The aim of this study is to establish the existence and describe the asymptotic behavior of negative eigenvalues of one-dimensional Schrödinger operators that serve as regularizations of formal Hamiltonians involving δ and δ′ potentials. These questions arise in the construction and analysis of exactly solvable models in quantum mechanics, a topic that continues to draw considerable attention in the literature (see Albeverio et al. [1] and Albeverio and Kurasov [2], as well as comprehensive reference lists therein, covering works up to the early 2000s).

Some point interactions (i.e., pseudopotentials supported on discrete sets) naturally lead to well-defined, exactly solvable models; others, however, exhibit essential ambiguities in defining the corresponding Hamiltonians. A notable example of this contrast is provided by the δ and δ′ potentials. In the case of the δ potential, the differential equation −y″+αδ(x)y = λy is well-posed in the space of distributions D′(ℝ) and has a two-dimensional solution space. In contrast, the equation −y″+αδ′(x)y = λy is ill-posed in D′(ℝ) and admits only the trivial solution when α ≠ 0. Moreover, while every reasonable regularization of Hamiltonians involving the δ potential yields the same exactly solvable model, the δ′ potential is sensitive to the regularization procedure, and different approximations may lead to different point interactions. As a result, the choice of the exactly solvable model for δ′ potentials is not determined by mathematical considerations alone. However, it must reflect the specifics of the particular physical experiment—a feature that only enriches the study of exactly solvable models.

In this study, we demonstrate a further distinction between the δ and δ′ potentials, this time concerning the spectral properties of their regularized Hamiltonians. Natural approximations of both the δ and δ′ potentials by regular potentials yield operator families that converge in the norm resolvent topology to semi-bounded limits. However, we show that in contrast to δ-like perturbations, δ′-like perturbations lead to operator families that are not uniformly bounded from below as the regularization parameter tends to zero. As a consequence, such regularized Hamiltonians can possess a finite (but arbitrarily large) number of low-lying eigenvalues that diverge to negative infinity. We explicitly determine the number of these eigenvalues and describe their asymptotic behavior in the singular limit.

The rest of the article is organized as follows. Section 2 gives a brief overview of studies on exactly solvable models for Hamiltonians with δ and δ′ potentials. In Section 3, we derive conditions under which Schrödinger operators with δ′-like potentials possess low-lying eigenvalues that diverge to negative infinity as the regularization parameter tends to zero. Section 4 introduces methods for estimating the number of such eigenvalues and shows, in particular, that the emergence of a discrete spectrum is closely related to zero-energy resonances for the corresponding Schrödinger operators. In Section 5, we investigate how the non-trivial interaction of δ-like and δ′-like perturbations leads to the emergence of a negative eigenvalue with a finite limit as the perturbation parameter tends to zero. Finally, Section 6 contains the proofs of Theorems 3 and 4 on the asymptotic behavior of eigenvalues.

In this section, we review existing approaches to constructing exactly solvable quantum mechanical models for one-dimensional Hamiltonians with pseudopotentials involving the Dirac δ-function and its derivative δ′.

The simplest case is the formal (pseudo-)Hamiltonian

Any reasonable method of associating a self-adjoint Hamiltonian to Equation 2.1—such as form-sum, generalized sum method, approximation by regular potentials—yields the same operator H, acting as Hy = −y″ on the domain

In other words, the distributional potential αδ(x) in the one-dimensional Schrödinger operator results in the point interaction imposing the interface condition

Moreover, this model serves as a good approximation in the norm resolvent sense of the Schrödinger operators with integrable potentials of special form. Specifically, given a real-valued function U of compact support such that

the scaled potentials Uε(x)=ε-1U(ε-1x) converge in the space of distributions D′ to the distribution αδ(x) as ε → 0, and the corresponding operators

converge to H in the norm resolvent sense [1, Theorem I.3.2.3], i.e., their resolvents converge in operator norm to the resolvent of H. Similar convergence results hold even in the presence of background potentials W, i.e., for operators of the form

One should not expect that every pseudopotential gives rise to a unique point interaction. Certain pseudopotentials are highly sensitive to the way they are approximated, and the δ′ potential is one of them. In physics, the symbol δ′ is often used to describe a strongly localized dipole-type potential, such as a high narrow barrier followed by a deep well. Let V be an integrable function with compact support and a finite first moment; then, the sequence ε⁻²V(ε⁻¹x) converges in D′, as ε → 0, if and only if ∫ℝV(x)dx=0, and in that case

where β=-∫ℝxV(x)dx. For this reason, we refer to such families of scaled potentials as δ′-like.

The question of how to correctly define the formal Hamiltonian

has a long and intricate history. As mentioned earlier, difficulties arise already at the level of interpreting the differential expression in Equation 2.4, since the equation −y″ + βδ′(x)y = λy admits only the trivial solution in the space of distributions D′. Indeed, the product δ′(x)ϕ(x) is well defined in D′ only if ϕ is continuously differentiable, and in that case, it is equal to the distribution ϕ(0)δ′(x) − ϕ′(0)δ(x). However, any non-trivial solution y of the above equation would have to be discontinuous at the origin since its second derivative y″ = βδ′(x)y − λy would necessarily include a δ′ term. In this case, the product δ′(x)y(x) is not defined in D′, making the equation invalid.

Moreover, the operator in Equation 2.4 cannot be rigorously defined using standard approaches such as the form-sum or generalized sum methods, or as a relatively bounded perturbation of the free Hamiltonian. For this reason, it is natural to approach this problem via regularization: one considers families of Schrödinger operators of the form

with δ′-like potentials ε⁻²V(ε⁻¹x) as a starting point for studying physical phenomena associated with zero-range dipoles. The construction of exactly solvable models for such dipole interactions is thus reduced to analyzing the limits of H_ε as ε → 0.

It has been shown that the operator family H_ε indeed converges in the norm resolvent sense as ε → 0. In a seminal paper, Šeba [3] argued that the limiting operator is the free Hamiltonian D₀ decoupled at the origin by the Dirichlet condition, namely

According to this result, no meaningful definition of a Schrödinger operator with a δ′-potential would be possible since the limit D₀ is completely impenetrable to a quantum particle and is independent of the specific form of the function V. However, this conclusion contradicts the findings of Zolotaryuk et al. [4–7], who analyzed transmission probabilities through piecewise constant δ′-like potentials and observed examples of quantum tunneling. These results prompted the revision of Šeba [3]; it was later rigorously proved in Golovaty and Hryniv [8] that the operator D₀ is the norm resolvent limit of H_ε only in the so-called non-resonant case, while in the resonant case, the situation is different.

We begin by recalling the relevant definitions [9–11]. The operator -d2dx2+V is said to have a zero-energy resonance if the equation v″ = Vv admits a non-trivial solution v that is bounded on the entire real line. Such a solution is called a half-bound state, and the potential V is then referred to as resonant. Every half-bound state v has finite, non-zero limits v_± at ±∞, and the ratio

is uniquely determined by V. As proved in [8] (see also Golovaty and Man'ko [12] and Golovaty et al. [13]), if the potential V is resonant, then the family H_ε converges in the norm resolvent sense as ε → 0 to the self-adjoint operator

We call H(θ) the Schrödinger operator with δθ′ potential, with θ specifying the above interface conditions.

Regardless of whether the family ε⁻²V(ε⁻¹x) converges in D′ as ε → 0 or not, the Schrödinger operators H_ε converge in the norm resolvent sense to either H(θ) or D₀ depending on whether V is resonant or non-resonant. Moreover, there is no functional dependence between the constant β appearing in the distributional limit (Equation 2.3) and the interface parameter θ in the point interaction

corresponding to H(θ). Two different resonant, zero-mean potentials V may produce the same β but different values of θ, and conversely, the same θ may arise for different β. It is worth noting that Kurasov [14, 15] was the first to establish a connection between the δ′-potential and the point interactions described by Equation 2.6.

In Golovaty [16, 17], it was proved that the approximations of pseudo-Hamiltonians -d2dx2+αδ(x)+βδ′(x) by the Hamiltonians

with every regular functions U and V also converge in the norm resolvent topology. If V is non-resonant, the operators converge to D₀. However, if V is resonant with a half-bound state v, then the limiting operator is associated with point interaction, producing the interface conditions

where

A comprehensive study of exactly solvable models with point interactions (Equation 2.7) has been done by Gadella et al. [18–20]. Besides the approximation of pseudopotentials by regular potentials, there are other methods to construct exactly solvable models: e.g., the method of self-adjoint extensions has been used by Nizhnik [21, 22], and the distributional approach has been proposed by Lunardi et al. [23, 24].

We note that the point interactions characterized by the interface conditions

commonly referred to as δ′-interactions, are also sometimes interpreted as models of the formal δ′ potential. Exner, Neidhardt, and Zagrebnov [25] proposed a refined potential approximation of such interactions using a family of three δ-like potentials centered at the points ±a and 0, with the separation distance a tending to zero in a carefully coordinated way with the coupling constants. Further contributions in this direction include the works of Cheon and Shigehara [26], Zolotaryuk [27], and Albeverio et al. [28–30]; see also the recent publication [31]. Although the potential families in Exner et al. [25] do not converge to δ′ in the sense of distributions, the term “δ′-interactions” can be partially justified by interpreting δ′ as a finite-rank perturbation; see Albeverio et al. [32] and Kuzhel and Nizhnik [33] for further discussion.

Let 〈·, ·〉 be the dual pairing between the Sobolev spaces W2-s(ℝ) and W2s(ℝ). Since δ(x)y(x) = y(0)δ(x) = 〈δ, y〉δ(x), the formal operator (Equation 2.1) can be written as

This shows that the δ-potential can be interpreted as a rank-one perturbation of the free Hamiltonian, and the standard theory of regular finite-rank perturbations yields the same exactly solvable model as in Equation 2.2. In the physical literature, the δ′-interaction is typically associated with rank-one perturbation of the free Hamiltonian as in Equation 2.9 but with δ′ in place of δ [1, Ch. 1.4]:

The more general results of Albeverio and Nizhnik [34] and Albeverio et al. [32] imply that there exist regular potentials ϕε,ψε∈L2(ℝ) converging to δ′ in D′ such that the rank-one perturbations of the free Hamiltonian,

converge to Equation 2.10 in the strong resolvent topology as ε → 0. The difference, in terms of spectral effects, between the perturbation of the so-called 1D conic oscillator consisting of a mixed potential αδ+βδ′ and the one with the δ′-interaction in Equation 2.10 was investigated in detail in Fassari et al. [35].

However, the model (Equation 2.10) is not directly related to the formal expression (Equation 2.4) with a δ′-potential. Indeed, if the product δ′(x)y(x) is well defined in the distributional sense, then

Using this identity, the formal expression (Equation 2.4) can be interpreted as a rank-two perturbation of the free Schrödinger operator:

In Golovaty [36, 37], the norm resolvent convergence of the regular Hamiltonians

was studied. Here, f_ε and g_ε are sequences of real- or complex-valued functions in C0∞(ℝ) such that fε→δ′ and g_ε → δ in the distributional sense, and (·, ·) denotes the inner product in L²(ℝ). Under suitable assumptions on f_ε, g_ε, and the potential U, such operators were shown to approximate the two-parameter family of point interactions defined by the interface conditions

Although there is no established theory of distributions on metric graphs, the notions of δ-like and δ′-like potentials can be naturally extended to this setting. The construction of exactly solvable models on quantum graphs, as well as the approximation of singular vertex couplings—including mixed αδ′+βδ interactions—has been explored in Cheon and Exner [38], Man'ko [39], Exner and Manko [40], and Golovaty [41].

The above results illustrate the richness of approaches to modeling point interactions and exactly solvable models in quantum mechanics. While δ-potentials admit a canonical interpretation, the situation becomes especially delicate when the formal δ′-potential is involved, as different approximations may lead to different exactly solvable models. The choice of the appropriate limit operator is, therefore, not unique and must be guided by the physical or mathematical context of the problem.

Let us consider the family of operators

with the domain W22(ℝ), where U, V, and W are compactly supported L^∞(ℝ)-potentials. This restriction on the potentials avoids unnecessary technical complications; however, the results remain valid for a significantly broader class of potentials (cf. [13] for an example of how this constraint can be relaxed). We are interested in the emergence of negative eigenvalues in Hamiltonians due to δ-like and δ′-like perturbations. Accordingly, we assume that W ≥ 0, so that the unperturbed operator H0=-d2dx2+W is non-negative and has a purely continuous spectrum.

As follows from the result of Golovaty [16], the operators Hε converge in the norm resolvent sense as ε → 0. If the potential V is resonant, i.e., possesses a half-bound state v (see Section 2), then Hε converge to the operator

where θ and η are given by Equation 2.8. In the non-resonant case, the family converges to the operator D0=-d2dx2+W subject to the Dirichlet boundary condition at the origin as in Equation 2.5. Both H and D₀ can be interpreted as perturbations of the operator H₀ by point interactions at the origin.

If the potential V is zero, then the family Hε is uniformly bounded from below as ε → 0. This follows from the fact that the δ-like perturbation is form-bounded relative H₀, with relative bound a < 1, so that there is a b > 0 such that

In this case, as we show below, the operators Hε may have at most one eigenvalue, and this eigenvalue converges to a finite limit as ε → 0.

In contrast, when V is not identically zero, although the limiting operators H and D₀ are semibounded from below, the family Hε is generally not uniformly bounded from below. Thus, the family Hε may exhibit eigenvalues that diverge to −∞ as ε → 0; we refer to such eigenvalues as low-lying eigenvalues.

The following result characterizes precisely when such eigenvalues may occur.

Theorem 1. Let Hε be the family of operators defined by Equation 3.1. Then, the operators Hε admit low-lying eigenvalues as ε → 0 if and only if the potential V is not identically zero and

Moreover, the number of such eigenvalues is finite.

Proof. Assume that the potential V is not identically zero and that ∫ℝVdx≤0. By [42, Th.XIII.110], the operator H=-d2dx2+V has then at least one negative eigenvalue −ω², and we let u be a corresponding normalized eigenfunction. Denote by Wε(x)=W(x)+ε-1U(ε-1x)+ε-2V(ε-1x) the perturbed potential in Equation 3.1 and introduce the quadratic form

Then, the scaled function uε(x)=ε-1/2u(xε) belongs to L²(ℝ) and has norm one. A direct computation shows that

as ε → 0. Therefore, for all sufficiently small ε, one has aε[uε]≤-12ω2ε-2, and we conclude from the minimax principle that there exists an eigenvalue λ_ε of Hε such that λε≤-12ω2ε-2.

Assume now that ∫ℝVdx>0. Then, for sufficiently small ε > 0, the integral

is positive, and again by [42, Th. XIII.110] the operator Hε has no negative eigenvalues.

Let N_ε be the number of negative eigenvalues of Hε. It is known (see, e.g., [43, Th. 5.3], [44, Th. 7.5], [9, 45]) that the inequality

holds, where f⁻ = min{f, 0} is the negative part of a function f. In view of the assumption W ≥ 0, the negative part Wε- comes only from the V and U terms, and we estimate the integral above as follows:

The right-hand side remains bounded uniformly in small ε, and thus N_ε is bounded as ε → 0, which completes the proof.     □

Relatively (form-) bounded symmetric perturbations preserve semi-boundedness of the perturbed operator; see [46, Th. IV.4.11, Th. VI.1.38]. However, even if a family of self-adjoint operators A_ε converges in the norm resolvent sense as ε → 0 to a self-adjoint operator A that is bounded from below, the family A_ε may fail to be uniformly bounded from below. Even if each operator A_ε is individually semi-bounded, its lower bound may diverge to −∞ as ε → 0. A classic example due to Rellich [46, Ex. IV.4.14] gives such an operator family A_ε with a single eigenvalue tending to −∞. The family of operators Hε with δ′-like perturbations provides a much stronger illustration of this effect. While Hε converges in the norm resolvent topology to a self-adjoint operator that is bounded from below, the number of eigenvalues that diverge to −∞ as ε → 0 can be arbitrary but finite. In the next section, we describe the procedure for counting these low-lying eigenvalues.

Let us consider the Schrödinger operators

with a real coupling constant α. We denote by R(V) the set of all values of α for which the potential αV is resonant. For each non-zero function V ∈ L^∞(ℝ) with compact support, the set R(V) is a countable subset of ℝ with accumulation points at +∞ and/or −∞ [47].

We now recall the following definition [9]. Let A and B be self-adjoint operators, with B relatively A-compact. Suppose that (a, b) is a spectral gap of A and that b ∈ σ_ess(A). If there exists an eigenvalue e_α of the perturbed operator A + αB in the gap (a, b) for all α > 0, and if e_α → b − 0 as α → 0, then α = 0, which is called a coupling constant threshold. Klaus [9] established a connection between resonant potentials and such coupling constant thresholds. Both phenomena are closely related to the emergence of negative eigenvalues in Schrödinger operators.

Suppose that a Schrödinger operator -d2dx2+V has a zero-energy resonance with a corresponding half-bound state v. According to Klaus [9, Th. 3.2], if U is a real-valued potential such that

then the perturbed operator Hϰ=-d2dx2+V+ϰU, ϰ > 0, has a coupling constant threshold at ϰ = 0 and possesses a unique threshold eigenvalue λ_ϰ obeying the asymptotics λϰ=-a2ϰ2+O(ϰ3), as ϰ → 0, where the coefficient a is given by

By reversing the direction of ϰ, we conclude that as ϰ increases from zero, the operator H_ϰ acquires a negative eigenvalue that detaches from the bottom of the continuous spectrum.

Without loss of generality, we may assume that the support of the potential V is contained in the interval (−1, 1). We now consider the spectral Regge problem with spectral parameter ω [48]:

A complex number ω is called an eigenvalue of the Regge problem if there exists a non-trivial solution u of Equation 4.3, in which case u is a corresponding eigenfunction.

Theorem 2. The number of low-lying eigenvalues of Hε is equal to each of the following:

Proof. (i) Consider the family of Schrödinger operators

with domain W22(ℝ). This family is uniformly bounded from below and converges to the operator T₁ in the norm-resolvent sense as ε → 0. Suppose that T₁ has n eigenvalues -ω12,…,-ωn2. Then, for sufficiently small ε, the operators Sε have exactly n eigenvalues -ω1,ε2,…,-ωn,ε2 such that ω_j,ε → ω_j. Since Hε is unitarily equivalent to ε-2Sε, it follows that Hε has exactly n negative eigenvalues -ε-2ω1,ε2,…,-ε-2ωn,ε2, each diverging to −∞ as ε → 0.

(ii)⇔(i) The operator T0=-d2dx2 has no eigenvalues. Suppose that the set R(V)∩(0,1) is non-empty, and let α₁ be its smallest element. We write

and let v₁ be a half-bound state corresponding to the resonant potential α₁V. Then,

and the operator T_α has an eigenvalue

for α > α₁, where a₁ is given by Equation 4.2 with v = v₁ and U=V.

As the parameter α increases, it may pass through further points in R(V)∩(0,1), and at each such crossing, the operator T_α acquires a new simple eigenvalue. Since no negative eigenvalue can get absorbed by the continuous spectrum as α increases [9], this gives a total count of the negative eigenvalues of T₁.

(iii)⇔(i) Suppose ω > 0 is an eigenvalue of the Regge problem with the corresponding eigenfunction u. Then, −ω² is an eigenvalue of the operator T₁, with the corresponding eigenfunction

Conversely, if ψ is an eigenfunction of T₁ corresponding to eigenvalue −ω², then, since suppV ⊂ (−1, 1), we have ψ(x)=a-eωx for x ≤ −1 and ψ(x)=a+e-ωx for x ≥ 1. This implies that ψ satisfies the boundary conditions in Equation 4.3, and its restriction to (−1, 1) is an eigenfunction of the Regge problem (Equation 4.3) with eigenvalue ω > 0.     □

Theorem 2 is of practical importance because solving the Regge problem on a finite interval or computing the resonance set R(V) is typically much easier than directly counting the eigenvalues of a Schrödinger operator on the real line. Another useful observation is that by replacing V with cV for sufficiently large c > 0, we can make the number of negative eigenvalues of T₁—and hence the number of low-lying eigenvalues of Hε—arbitrarily large.

The following theorem describes the two-term asymptotic expansion of the low-lying eigenvalues, which are constructed and justified in Section 6.

Theorem 3. Assume that the Schrödinger operator T1=-d2dx2+V has n eigenvalues -ω12<-ω22<⋯<-ωn2<0 with eigenfunctions v₁, v₂, …, v_n. Then, the operator family Hε has n low-lying eigenvalues λ1ε<λ2ε<⋯<λnε with asymptotics

The corresponding eigenfunctions v_k,ε converge to zero in the weak topology.

We mention that one of the reasons why low-lying eigenvalues do not obstruct the norm resolvent convergence of Hε is that the corresponding eigenfunctions converge weakly to zero in L²(ℝ).

As shown in the previous section, the emergence of low-lying eigenvalues is caused by a δ′-like perturbation, and the number of these eigenvalues is determined by the profile V of the approximating δ′-like potential. However, negative eigenvalues may also arise from δ-like perturbations, whether or not a δ′-like component is present in the operators Hε. In such cases, at most one negative eigenvalue may appear, and it always has a finite limit as ε → 0.

The Schrödinger operator

with a δ-potential of intensity α ∈ ℝ acts by Sαy=-y″+Wy on its natural domain

So defined S_α is self-adjoint and has an absolutely continuous spectrum filling the positive half-line ℝ₊, while its negative spectrum consists of at most one eigenvalue. We recall that the unperturbed operator S0=-d2dx2+W(x) is non-negative.

Lemma 1. Assume W ∈ L^∞(ℝ) is a non-negative function of compact support. Then, there exists α₀ ∈ (−∞, 0) such that, for all α < α₀, the operator S_α has exactly one negative eigenvalue.

Proof. If W = 0, then the operator S_α is non-negative for α ≥ 0, while for α < 0, it has a unique eigenvalue λ=-α24 with the normalized eigenfunction

see Albeverio et al. [1, Th.3.1.4]. For a generic W, we take an α < 0 and find that

Since ∫ℝW(x)eα|x|dx→0 as α → −∞ by the Lebesgue dominated convergence theorem, we conclude that the value

becomes negative for negative α of large enough absolute value. As a result, for such α, the operator S_α has a negative eigenvalue. This eigenvalue is unique because S_α is a rank-one perturbation of the non-negative operator S₀.     □

Lemma 1 remains valid for positive potentials W such that

for example, for potentials with polynomial growth at infinity. Moreover, the above arguments suggest an explicit way to construct α₀. The function

is monotonically increasing in α ∈ (−∞, 0], f(0) > 0 and f becomes negative as α → −∞. Thus, f has a unique non-positive zero α₀. Since (Sαψα,ψα)=|α|4f(α), we conclude that the operator S_α has a unique negative eigenvalue for all α < α₀.

Example 1. Consider the family of operators

Since ∫ℝ(1+sinx)eα|x|dx=-2α for α < 0, the zero of f in Equation 5.1 satisfies α² = 4b². Hence, S_α has a unique negative eigenvalue for all α < −2|b|.

Example 2 (Cf. [49, 50]). Let S_α be the harmonic oscillator perturbed by the δ potential:

In this case, the zero of f is a negative root of α⁴ = 8k, since

Therefore, the operator S_α has a unique negative eigenvalue for all α < −2^3/4k^1/4.

When V = 0, the operators

are uniformly bounded from below and converge in the norm resolvent sense to S_α with α=∫ℝUdx. This convergence, in particular, implies the convergence of negative eigenvalues; our next objective is to obtain a more precise asymptotic formula (proved in Section 6).

Theorem 4. Suppose that W and U are L^∞(ℝ)-functions of compact support, and that W is non-negative. If ∫ℝUdx<α0, where the threshold value α₀ is the root of Equation 5.1, then the operator Hε of Equation 5.2 has a unique negative eigenvalue λ_ε satisfying the asymptotics

Here, ψ is a real-valued, L²(ℝ)-normalized eigenfunction of S_α, with α=∫ℝUdx, corresponds to the unique negative eigenvalue λ.

Moreover, the normalized eigenfunctions ψ_ε of S_ε can be chosen in such a way that ψ_ε → ψ in L²(ℝ).

If the potential W is even, then the ground state λ_ε has asymptotics

since the eigenfunction ψ is also even and therefore ψ′(−0)+ψ′(+0) = 0. If W = 0 and ∫ℝUdx<0, the asymptotic formula (Equation 5.3) becomes

and coinsides with the Abarbanel–Callan–Goldberger formula up to the factor ε². The formula arises when studying the weakly coupled Hamiltonians -d2dx2+γU, their negative eigenvalues, and the absorption of such eigenvalues, as γ → 0, by a continuous spectrum [51].

Now, suppose that the potential V is non-zero. If V is non-resonant, then the behavior of the negative spectrum of Hε is described by Theorem 3. However, if the shape V of the δ′-perturbation is resonant, then under certain conditions on the δ-perturbation, the operator Hε may have—in addition to low-lying eigenvalues—an extra eigenvalue that has a finite limit as ε → 0. We recall that in the resonant case, the norm resolvent limit of Hε as ε → 0 is the operator H given by Equation 3.2, with constants θ and η determined by V and U via Equation 2.8.

Lemma 2. Let W be a non-negative function in L^∞(ℝ) of compact support. If ηθ < 0 and the condition

holds, then the operator H defined by Equation 3.2 has a unique negative eigenvalue.

Proof. Assume first that W = 0. Integration by parts, on account of the interface conditions, yields

thus for θη ≥ 0, the operator H is non-negative. In contrast, if θη < 0, then H has a unique eigenvalue

with the normalized eigenfunction

as can be verified by straightforward calculations.

Now consider the case of arbitrary W ∈ L^∞(ℝ), and let the constants θ and η from Equation 2.8 satisfy θη < 0. Using the function Ψ defined above, we find that

Therefore, H has a negative eigenvalue if

This inequality is equivalent to

which is guaranteed under condition (Equation 5.4).     □

Theorem 5. Assume that V is resonant with a half-bound state v, and that the potentials W and U satisfy the conditions W ≥ 0 and

Then, for ε small enough, the operator Hε has a negative eigenvalue λ_ε converging, as ε → 0, to the negative eigenvalue of the operator H defined by Equation 3.2, where the parameters θ and η are given by Equation 2.8.

If W = 0 and ∫ℝUv2dx<0, then this eigenvalue λ_ε has asymptotics

Proof. Inequality (Equation 5.6) and asymptotic formula (Equation 5.7) are equivalent forms of Equations 5.4, 5.5 when evaluated for the specific values θ and η. In addition, inequality (Equation 5.6) ensures that ηθ < 0. Indeed, it implies that ∫ℝUv2dx<0, and since

we conclude that ηθ < 0. The convergence Hε→H in the norm resolvent sense as ε → 0 then guarantees that Hε has a negative eigenvalue λ_ε approaching the unique negative eigenvalue of H.     □

In this section, we derive asymptotic formulas (Equations 4.5, 5.3) by constructing and justifying formal asymptotic expansions of the eigenvalues. For the sake of definiteness, we assume that the supports of U and V are contained in (−1, 1).