LOWER BOUND ON THE SPECTRUM OF THE TWO-DIMENSIONAL

Theoretical and Mathematical Physics, 162(3): 332–340 (2010)

LOWER BOUND ON THE SPECTRUM OF THE TWO-DIMENSIONAL ¨ SCHRODINGER OPERATOR WITH A δ-PERTURBATION ON A CURVE c I. S. Lobanov,∗ V. Yu. Lotoreichik,∗ and I. Yu. Popov∗


We consider the two-dimensional Schr¨ odinger operator with a δ-potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the considered operator that are expressed explicitly in terms of the interaction strength and a parameter characterizing the curve geometry. We estimate the bottom of the spectrum for a piecewise smooth curve using parameters characterizing the geometry of the separate pieces. As applications of the obtained results, we consider curves with a finite number of cusps and general “leaky” quantum graph as the support of the δ-potential.

Keywords: Schr¨ odinger operator, singular potential, spectral estimate, Birman–Schwinger transformation

1. Introduction We consider the operator Hα,Γ = −∆ − αδ(· − Γ),

dom Hα,Γ ⊂ L2 (R2 ),

with an attractive δ-potential of strength α > 0 supported by a curve Γ. The operator Hα,Γ is the model Hamiltonian of an attractive quantum wire. Such operators have been studied during the last two decades; a review of the latest results in the theory of such operators can be found in [1]. The operator Hα,Γ is rigorously defined by the quadratic form (see [2]) qα,Γ [u] = ∇u2L2 (R2 ) − αu2L2 (Γ) ,

dom qα,Γ = W 1,2 (R2 ).

It is well known that if Γ is a straight line, then Hα,Γ has no eigenvalues, and its spectrum (under the assumption that α is positive constant) is 
α2 σ(Hα,Γ ) = σess (Hα,Γ ) = − , +∞ . 4

(1.1)

It was shown in [3] that for a small deformation of the curve Γ preserving some smoothness, the essential spectrum does not change, but at least one eigenvalue appears below the essential spectrum. The essential spectrum of the Schr¨odinger operator with a δ-potential supported by a closed curve in accordance with [2] is σess (Hα,Γ ) = [0, +∞), (1.2) ∗

St. Petersburg State University of Information Technologies, Mechanics, and Optics, St. Petersburg, Russia, e-mail: [email protected], [email protected], [email protected]. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 162, No. 3, pp. 397–407, March, 2010. Original article submitted August 1, 2009. 332

0040-5779/10/1623-0332

Moreover, there exists at least one negative eigenvalue. Here, we obtain lower bounds for the spectrum of the operator Hα,Γ in the cases of infinite and closed finite curves. Let α ∈ R+ and Γ be a continuous piecewise-C 1 curve in R2 . We need the geometric characteristic of the curve Γ distR2 (x, y) , (1.3) c(Γ) = inf x,y∈Γ distΓ (x, y) where distR2 is the distance between points on the plane and distΓ is the distance between the points along the curve. We note that Γ has no cusps or self-crossings if c(Γ) > 0 and that its possible asymptotic lines are not parallel if Γ is infinite. Theorem 1. Let Γ be a closed curve of finite length and c(Γ) > 0. Then inf σ(Hα,Γ ) ≥ −

α2 . 4(c(Γ))2

(1.4)

In particular, we have c(Γ0 ) = 2/π for the circle Γ0 for simple geometric reasons, and then inf σ(Hα,Γ0 ) ≥ −

α2 π 2 . 16

Theorem 2. Let Γ be an infinite curve and c(Γ) > 0. Then 1. the estimate of the discrete spectrum inf σd (Hα,Γ ) ≥ −

α2 4(c(Γ))2

(1.5)

holds in the most general case and 2. if Γ satisfies condition (2.9) below, which, in particular, requires that asymptotic lines exist, then inf σ(Hα,Γ ) ≥ −

α2 . 4(c(Γ))2

(1.6)

Estimate (1.6) is exact if Γ is a straight line. For two rays forming an angle Γβ of magnitude β, simple geometric reasoning gives c(Γβ ) = sin(β/2), and the estimate inf σ(Hα,Γβ ) ≥ −α2 /(4 sin2 (β/2)) consequently becomes very rough as β → 0, as is seen below. But estimates (1.4) and (1.6) can be improved if Γ is a composition of curves with positive parameters c determined by (1.3). The composition of the curves Γi is a curve Γ that can be divided by a finite number of points on it into the pieces Γi . At the same time, we allow that the characteristic c(Γ) can be zero; this case is not described by the results in [2]. Theorem 3. Let the curve Γ be a composition of curves Γ1 , . . . , ΓN , N < ∞, such that each Γi has a characteristic c(Γi ) > 0. Then the quadratic form qα,Γ is semibounded below, is closed, and corresponds to the self-adjoint operator Hα,Γ . Moreover, if each Γi has a continuous piecewise-C 1 extension Γext ⊃ Γi with i ext ext the parameter c(Γi ) > 0 and, moreover, if Γi is a closed finite or infinite curve satisfying condition (2.9), then the estimate N  α2 inf σ(Hα,Γ ) ≥ −N (1.7) 2 4(c(Γext i )) i=1 holds. 333

Consequently, for the abovementioned angle Γβ , we obtain inf σ(Hα,Γβ ) ≥ −α2 . Indeed, it suffices to divide Γβ into two rays, extend each ray to a straight line, and then use Theorem 3. The result of Theorem 3 can be applied to curves with a finite number of cusps. It can be proved that the operator Hα,Γ is also self-adjoint in this case, which allows extending the class of curves for which a δ-potential can be considered beyond the class described in [2]. In what follows, we outline the necessary theoretical background and then consecutively prove the theorems formulated in the introduction. In the last section, we apply Theorem 3 to curves with cusps and generalize it to arbitrary “leaky” quantum graphs.

2. Theoretical background In this section, we present necessary facts concerning the Schr¨ odinger operator with the δ-potential on a curve. Proofs of the statements in this section can be found in [2], [3]. We consider the Dirac measure mΓ related to the curve Γ and determined by mΓ (M ) = l1 (M ∩ Γ)

(2.1)

for each Borel set M ⊂ R2 , where l1 is the one-dimensional Hausdorf measure; in our case, l1 is the length of the part of the curve Γ that is inside M . Accordance to [2], for α > 0 and c(Γ) > 0, the relation    (1 + α) |ψ(x)|2 dmΓ (x) ≤ a |∇ψ(x)|2 dx + b |ψ(x)|2 dx (2.2) R2

R2

R2

holds for all ψ ∈ S(R2 ) and some a < 1 and b. Here, S(R2 ) is the Schwartz class of rapidly decaying functions in R2 . The map IΓ : S(R2 ) → L2 (Γ) defined as IΓ ψ = ψ can be uniquely extended to the continuous map (2.3) IΓ : W 1,2 (R2 ) → L2 (Γ) := L2 (R2 , mΓ ). We use the same symbol ψ for a function in W 1,2 (Γ) and for its trace on the curve Γ. Relation (2.2) can be extended to W 1,2 (R2 ) by replacing ψ in the left-hand side with IΓ ψ. The operator we consider is given by the quadratic form   2 qα,Γ [ψ] := |∇ψ(x)| dx − α |IΓ ψ(x)|2 dmΓ (x) (2.4) R2

R2

with the domain dom qα,Γ = W 1,2 (R2 ). According to [2], this form with the core C0∞ (R2 ) is closed and bounded under condition (2.2), which implies that it corresponds to a unique self-adjoint operator, denoted here by Hα,Γ . We can apply the Birman–Schwinger technique to the operator Hα,Γ corresponding to quadratic form (2.4) and obtain the formula for the resolvent. We let R0k denote the resolvent of the Schr¨ odinger operator without potential in L2 (R2 ), where Im k > 0 and (−∆ − k 2 )−1 = R0k . The resolvent R0k is an integral operator with the kernel i (1) Gk (x − y) = H0 (k|x − y|). (2.5) 4 Let µ and ν be arbitrary positive Radon measures in R2 satisfying the condition µ(x) = ν(x) = 0 for k denote the integral operator from L2 (µ) := L2 (R2 , µ) to L2 (ν) with the kernel any point x ∈ R2 . Let Rν,µ Gk , i.e., k Rν,µ φ = Gk ∗ φµ (2.6) 334

k holds almost everywhere in respect to the measure ν for all φ ∈ dom Rν,µ ⊂ L2 (µ). There are two measures, the previously introduced measure mΓ and the Lebesque measure dx in R2 . According to [2], we can formulate the following proposition using this notation. iκ Proposition. 1. There exists κ0 > 0 such that the operator I − αRm on L2 (Γ) has a bounded Γ ,mΓ inverse for all κ > κ0 . k 2. Let Im k > 0, the operator I − αRm be invertible, and the operator Γ ,mΓ k k k [I − αRm ]−1 Rm Rk := R0k + αRdx,m Γ Γ ,mΓ Γ ,dx

(2.7)

from L2 (R2 ) to L2 (R2 ) be defined everywhere. Then k 2 ∈ ρ(Hα,Γ ) and (Hα,Γ − k 2 )−1 = Rk . k 3. For all k such that Im k > 0, dim ker(Hα,Γ − k 2 ) = dim ker(I − αRm ). Γ ,mΓ We now recall some known results for the spectrum of Hα,Γ in the case of closed finite or infinite curves. Let the curve Γ be determined by a continuous piecewise-C 1 map γ = (γx , γy ) either from R to R2 in the case of an infinite curve or from [0, L] to R2 with γ(L) = γ(0) in the case of a closed curve of finite length L. We assume that the curve has a natural parameterization, i.e., |γ(s)| ˙ = 1. Because γ is continuous and has a natural parameterization, it is obvious that the inequality |γ(s) − γ(s )| ≤ |s − s |

(2.8)

holds for all s, s ∈ dom γ. Further, we impose some conditions, suggested in [3], on the curve. I. The constant c(Γ) is strictly positive. In particular, we have |γ(s) − γ(s )| ≥ c(Γ)|s − s | for an infinite curve and |γ(s) − γ(s )| ≥ c(Γ) min{|s − s |, L − |s − s |} for finite closed curve. II. This condition is reasonable only for infinite curves. We assume that Γ is asymptotically straight in the sense that there exist positive constants d, µ > 1/2 and ω ∈ (0, 1) such that the inequality 1−

|γ(s) − γ(s )| ≤ d[1 + |s + s |2µ ]−1/2 |s − s |

(2.9)

holds in the sector Sω = {(s, s ) : ω ≤ s/s ≤ ω −1 }. We can now formulate some results in [2] and [3] that were given informally in the introduction. Theorem 4. Let Γ be a continuous piecewise-C 1 curve and Condition I be satisfied. Then 1. if Γ is a closed finite curve, then σess (Hα,Γ ) = [0, +∞),

(2.10)

and there exists at least one negative eigenvalue; 2. if Γ is an infinite curve and Condition II is satisfied, then 
α2 σess (Hα,Γ ) = − , +∞ . 4

(2.11)

Moreover, if Γ is not straight, then there exists at least one eigenvalue below the essential spectrum. 335

3. Proof of Theorem 1 iκ κ We introduce the integral operator Rκα,Γ := αRm in the space L2 [0, L], where Rν,µ is defined Γ ,mΓ 2 by (2.6). According to the proposition (statement 3), a negative number λ = −κ is an eigenvalue of the operator Hα,Γ if and only if the integral equation

Rκα,Γ ψ = ψ

(3.1)

has a nontrivial solution. The integral kernel Rκα,Γ has the form α K0 (κ|γ(s) − γ(s )|), 2π

Rκα,Γ (s, s ) =

s, s ∈ [0, L],

(3.2)

(1)

where K0 is the McDonald function. We recall that K0 (z) = (πi/2)H0 (iz) (see [4]). The condition Rκα,Γ  < 1 suffices only for Eq. (3.1) to have a trivial solution. It follows from Condition I and the monotonic decrease of the McDonald function that 0 ≤ K0 (κ|γ(s) − γ(s )|) ≤ K0 (κc(Γ) distΓ (γ(s), γ(s ))).

(3.3)

We introduce the operator Rκc in L2 (0, L) with the kernel α K0 (κc(Γ) distΓ (γ(s), γ(s ))) = Rκc (s, s ) = 2π α K0 (κc(Γ) min{|s − s |, L − |s − s |}). = 2π By (3.3), Rκα,Γ  ≤ Rκc .

(3.4)

Because the integral kernel of the operator Rκc is positive, we can use Schur’s lemma (see Theorem 5.2 in [5]) to estimate the norm from above:   L  L κ Rc  ≤ sup Rκc (s, s ) ds · sup Rκc (s, s ) ds. (3.5) s∈[0,L]

s
∈[0,L]

0

0

Rκc

ia a difference kernel, the supremums are equal. Let s < L/2. Then Because the operator  L  L α κ   Rc (s, s ) ds = K0 (κc(Γ) min{|s − s |, L − |s − s |}) ds = 2π 0 0  s α = K0 (κc(Γ) min{|t|, L − |t|}) dt = 2π s−L α = 2π α = 2π =

α 2π



−L/2

 K0 (κc(Γ)(L − |t|)) dt +

s−L





L/2

K0 (κc(Γ)t) dt + s



L/2

−L/2

K0 (κc(Γ)|t|) dt
L/2. Hence, Rκα,Γ  < α/(2c(Γ)κ), and if κ > α/(2c(Γ)), then the right-hand side of the estimate is less than unity. Therefore, if λ < −α2 /(4(c(Γ))2 ), then λ cannot be an eigenvalue. By statement 1 in Theorem 4, we obtain inf σ(Hα,Γ ) ≥ − 336

α2 . 4(c(Γ))2

4. Proof of Theorem 2 In the spirit of the proof of the preceding theorem, we introduce the integral operator Rκα,Γ := κ L2 (R), where Rν,µ is defined by (2.6). According to the proposition (statement 3), a nega2 tive number λ = −κ is an eigenvalue of the operator Hα,Γ if and only if the integral equation iκ αRm in Γ ,mΓ

Rκα,Γ ψ = ψ

(4.1)

has a nontrivial solution. The integral kernel of the operator Rκα,Γ has the form Rκα,Γ (s, s ) =

α K0 (κ|γ(s) − γ(s )|), 2π

s, s ∈ R.

(4.2)

The condition Rκα,Γ  < 1 suffices only for Eq. (3.1) to have a trivial solution. It follows from Condition I and the monotonic decrease of the McDonald function that 0 ≤ K0 (κ|γ(s) − γ(s )|) ≤ K0 (κc(Γ)|s − s |).

(4.3)

We introduce the operator Rκc with the integral kernel Rκc (s, s ) = (α/(2π))K0 (κc(Γ)|s − s |). By (4.3), Rκα,Γ  ≤ Rκc .

(4.4)

We can improve the estimate in the case of an infinite curve. We calculate the norm Rκc exactly. Let F : L2 (R) → L2 (R) be the Fourier transformation on the real axis (see [6]) defined by the formula  1 f (ξ)e−iξp dξ fˆ(p) = (F f )(p) = √ 2π R with the inverse map 1 fˇ(p) = (F −1 f )(p) = √ 2π

 f (ξ)eiξp dξ. R

 κ := F Rκ F −1 (momentum representation). The norm of the introduced Further, we consider the operator R c c operator coincides with the norm Rκc because the Fourier transformation is unitary. Moreover, α  F K0 (κc|ξ|) ∗ (F −1 f ) = 2π  α α = √ F K0 (κc(Γ)|ξ|) · F(F −1 f ) =

f. 2π 2 (c(Γ))2 κ2 + p2

F Rκc F −1 f =

√ In calculating, we use the formula F (f ∗ g) = 2π fˆgˆ and the explicit form of the Fourier transform of the  κ is determined as McDonald function (see [6]). The operator R c  κ : L2 (R) → L2 (R), R c and then

α κf =

R f (p), c 2 (c(Γ))2 κ2 + p2

(4.5)

α α  κc  = sup

= R . 2 2 2 2c(Γ)κ p∈R 2 (c(Γ)) κ + p

This norm is less than unity if κ > α/(2c(Γ)). Consequently, if λ < −α2 /(4c(Γ)2 ), then this number cannot be an eigenvalue of Hα,Γ , and statement 1 in Theorem 2 is proved. To prove statement 2, we use statement 2 in Theorem 4 and obtain α2 inf σ(Hα,Γ ) ≥ − . 4(c(Γ))2 337

5. Proof of Theorem 3 We first prove that the operator Hα,Γ is self-adjoint. For each piece Γi of the initial curve Γ, we have c(Γi ) > 0. We can hence write the inequality of type (2.2) N (1 + α)u2L2 (Γi ) ≤ ai ∇u2L2 (R2 ) + bi u2L2 (R2 ) ,

(5.1)

where 0 < ai < 1 and b > 0. Here, we use the validity of inequality (2.2) for each α ∈ R. Summing the inequalities over all pieces Γi and dividing by N , we obtain N (1 + α)uL2 (Γ) ≤

i=1

N

ai

N ∇u2L2 (R2 )

+

i=1 bi

N

u2L2 (R2 ) .

N According to [2], because 0 < i=1 ai /N < 1, the form qα,Γ with the domain W 1,2 (R2 ) is bounded below and closed. The form qα,Γ generates a unique self-adjoint operator Hα,Γ . Further, we prove estimate (1.7). We consider quadratic forms for each piece Γi of the initial curve Γ: qαN,Γi [u] = ∇u2 − αN u2L2 (Γi ) ,

dom qαN,Γi = W 1,2 (R2 ).

(5.2)

We consider quadratic forms for each extension Γext i : qαN,Γext [u] = ∇u2 − αN u2L2 (Γext ) , i i

dom qαN,Γext = W 1,2 (R2 ). i

Because Γext ⊃ Γi , i [u] qαN,Γi [u] ≥ qαN,Γext i for each u ∈ W 1,2 (R2 ). The quadratic form of the Schr¨ odinger operator with a δ-potential of strength α supported by the whole curve Γ can be expressed as qα,Γ [u] =

N N 1  1  ext [u]. qαN,Γi [u] ≥ q N i=1 N i=1 αN,Γi

(5.3)

The lower bound of the form qαN,Γext coincides with inf σ(HαN,Γext ). If Γext is a closed finite curve, i i i then we use Theorem 1; if it is infinite, then we use statement 2 in Theorem 2. Because inf σ(Hα,Γ ) = inf u∈W 1,2 (R2 ) qα,Γ [u], N  α2 inf σ(Hα,Γ ) ≥ −N . (5.4) 2 4(c(Γext i )) i=1 The theorem is proved. Remark. The lower estimate for the spectrum is approximately N times rougher than the spectrum bottom. Here, N is the number of pieces of the initial curve. Indeed, if we cut a straight line Γ into N −2 segments and two rays and extend them to the whole line, then according to Theorem 3, we obtain inf σ(Hα,Γ ) ≥ −

α2 N . 4

It is interesting to apply Theorem 3 to pathological cases where other ways of proving the self-adjointness and spectral lower estimates are difficult to use, in particular, to the cases where c(Γ) = 0 and Theorems 1 and 2 cannot be applied. 338

6. Applications and possible generalizations 6.1. Application to curves with cusps. We first define a cusp (see [7]). Definition. For a curve defined as the geometric locus of points for which a function F (x, y) ∈ C ∞ of two variables vanishes, the point (x0 , y0 ) is called a cusp (cusp point) if 1. F (x0 , y0 ) = 0, 2.

∂F ∂F (x0 , y0 ) = (x0 , y0 ) = 0, and ∂x ∂y

3. the Hessian matrix has a zero determinant at this point, i.e., 2 ∂ F ∂ 2 F ∂x2 ∂x ∂y (x , y ) = 0. 2 ∂ F ∂ 2 F 0 0 ∂x ∂y ∂y 2 Examples of cusps on rational curves are the spinode on γ(t) = (t2 , t3 ), the rhamphoid on γ(t) = (t2 , t5 ), and also the cusp on γ(t) = γm,n (t) := (t2n , t2m+1 ), where m ≥ n. All the cusps on these curves are at the origin. This can be easily verified by constructing appropriate functions F (x, y) satisfying the defining conditions. We show that in the case of the spinode, c(Γ) = 0:

(t20 − t20 )2 + (t30 + t30 )2 2t30 = lim c(Γ) ≤ lim

t0 √  = 2 + 9t4 dt t0 →0+ t0 →0+ 2 3/2 − 1 4t (16/27) 1 + (9/4)t −t0 0 2t30 2 = 0. t0 →0+ 2t2 0 + o(t0 )

= lim

This calculation shows that when considering curves with cusps, we cannot use Theorems 1 and 2 (they do not give any information about the spectrum). We consider a finite closed curve Γ of length L without self-intersections. We assume that Γ is determined by the map γ(t) and has a finite number of cusps at the points 0 ≤ t1 < t2 < · · · < tN ≤ L. We add the points t0 = 0 and tN +1 = L to this sequence. We cut the curve Γ into pieces Γi = γ([ti , ti+1 ]), i = 0, . . . , N . If each piece is such that c(Γi ) > 0, then we can use Theorem 3 to prove that Hα,Γ is ⊃ Γi such that c(Γext self-adjoint, and if there exist extensions Γext i i ) > 0, then we can obtain the lower bound for the spectrum. We can also reason analogously for an infinite curve. 6.2. Generalization to arbitrary “leaky” quantum graphs. The result of Theorem 3 can be generalized to arbitrary “leaky” quantum graphs. We recall that the model Hamiltonian of a “leaky” quantum graph is defined as the self-adjoint operator Hα,Γ corresponding to the quadratic form qα,Γ (see Eq. (2.4)), where Γ is a metric graph (a composition of segments intersecting only in the points of common ends). We assume that the graph is finite, i.e., it consists of a finite number of segments and that all infinite parts are asymptotically straight. Then Theorem 3 can be reformulated as follows. Theorem 5. Let each segment Γi of the metric graph Γ be such that c(Γi ) > 0. Then the quadratic form qα,Γ is bounded from below, is closed, and has a unique corresponding self-adjoint operator Hα,Γ . If it is possible to construct continuous and piecewise-C 1 extensions Γext ⊃ Γi , where each Γext is either an i i ext infinite curve satisfying inequality (2.9) or a closed finite curve, such that c(Γi ) > 0, then the estimate inf σ(Hα,Γ ) ≥ −N

N  i=1

α2 2 4(c(Γext i ))

(6.1)

holds, where N is the number of edges of the graph. 339

Acknowledgments. This work is supported by the program ”Development of the potential of higher schools of Russia 2009–2010” (Grant No. 2.1.1/4215) and in part (V. Yu. L.) by the St. Petersburg government (Personal Grant No. 2.1/30-04/035).

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