0 , z), Fb(0 , z)

), (4.55)

thus proving (4.48) for z ∈ C\σ(Tb) and therefore for z ∈ C\σess(Tb) as both sidesof (4.48) are not defined at z ∈ σd(Tb) by Lemma 4.7.

Using (4.53), its analogue for a, and (4.3), a short calculation yields (4.49).Formula (4.50) follows from formulae (4.51), (4.54), from Theorem 4.8, and fromthe following computation:[

Fa(0, z) Fb(0, z)]

= Y(0, z)[−W

(φ(0, z), Fa(0, z)

)−W

(φ(0, z), Fb(0, z)

)W(θ(0, z), Fa(0, z)

)W(θ(0, z), Fb(0, z)

) ]= −Y(0, z)

[I I

Ma(z) Mb(z)

] [W(φ(0 , z), Fa(0 , z)

)0

0 W(φ(0 , z), Fb(0 , z)

)]= −Y(0, z)

[I 0

Ma(z) Mb(z)−Ma(z)

]·[I I0 I

]


×[W(φ(0 , z), Fa(0 , z)

)0

0 W(φ(0 , z), Fb(0 , z)

)] .�

We conclude the section with several examples of Hamiltonian systems wherethe assumptions in Lemma 4.7 and Theorem 4.8 are satisfied.

Example 1. The Dirac-type system (see, e.g., [8]) is obtained by setting A(t) =I2n×2n in (4.1).

Lemma 4.9. [8, Lemma 2.15] The limit point case holds for Dirac-type systems ata = −∞ and b = +∞.

Example 2. The Schrodinger equation with matrix-valued potential,

−y′′(t) +Q(t)y(t) = k2y(t), −∞ < t <∞; (4.56)

for which we impose the following assumption.

Hypothesis 4.10. Assume that Q = Q∗ ∈ L1(R)n×n.

Equation (4.56) may be re-written as the system

JY ′(t) = (k2A(t) +B(t))Y (t), t ∈ (−∞,+∞), (4.57)

where

Y (t) =[y(t)y′(t)

], A(t) =

[In×n 0

0 0

], B(t) =

[−Q(t) 0

0 In×n

]. (4.58)

As it is well known (see the proof of Theorem 1.4.1 [2]), equation (4.56) has asystem of matrix valued solutions f+(·, k) and f+(·, k) satisfying the asymptoticboundary conditions

limt→∞

e−iktf+(t, k) = In×n, limt→∞

eiktf+(t, k) = In×n, Im(k) > 0. (4.59)

If F+(·, k) =[f+(·, k) f ′+(·, k)

]> and F+(·, k) =[f+(·, k) f ′+(·, k)

]>, then F (·, k)

and F (·, k) form a fundamental system of solutions of (4.57) satisfying asymptoticboundary conditions

limt→∞

e−iktF+(t, k) =[In×n ikIn×n

]>,

limt→∞

eiktF+(t, k) =[In×n −ikIn×n

]>, Im(k) > 0.

(4.60)

Lemma 4.11. Assume Hypothesis 4.10. Then equation (4.57) is in the limit pointcase at both −∞ and +∞.

Proof. Equation (4.57) is in the limit point case at +∞ since the columns of F+(·, k)are A-square integrable solutions of (4.57) while the columns of F+(·, k) are not A-square integrable solutions of (4.57) which is clear from (4.60) and the formula∫

F ∗(t, k)A(t)F (t, k)dt =∫f∗(t, k)f(t, k)dt, (4.61)

which holds for any solution F (·, k) =[f(·, k) f ′(·, k)

]> of (4.57). Similarly, itcan be proved that (4.57) is in the limit point case at −∞. �


Example 3. The Zakharov-Shabat problem (see, e.g., [1]) is given by

v′ =[−ik r(t)q(t) ik

]v, −∞ < t <∞, (4.62)

where we assume that the scalar functions r, q ∈ L1(R) satisfy r(t) = q(t) for allt ∈ (−∞,∞). Let us make the change of variables

u =1√2

[i i1 −1

]v. (4.63)

Then (4.62) becomes

Ju′ =(kI2×2 +

12

[i(q − r) r + qr + q i(r − q)

])u. (4.64)

Since r(t) = q(t),

Ju′(t) = (kI2×2 +B(t))u(t), where B = B∗ =12

[i(q − q∗) q∗ + qq∗ + q i(q∗ − q)

]. (4.65)

Since q ∈ L1(R) and r(t) = q(t), there exist four solutions f±(·, k), f±(·, k) of (4.62)(see [1, Chapter 1.3]) defined by the following boundary conditions:

limt→∞

f+(t, k)e−ikt =[0 1

]>, lim

t→−∞f−(t, k)eikt =

[1 0

]>, (4.66)

limt→∞

f+(t, k)eikt =[1 0

]>, lim

t→−∞f−(t, k)e−ikt =

[0 1

]>. (4.67)

Therefore, via (4.63) there exist four solutions F±(·, k), F±(·, k) of (4.65) definedby the following boundary conditions:

limt→∞

F+(t, k)e−ikt =1√2

[i −1

]>, limt→−∞

F−(t, k)eikt =1√2

[i −1

]>, (4.68)

limt→∞

F+(t, k)eikt =1√2

[i 1

]>, limt→−∞

F−(t, k)e−ikt =1√2

[i −1

]>. (4.69)

It is clear from (4.68) and (4.69) that (4.65) is in lpc at both −∞ and +∞.Example 4. The nearly complex Ginzburg-Landau equation (see [24]) is

ut −12uxx − u+ |u|2u = iε(

12d1uxx + d2u+ d3|u|2u+ d4|u|4u), (4.70)

where u = u(t, x), t ≥ 0, x ∈ R, ε > 0 is small, and the other parameters are realand of O(1) as ε → 0. Assuming ε = 0 and introducing v(t, x) = u(t, x), equation(4.70) can be rewritten as the system

ut −12uxx − u+ u2v = 0,

vt −12vxx − v + v2u = 0.

(4.71)

Linearizing (4.71) around the hole solution U(x) = tanhx yields

ut −12uxx − u+ 2U2(x)u+ U2(x)v = 0,

vt −12vxx − v + U2(x)u+ 2U2(x)v = 0,

(4.72)


for u = u(x), v = v(x), which induces the eigenvalue problem for z ∈ C\R+,

− 12uxx − (1− 2U2(x))u+ U2(x)v = zu,

− 12vxx − (1− 2U2(x))v + U2(x)u = zv.

(4.73)

Letting Y (x) =[u(x) v(x) u′(x) v′(x)

]>, we rewrite (4.73) as the first ordersystem

∂xY (x) = M(x, z)Y (x), (4.74)where

M(x, z) =

0 0 1 00 0 0 1

2(−z − 1 + 2U2(x)) 2U2(x) 0 02U2(x) 2(−z − 1 + 2U2(x)) 0 0

. (4.75)

Note that limx→±∞M(x, z) = M∞(z) exponentially fast, where

M∞(z) =

0 0 1 00 0 0 1

2(−z + 1) 2 0 02 2(−z + 1) 0 0

. (4.76)

The eigenvalues of M(z) are µ1,2 = ±i√

2z, µ3,4 = ±i√

2z + 4 and thus nonimagi-nary if and only if z ∈ C\R+. Since M(·, z)−M∞(z) ∈ L1(R)4×4, by the standardasymptotic theory (see, e.g., [11, Chapter I] or [9, Chapter 3, Problem 29]), thereexist four solutions Y (j), j = 1, . . . , 4, of (4.74) defined by the boundary conditions

limx→+∞

Y (j)(x, z)e−µjx = pi, Re(µj) 6= 0, z ∈ C\R+, (4.77)

where pj is the eigenvector of M∞(z) corresponding to the eigenvalue µj . We canrewrite (4.74) as

J(∂xY )(x) = (zA+B(x))Y (x), (4.78)where

A =

2 0 0 00 2 0 00 0 0 00 0 0 0

, B =

2(1− 2U2) −2U2 0 0−2U2 2(1− 2U2) 0 0

0 0 1 00 0 0 1

, J =[

0 −I2I2 0

].

It is clear from (4.77) that (4.78) is in lpc at +∞. Similarly, it can be shown that(4.78) is in lpc at −∞.

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Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

E-mail address: [email protected]

URL: http://www.math.missouri.edu/personnel/faculty/latushkiny.html

E-mail address: [email protected]

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