ORIGINAL RESEARCH article

Front. Appl. Math. Stat., 10 July 2025

Sec. Mathematical Physics

Volume 11 - 2025 | https://doi.org/10.3389/fams.2025.1615447

This article is part of the Research TopicRecent Mathematical and Theoretical Progress in Quantum MechanicsView all 3 articles

On negative eigenvalues of 1D Schrödinger operators with δ′-like potentials

  • 1Faculty of Mathematics and Mechanics, Ivan Franko National University of Lviv, Lviv, Ukraine
  • 2Faculty of Applied Sciences, Ukrainian Catholic University, Lviv, Ukraine
  • 3Faculty of Exact and Technical Sciences, University of Rzeszów, Rzeszów, Poland

In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with δ′-like potentials used to represent localized dipoles. These operators arise as norm resolvent limits of Schrödinger operators with suitably regularized potentials. Although the limiting operator is bounded below, we show that the approximating operators may possess a finite but arbitrarily large number of negative eigenvalues that diverge to −∞ as the regularization parameter vanishes. This phenomenon illustrates a spectral instability of Schrödinger operators with δ′-like singularities.

1 Introduction

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() and has a two-dimensional solution space. In contrast, the equation −y″+αδ′(x)y = λy is ill-posed in D() 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.

2 Short review of exactly solvable models for δ and δ′ potentials

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

-d2dx2+αδ(x),α.    (2.1)

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

domH={yW22(\0):y(+0)=y(-0),      y(+0)-y(-0)=αy(0)}.

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

(y(+0)y(+0))=(10α1)(y(-0)y(-0)).    (2.2)

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

α=U(x)dx,

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

-d2dx2+ε-1U(ε-1x)

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

-d2dx2+W(x)+αδ(x).

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−1x) converges in D, as ε → 0, if and only if V(x)dx=0, and in that case

ε-2V(ε-1x)βδ(x),    (2.3)

where β=-xV(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

-d2dx2+βδ(x)    (2.4)

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

Hε=-d2dx2+ε-2V(ε-1x),  domHε=W22(),

with δ′-like potentials ε−2V−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 Hε as ε → 0.

It has been shown that the operator family Hε 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

D0y=-y,  domD0={yW22(\0):y(0)=0}.    (2.5)

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. [47], 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 Hε only in the so-called non-resonant case, while in the resonant case, the situation is different.

We begin by recalling the relevant definitions [911]. 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

θ=v+v-

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 Hε converges in the norm resolvent sense as ε → 0 to the self-adjoint operator

H(θ)=-d2dx2,domH(θ)={yW22(\0):              y(+0)=θy(-0),θy(+0)=y(-0)}.

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

Regardless of whether the family ε−2V−1x) converges in D as ε → 0 or not, the Schrödinger operators Hε 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

(y(+0)y(+0))=(θ00θ-1)(y(-0)y(-0))    (2.6)

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

-d2dx2+ε-1U(ε-1x)+ε-2V(ε-1x)

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

(y(+0)y(+0))=(θ0ηθ-1)(y(-0)y(-0)),    (2.7)

where

θ=v+v-,  η=1v-v+Uv2dx.    (2.8)

A comprehensive study of exactly solvable models with point interactions (Equation 2.7) has been done by Gadella et al. [1820]. 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

(y(+0)y(+0))=(1β01)(y(-0)y(-0)),

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. [2830]; 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() and W2s(). Since δ(x)y(x) = y(0)δ(x) = 〈δ, y〉δ(x), the formal operator (Equation 2.1) can be written as

-d2dx2+αδ,·δ(x).    (2.9)

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]:

-d2dx2+βδ,·δ(x).    (2.10)

The more general results of Albeverio and Nizhnik [34] and Albeverio et al. [32] imply that there exist regular potentials ϕε,ψεL2() converging to δ′ in D such that the rank-one perturbations of the free Hamiltonian,

-d2dx2+βϕε, · ψε(x),

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

δ(x)y(x)=y(0)δ(x)-y(0)δ(x).

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

-d2dx2+βδ,·δ(x)+βδ,·δ(x).

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

-d2dx2+(gε,·)fε+(fε,·)gε+ε-1U(xε).

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

(y(+0)y(+0))=(μϰ0μ-1)(y(-0)y(-0)).

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.

3 Existence of low-lying eigenvalues for δ′-like potentials

Let us consider the family of operators

Hε=-d2dx2+W(x)+ε-1U(ε-1x)+ε-2V(ε-1x),    (3.1)

with the domain W22(), 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

H=-d2dx2+W, domH={ϕW22(\{0}):ϕ(+0)=θϕ(-0),ϕ(+0)=θ-1ϕ(-0)+ηϕ(-0)},    (3.2)

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

ϕW22():  ε-1(U(ε-1·)ϕ,ϕ)a(H0ϕ,ϕ)+b(ϕ,ϕ).

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

Vdx0.

Moreover, the number of such eigenvalues is finite.

Proof. Assume that the potential V is not identically zero and that Vdx0. 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

aε[ϕ]=(|ϕ(x)|2+Wε(x)|ϕ(x)|2)dx.

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

ε2aε[uε]=(|u(t)|2+V(t)|u(t)|2)dt+εU(xε)|uε(x)|2dx+ε2W(x)|uε(x)|2dx=-ω2(1+O(ε)),

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 λε of Hε such that λε-12ω2ε-2.

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

Wε(x)dx=ε-1V(x)dx+U(x)dx+W(x)dx

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

Let Nε 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

Nε1+|x| |Wε(x)|dx

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:

|x| |Wε(x)|dx=ε2|x| |V(xε)+εU(xε)|dx                                              |t| |V(t)|dt+ε|t| |U(t)|dt.

The right-hand side remains bounded uniformly in small ε, and thus Nε 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 Aε converges in the norm resolvent sense as ε → 0 to a self-adjoint operator A that is bounded from below, the family Aε may fail to be uniformly bounded from below. Even if each operator Aε 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 Aε 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.

4 Counting the number of low-lying eigenvalues

Let us consider the Schrödinger operators

Tα=-d2dx2+αV(x),  domTα=W22(),    (4.1)

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 VL(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 ∈ σess(A). If there exists an eigenvalue eα of the perturbed operator A + αB in the gap (a, b) for all α > 0, and if eαb − 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

Uv2dx<0,

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

a=1v-2+v+2Uv2dx.    (4.2)

By reversing the direction of ϰ, we conclude that as ϰ increases from zero, the operator Hϰ 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]:

d2udx2+(V(x)+ω2)u=0,           x(1,1),dudx(1)ωu(1)=0,   dudx(1)+ωu(1)=0.    (4.3)

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:

(i) the number of negative eigenvalues of the operator T1=-d2dx2+V(x);

(ii) the number of points in the set R(V) belonging to the interval (0, 1);

(iii) the number of positive eigenvalues ω of the Regge problem (Equation 4.3).

Proof. (i) Consider the family of Schrödinger operators

Sε=-d2dx2+V(x)+εU(x)+ε2W(εx)

with domain W22(). 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 ωj → ωj. 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

Tα=-d2dx2+α1V(x)+(α-α1)V(x),

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

Vv12dx=-1α1v12dx<0,

and the operator Tα has an eigenvalue

λα=-a12(α-α1)2+O((α-α1)3)

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 Tα 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

ψ(x)={u(-1)eω(x+1),if x<-1,u(x),if |x|1,u(1)e-ω(x-1),if x>1.    (4.4)

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, …, vn. Then, the operator family Hε has n low-lying eigenvalues λ1ε<λ2ε<<λnε with asymptotics

λkε=-ε-2(ωk+εU|vk|2dx2ωkvk||2)2+O(1), asε+0.    (4.5)

The corresponding eigenfunctions vk 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).

5 Negative eigenvalues generated by δ-like potentials

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

Sα=-d2dx2+W(x)+αδ(x)

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

domSα={yW22(\0):y(+0)=y(-0),        y(+0)-y(-0)=αy(0)}.

So defined Sα 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 WL(ℝ) is a non-negative function of compact support. Then, there exists α0 ∈ (−∞, 0) such that, for all α < α0, the operator Sα has exactly one negative eigenvalue.

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

ψα(x)=|α|2eα|x|2,

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

(Sαψα,ψα)=(-ψα,ψα)+(Wψα,ψα)       =-α24+|α|2W(x)eα|x|dx.

Since W(x)eα|x|dx0 as α → −∞ by the Lebesgue dominated convergence theorem, we conclude that the value

(Sαψα,ψα)=|α|4(α+2W(x)eα|x|dx)

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

Lemma 1 remains valid for positive potentials W such that

W(x)eα|x|dx<, for all α<0;

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

f(α)=α+2W(x)eα|x|dx    (5.1)

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 Sα has a unique negative eigenvalue for all α < α0.

Example 1. Consider the family of operators

Sα=-d2dx2+b2(1+sinx)+αδ(x)

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

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

Sα=-d2dx2+kx2+αδ(x),k>0.

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

x2eα|x|dx=-4α3,α<0.

Therefore, the operator Sα has a unique negative eigenvalue for all α < −23/4k1/4.

When V = 0, the operators

Hε=-d2dx2+W(x)+ε-1U(ε-1x),    (5.2)

are uniformly bounded from below and converge in the norm resolvent sense to Sα with α=Udx. 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 Udx<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 λε satisfying the asymptotics

λε=λ+ε(12ψ(0)22U(t)|t-τ|U(τ)dτdt+ψ(0)(ψ(-0)+ψ(+0))tU(t)dt)+O(ε2),ε0.    (5.3)

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

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

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

λε=λ+12εψ(0)22U(t)|t-τ|U(τ)dτdt+O(ε2),

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

λε=14(U(t) dt+12 ε2U(t)|tτ|U(τ)dτdt)2+O(ε2)

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

0+(W(-x)+θ2W(x))dx<|ηθ|2    (5.4)

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

(Hy,y)=y||2+θη|y(-0)|2,

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

λ=-η2θ2(θ2+1)2    (5.5)

with the normalized eigenfunction

Ψ(x)=2|ηθ|θ2+1·{e-ηθθ2+1x for x<0,θeηθθ2+1x for x>0,

as can be verified by straightforward calculations.

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

(HΨ,Ψ)=(WΨ,Ψ)-η2θ2(θ2+1)2.

Therefore, H has a negative eigenvalue if

(WΨ,Ψ)<η2θ2(θ2+1)2.

This inequality is equivalent to

0+(W(-x)+θ2W(x))e-2|ηθ|θ2+1xdx<|ηθ|2,

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

0+(v-2W(-x)+v+2W(x))dx<-12Uv2dx.    (5.6)

Then, for ε small enough, the operator Hε has a negative eigenvalue λε 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 Uv2dx<0, then this eigenvalue λε has asymptotics

λε=-1(v-2+v+2)2(Uv2dx)2+O(ε),ε0.    (5.7)

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 Uv2dx<0, and since

ηθ=1v-2Uv2dx,

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

6 Asymptotic expansions of eigenvalues

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).

6.1 Formal asymptotics

We start with asymptotics (Equation 5.3). The equation

-d2yεdx2+(W(x)+ε-1U(ε-1x))yε=λεyε

on ℝ\(−ε, ε) reads

-d2yεdx2+W(x)yε=λεyε,    (6.1)

while after rescaling (−ε, ε) to (−1, 1) and introducing wε(t) = yεt), one gets

-d2wεdt2+ε2W(εt)wε+εU(t)wε=ε2λεwε    (6.2)

on (−1, 1). In addition, the components yε and wε must satisfy the matching conditions

wε(±1)=yε(±ε),  wε(±1)=εyε(±ε).

We look for approximations of eigenvalues and eigenfunctions of the form

λε~λ0+ελ1,    (6.3)
yε(x)~Yε(x)={y0(x)+εy1(x),if|x|>ε,w0(ε-1x)+εw1(ε-1x)+ε2w2(ε-1x),if|x|<ε,    (6.4)

where y0 ≠ 0. Substituting the approximations into Equations 6.1, 6.2), we find that y0 and y1 satisfy the equations

-d2y0dx2+W(x)y0=λ0y0,  -d2y1dx2+W(x)y1=λ0y1+λ1y0

on ℝ\{0}, while the fast-variable components w0, w1, and w2 are solutions to the boundary value problems

d2w0dt2=0,  dw0dt(-1)=0,dw0dt(1)=0;    (6.5)
d2w1dt2=U(t)w0,  dw1dt(-1)=y0(-0),dw1dt(1)=y0(+0);    (6.6)
d2w2dt2=U(t)w1+(W(0)-λ0)w0,
dw2dt(-1)=y1(-0)-y0(-0),    (6.7)
dw2dt(1)=y1(+0)+y0(+0).

Furthermore, the equalities

w0(-1)=y0(-0),w0(1)=y0(+0),    (6.8)
w1(-1)=y1(-0)-y0(-0),w1(1)=y1(+0)+y0(+0)    (6.9)

hold. In view of Equations 6.5, 6.8, w0 is a constant function and therefore y0(+0) = y0(−0). Set w0(t) = y0(0). Problem (Equation 6.6) can be solved if and only if

dw1dt(1)-dw1dt(-1)=y0(0)-11U(τ)dτ,

which yields the second interface condition for y0:

y0(+0)-y0(-0)=αy0(0),  α=U(τ)dτ.    (6.10)

Therefore, the leading terms y0 and λ0 of Equations 6.3, 6.4 solve the problem

-d2y0dx2+W(x)y0=λ0y0 on \{0},y0(+0)=y0(-0),y0(+0)-y0(-0)=αy0(0).

Since y0 must be a non-trivial solution, we conclude that λ0 is an eigenvalue of Sα and y0 is the corresponding (real-valued) eigenfunction; we denote it by ψ and normalize by ||ψ|| = 1. Moreover, problem (Equation 6.6) is solvable now, and the solution w1 is defined up to a constant.

Integrating twice the equation for w1 and using the relations w1(-1)=ψ(-0) and w1(-1)=y1(-0)-ψ(-0), we arrive at the formula

w1(t)=ψ(0)-1t(t-τ)U(τ)dτ+ψ(-0)t+y1(-1),    (6.11)

which on account of w1(1)=y1(+0)+ψ(+0) yields

y1(+0)+ψ(+0)=ψ(0)-11(1-τ)U(τ)dτ+ψ(-0)+y1(-0).

Set α1=τU(τ)dτ. Then, y1(+0)−y1(−0) = −α1ψ(0), by Equation 6.10.

To get the second interface relation for y1, we integrate the equation for w2 and find that

w2(1)-w2(-1)=-11U(t)w1(t)dt+2(W(0)-λ0)ψ(0).

We assume that W is continuous in the vicinity of x = 0, and this implies that

ψ(+0)=ψ(-0)=(W(0)-λ0)ψ(0).

Combining this with the boundary conditions in Equations 6.7, 6.11, we obtain

y1(+0)-y1(-0)=αy1(-0)+α1ψ(-0)+γψ(0),

where

γ=-tU(t)(t-τ)U(τ)dτdt.

So, we get the boundary value problem for y1:

-d2y1dx2+W(x)y1=λ0y1+λ1ψon\{0},      (6.12)
y1(+0)-y1(-0)=-α1ψ(0),    (6.13)
y1(+0)-y1(-0)-αy1(-0)=α1ψ(-0)+γψ(0).    (6.14)

Observe that the solution y1, if it exists, is determined up to adding a multiple of ψ; therefore, by the Fredholm alternative, the above non-homogeneous problem is solvable only when some extra conditions are met. To derive them, we multiply Equation 6.12 by the eigenfunction ψ and then integrate by parts twice to get

ψ(0)(y1(+0)-y1(-0)-αy1(-0))-ψ(+0)(y1(+0)-y1(-0))      =λ1ψ2(x)dx.

Relations (Equations 6.13, 6.14) result in the expression

λ1=γψ2(0)+α1ψ(0)(ψ(-0)+ψ(+0))

for the second term in asymptotics (Equation 5.3).

We seek an approximation of the low-lying eigenvalues and the corresponding eigenfunctions of the form

-λε~ωε=ε-1(ω+εϰ),    (6.15)
yε(x)~Yε(x)={e(ω+εϰ)xε,if x<-ε,u(ε-1x)+εw(ε-1x),if |x|<ε,(a+bε)e-(ω+εϰ)xε,if x>ε,    (6.16)

where ω > 0. On the region ℝ\(−ε, ε), the function Yε satisfies (Equation 6.1) up to terms with a small norm in L2(ℝ). For instance, only the term W(x)e(ω+εϰ)xε remains if x < −ε, and its norm in L2(−∞, 0) is of order O1/2). By substituting Yε into Equation 6.2 and matching the terms at x = ±ε, we obtain

d2udt2+(V(t)+ω2)u=0,d2wdt2+(V(t)+ω2)w=(2ωκ+U(t))u,u(1)=eω,          dudt(1)=ωeω,u(1)=aeω,          dudt(1)=aωeω,w(1)=κeω,          dwdt(1)=κ(1ω)eω,w(1)=(baκ)eω,          dwdt(1)=(aκ(ω1)bω)eω.    (6.17)

The relations for u can be written as

-d2udt2+(V(t)+ω2)u=0,  t(-1,1),    (6.18)
dudt(-1)-ωu(-1)=0,dudt(1)+ωu(1)=0,    (6.19)

which is the Regge problem (Equation 4.3). We assume that ω is an eigenvalue of the problem with real-valued eigenfunction u. Recall that −ω2 is an eigenvalue of the operator T1 of Equation 4.1 with the eigenfunction ψ given by Equation 4.4. We also have a = u(1)/u(−1).

From Equation 6.17, we similarly obtain the problem for w:

-d2wdt2+(V(t)+ω2)w=-(2ωϰ+U(t))u,  t(-1,1),    (6.20)
dwdt(-1)-ωw(-1)=ϰu(-1),dwdt(1)+ωw(1)=-ϰu(1).    (6.21)

The problem is generally unsolvable because ω is an eigenvalue of the homogeneous problem (Equations 6.18, 6.19). In this situation, however, the free parameter ϰ can be chosen so that the problem admits solutions.

Solvability condition of Equations 6.20, 6.21 has the form

ϰ=--11Uu2dtu2(-1)+u2(1)+2ωu2.

When ω > 0, then the denominator can be written as 2ω||ψ||2, with the eigenstate ψ of Equation 4.4 resulting in

ϰ=-U|ψ2(x)|dx2ωψ2.

With ϰ as above, there exists a solution w of Equations 6.20, 6.21 defined up to the additive term cu. Finally, we can calculate

b=u(-1)w(1)-u(1)w(-1)u2(-1).

Observe that the right hand side of the latter expression is independent of the chosen partial solution w. Hence, we have formally obtained asymptotics (Equation 4.5).

6.2 Justification of asymptotics

We now justify the asymptotic representations for λε and yε by constructing a so-called quasimode for the operator Hε. Let A be a self-adjoint operator in a Hilbert space L. A pair (μ, ϕ) ∈ ℝ × domA is called a quasimode of A with accuracy ϵ if ||ϕ||L = 1 and ||(A − μI)ϕ||L ≤ ϵ.

Lemma 3 ([52, p.139]). Assume (μ, ϕ) is a quasimode of A with accuracy ϵ > 0 and that the spectrum of A in the interval [μ−ϵ, μ+ϵ] is discrete. Then there exists an eigenvalue λ of A such that |λ−μ| ≤ ϵ.

Moreover, if the interval [μ−Δ, μ+Δ] contains precisely one simple eigenvalue λ with normalized eigenvector u, then

ϕ-eiau2ϵΔ-1    (6.22)

for some real number a.

A quasimode of Hε can be constructed based on the approximation Yε. Note, however, that the function Yε defined by Equation 6.4 is not smooth enough to belong to the domain domHε, as it has jump discontinuities at the points x = ±ε. Nevertheless, all these jumps are small due to the construction; namely,

|[Yε]-ε|+|[Yε]ε|+|[Yε]-ε|+|[Yε]ε|cε2,    (6.23)

where [·]x denotes the jump of a function at the point x.

Suppose the functions ζ and η are smooth outside the origin, have compact supports contained in [0, ∞), and ζ(+0) = 1, ζ′(+0) = 0, η(+0) = 0, η′(+0) = 1. We introduce the function

rε(x)=[Yε]-εζ(-x-ε)-[Yε]-εη(-x-ε)   -[Yε]εζ(x-ε)-[Yε]εη(x-ε),

which has the jumps at ±ε that are negative of those of Yε. Therefore, the function ŷε = Yε + rε is continuous on ℝ along with its derivative and consequently belongs to W22(). Moreover, ||ŷε|| = 1+O(ε) as ε → 0, because the main term y0 = ψ is normalized in L2(ℝ). The corrector function rε is small, because rε is identically zero on (−ε, ε), and Equation 6.23 makes it obvious that

max|x|ε|rε(k)(x)|cε2,  k=0,1,2.

A straightforward computation shows that a pair (λ0 + λ1ε, ŷε) is a quasimode of Hε with accuracy of order O2), i.e., Hεŷε-(λ0+λ1ε)ŷεcε2ŷε. Hence,

|λε-(λ0+λ1ε)|Cε2,

where λε is an eigenvalue of Hε. Since λε is a simple eigenvalue, the corresponding eigenfunction yε can be chosen so that yεy0 in L2(ℝ), by Equation 6.22. Theorem 4 is proved.

The proof of Theorem 3 is similar, with one key difference. The approximation given by Equations 6.15, 6.16 is not sufficient to construct a quasimode of Hε with sufficiently small accuracy. It is, therefore, necessary to refine the approximation as follows:

-λε~ωε=ε-1(ω+εϰ1+ε2ϰ2+ε3ϰ3),yε(x)~Yε(x)={eωεx,if x<-ε,u(xε)+εu1(xε)+ε2u2(xε)+ε3u3(xε),if |x|<ε,(a+a1ε+a2ε2+a3ε3)e-ωεx,if x>ε,

after which the construction proceeds as in the case of Theorem 4.

7 Concluding remarks

Exactly solvable models of quantum mechanics with point interactions provide useful approximations for short-range interactions between particles. In particular, δ- and δ′-type potentials serve as mathematical abstractions representing idealized phenomena such as sharply localized charges or dipoles. However, while these models facilitate rigorous quantitative analysis, they do not always preserve the qualitative behavior of the corresponding regular systems.

For example, Hamiltonians with δ-potentials capture well the spectral and qualitative behavior of regular Hamiltonians with sharply localized attractive wells; in particular, the unique bound state of the regular system persists in the limit model. In contrast, Hamiltonians with localized dipoles may possess arbitrarily many negative eigenvalues, while the corresponding exactly solvable model with δθ-potential has none. We stress that, in both settings, the exactly solvable models arise as norm resolvent limits of families of regular Schrödinger operators as the regularization parameter tends to zero.

Our results show that despite norm resolvent convergence to a bounded below operator with δθ-potential, the family of approximating Hamiltonians with localized dipoles is not uniformly bounded below: their low-lying eigenvalues diverge to −∞ as the regularization parameter tends to zero. This demonstrates that even the strongest form of convergence of Hamiltonians does not ensure that the spectral or qualitative properties of the real physical models are reflected in the idealized limit. Therefore, while exactly solvable models offer powerful tools for quantitative analysis, caution is needed when interpreting their qualitative features as representative of the physical systems they are meant to approximate.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

YG: Investigation, Conceptualization, Writing – original draft. RH: Investigation, Writing – original draft, Conceptualization.

Funding

The author(s) declare that no financial support was received for the research and/or publication of this article.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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The author(s) declare that no Gen AI was used in the creation of this manuscript.

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Keywords: 1D Schrödinger operator, point interaction, δ-potential, δ′-potential, exactly solvable model, discrete spectrum

Citation: Golovaty Y and Hryniv R (2025) On negative eigenvalues of 1D Schrödinger operators with δ′-like potentials. Front. Appl. Math. Stat. 11:1615447. doi: 10.3389/fams.2025.1615447

Received: 21 April 2025; Accepted: 17 June 2025;
Published: 10 July 2025.

Edited by:

Luis M. Nieto, University of Valladolid, Spain

Reviewed by:

Osman Teoman Turgut, Boğaziçi University, Türkiye
Fabio Rinaldi, Universitá degli Studi Guglielmo Marconi, Italy
Ahmad Qazza, Zarqa University, Jordan

Copyright © 2025 Golovaty and Hryniv. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Yuriy Golovaty, eXVyaXkuZ29sb3ZhdHlAbG51LmVkdS51YQ==

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