Abstract
In this paper we study the spectrum of self-adjoint Schrödinger operators in \(L^2(\mathbb {R}^2)\) with a new type of transmission conditions along a smooth closed curve \(\Sigma \subseteq \mathbb {R}^2\). Although these oblique transmission conditions are formally similar to \(\delta '\)-conditions on \(\Sigma \) (instead of the normal derivative here the Wirtinger derivative is used) the spectral properties are significantly different: it turns out that for attractive interaction strengths the discrete spectrum is always unbounded from below. Besides this unexpected spectral effect we also identify the essential spectrum, and we prove a Krein-type resolvent formula and a Birman-Schwinger principle. Furthermore, we show that these Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar \(\delta \)-interactions justifying their usage as models in quantum mechanics.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In many quantum mechanical applications one considers particles moving in an external potential field which is localized near a set \(\Sigma \) of measure zero. Such strongly localized fields can be modeled by singular potentials that are supported on \(\Sigma \) only; of particular importance in this regard are \(\delta \) and \(\delta '\)-interactions. To be more precise, assume that \(\Sigma \) splits \({\mathbb R}^2\) into a bounded domain \(\Omega _{+}\) and an unbounded domain \(\Omega _{-} = {\mathbb R}^2 {\setminus } \overline{\Omega _{+}}\), and consider the formal Schrödinger differential expressions
These singular perturbations of the free Schrödinger operator \(-\Delta \) are characterized by certain transmission conditions along the interface \(\Sigma \) for the functions in the operator domain. For \(\delta \)-interactions one considers functions \(f: {\mathbb R}^2 \rightarrow {\mathbb C}\) such that the restrictions \(f_{\pm } = f \upharpoonright \Omega _{\pm }\) satisfy the transmission conditions
while \(\delta '\)-interactions are modeled by the transmission conditions
here \(\partial _{\nu } f_{\pm }\) is the normal derivative and \(\nu = (\nu _1, \nu _2)\) the unit normal vector field on \(\Sigma \) pointing outwards of \(\Omega _{+}\). The spectra and resonances of the self-adjoint realizations associated with the formal expressions (1.1) in \(L^2({\mathbb R}^2)\) are well understood, see, e.g., [8, 9, 12, 13, 15,16,17,18, 24]. In particular, the essential spectrum is given by \([0, \infty )\) and the discrete spectrum consists of at most finitely many points for every interaction strength \(\alpha < 0\), while there is no negative spectrum if \(\alpha \ge 0\).
In contrast to the transmission conditions (1.2) and (1.3) we are interested in a new type of transmission conditions of the form
where \(\alpha \in {\mathbb R}\) and \(\partial _{\overline{z}} = \frac{1}{2} ( \partial _1 + i \partial _2)\) is the Wirtinger derivative. In the sequel such jump conditions will be referred to as oblique transmission conditions. Note that the conditions (1.4) can be rewritten as
where \(\partial _t\) denotes the tangential derivative. Thus, on a formal level there is some analogy to the \(\delta '\)-transmission conditions in (1.3), but it will turn out that the properties of the corresponding self-adjoint realization in \(L^2({\mathbb R}^2)\) differ significantly from those of Schrödinger operators with \(\delta '\)-interactions.
To make matters mathematically rigorous, assume that the curve \(\Sigma \) is the boundary of a bounded and simply connected \(C^\infty \)-domain \(\Omega _+\) with open complement \(\Omega _{-} = {\mathbb R}^2 {\setminus } \overline{\Omega _{+}}\), denote the \(L^2\)-based Sobolev space of first order by \(H^1\), let \(\gamma _D^\pm :H^1(\Omega _{\pm })\rightarrow L^2(\Sigma )\) be the Dirichlet trace operators, and define for \(\alpha \in {\mathbb R}\) the Schrödinger operator with oblique transmission conditions by
The next theorem is the main result in this paper. We discuss the spectral properties of the Schrödinger operators \(T_\alpha \) and, in particular, we show in item (ii) that for every \(\alpha <0\) the operator \(T_\alpha \) is necessarily unbounded from below and the discrete spectrum in \((-\infty ,0)\) is infinite and accumulates to \(-\infty \). In items (iii) and (iv) we shall make use of the potential operator \(\Psi _{\lambda }:L^2(\Sigma ) \rightarrow L^2({\mathbb R}^2)\) and the single layer boundary integral operator \(S(\lambda ):L^2(\Sigma ) \rightarrow L^2(\Sigma )\) defined in (2.2) and (2.4), respectively.
Theorem 1.1
For any \(\alpha \in {\mathbb R}\) the operator \(T_{\alpha }\) is self-adjoint in \(L^2({\mathbb R}^2)\) and the essential spectrum is given by
Furthermore, the following statements hold:
-
(i)
If \(\alpha \ge 0\), then \(\sigma _{\textrm{disc}}(T_{\alpha }) = \emptyset \) and \(T_\alpha \) is a nonnegative operator in \(L^2({\mathbb R}^2)\).
-
(ii)
If \(\alpha < 0\), then \(\sigma _{\textrm{disc}}(T_{\alpha }) \) is infinite, unbounded from below, and does not accumulate to 0. Moreover, for every fixed \(n \in \mathbb {N}\) the n-th discrete eigenvalue \(\lambda _n \in \sigma _{\textrm{disc}}(T_{\alpha })\) (ordered non-increasingly) admits the asymptotic expansion
$$\begin{aligned} \lambda _n = - \frac{4}{\alpha ^2} + \mathcal {O}(1) \quad \text {for } \alpha \rightarrow 0^{-}, \end{aligned}$$where the dependence on n appears in the \(\mathcal {O}(1)\)-term.
-
(iii)
For \(\lambda \in {\mathbb C}{\setminus } [0, \infty )\) the Birman-Schwinger principle is valid:
$$\begin{aligned} \lambda \in \sigma _\textrm{p}(T_{\alpha }) \quad \Longleftrightarrow \quad 1 \in \sigma _\textrm{p} \big (\alpha \lambda S(\lambda ) \big ). \end{aligned}$$ -
(iv)
For \(\lambda \in \rho (T_{\alpha })={\mathbb C}{\setminus } ( [0, \infty ) \cup \sigma _\textrm{p}(T_{\alpha }))\) the operator \(I - \alpha \lambda S(\lambda )\) is boundedly invertible in \(L^2(\Sigma )\) and the resolvent formula
$$\begin{aligned} (T_{\alpha } - \lambda )^{-1} = (- \Delta - \lambda )^{-1} +\alpha \Psi _{\lambda } \bigl ( I - \alpha \lambda S(\lambda ) \bigr )^{-1} \Psi _{\overline{\lambda }}^{*} \end{aligned}$$holds, where \(- \Delta \) is the free Schrödinger operator defined on \(H^2({\mathbb R}^2)\).
To illustrate the significance of Theorem 1.1 we show that Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar \(\delta \)-interactions. To motivate this, consider one-dimensional Dirac operators with \(\delta '\)-interactions of strength \(\alpha \in \mathbb {R}\) supported in the point \(\Sigma =\{0\}\). These are first order differential operators in \(L^2(\mathbb {R})^2\) and the singular interaction is modeled by transmission conditions for functions in the operator domain, which for sufficiently smooth \(f = (f_1,f_2) \in L^2(\mathbb {R})^2\) are given by
where \(c>0\) is the speed of light. It is known that the associated self-adjoint Dirac operators converge in the non-relativistic limit to a Schrödinger operator with a \(\delta '\)-interaction of strength \(\alpha \); cf. [2, 19] and also [10, 11] for generalizations. It is not difficult to see that (1.7) can be rewritten as the transmission conditions associated with a Dirac operator with a combination of an electrostatic and a Lorentz scalar \(\delta \)-interaction of strengths \(\eta =- \frac{\alpha c^2}{2}\) and \(\tau = \frac{\alpha c^2}{2}\), respectively, as they were studied in dimension one recently in [7] and in higher space dimensions in, e.g., [3, 5,6,7].
To find a counterpart of the above result in dimension two, consider a Dirac operator with electrostatic and Lorentz scalar \(\delta \)-shell interactions of strength \(\eta \) and \(\tau \), respectively, supported on \(\Sigma \), which is formally given by
here \(A_0\) is the unperturbed Dirac operator, \(I_2\) is the \(2 \times 2\)-identity matrix and \(\sigma _3 \in \mathbb {C}^{2 \times 2}\) is given in (3.1). The differential expression \(\mathcal {A}_{\eta , \tau }\) gives rise to a self-adjoint operator \(A_{\eta , \tau }\) in \(L^2(\mathbb {R}^2)^2\), see (3.3). If one chooses, as above, \(\eta = -\frac{\alpha c^2}{2}\) and \(\tau = \frac{\alpha c^2}{2}\) and computes the non-relativistic limit, then instead of a Schrödinger operator with a \(\delta '\)-interaction one gets the somewhat unexpected limit \(T_\alpha \). Of course, this is compatible with the one-dimensional result described above, as the one-dimensional counterparts of (1.3) and (1.5) coincide, since there are no tangential derivatives in \(\mathbb {R}\). However, in higher dimensions Schrödinger operators with oblique transmission conditions should be viewed as the non-relativistic counterparts of Dirac operators with transmission conditions generalizing (1.7). Related results on non-relativistic limits of three-dimensional Dirac operators with singular interactions can be found in [4, 5, 21]. The precise result about the non-relativistic limit described above is stated in the following theorem and shown in Sect. 3.
Theorem 1.2
Let \(\alpha \in {\mathbb R}\). Then for all \(\lambda \in \mathbb {C} {\setminus } \mathbb {R}\) one has
where the convergence is in the operator norm and the convergence rate is \(\mathcal {O}\left( \frac{1}{c} \right) \).
Notations Throughout this paper \(\Omega _{+} \subseteq {\mathbb R}^2\) is a bounded and simply connected \(C^{\infty }\)-domain and \(\Omega _{-} = {\mathbb R}^2 {\setminus } \overline{\Omega _{+}}\) is the corresponding exterior domain with boundary \(\Sigma = \partial \Omega _{-} = \partial \Omega _{+}\). The unit normal vector field on \(\Sigma \) pointing outwards of \(\Omega _+\) is denoted by \(\nu \). Moreover, for \(z \in {\mathbb C}{\setminus }[0, \infty )\) we choose the square root \(\sqrt{z}\) such that \(\text {Im} \sqrt{z} > 0\) holds. The modified Bessel function of order \(j \in {\mathbb N}_0\) is denoted by \(K_j\).
For \(s \ge 0\) the spaces \(H^s({\mathbb R}^2)^n\), \(H^s(\Omega _{\pm })^n\), and \(H^s(\Sigma )^n\) are the standard \(L^2\)-based Sobolev spaces of \({\mathbb C}^n\)-valued functions defined on \({\mathbb R}^2\), \(\Omega _{\pm }\), and \(\Sigma \), respectively. If \(n=1\) we simply write \(H^s({\mathbb R}^2)\), \(H^s(\Omega _{\pm })\), and \(H^s(\Sigma )\). For negative \(s < 0\) we define the spaces \(H^s({\mathbb R}^2)^n\) and \(H^s(\Sigma )^n\) as the anti-dual spaces of \(H^{-s}({\mathbb R}^2)^n\) and \(H^{-s}(\Sigma )^n\), respectively. We denote the restrictions of functions \(f: \mathbb {R}^2 \rightarrow \mathbb {C}^n\) onto \(\Omega _\pm \) by \(f_\pm \); in this sense we write \(H^1(\mathbb {R}^2 {{\setminus }} \Sigma )^n = H^1(\Omega _+)^n \oplus H^1(\Omega _-)^n\) and identify \(f \in H^1(\mathbb {R}^2 {{\setminus }} \Sigma )^n\) with \(f_+ \oplus f_-\), where \(f_\pm \in H^1(\Omega _\pm )^n\). In the following \(\gamma _D^\pm :H^1(\Omega _{\pm })\rightarrow L^2(\Sigma )\) denote the Dirichlet trace operators and we shall write \(\gamma _D:H^1(\mathbb R^2)\rightarrow L^2(\Sigma )\) for the Dirichlet trace on \(H^1(\mathbb R^2)\); sometimes these trace operators are also viewed as bounded mappings to \(H^{1/2}(\Sigma )\).
For a Hilbert space \(\mathcal {H}\) we write \(\mathcal {L}(\mathcal {H})\) for the space of all everywhere defined, linear, and bounded operators on \(\mathcal {H}\). Furthermore, the domain, kernel, and range of a linear operator T from a Hilbert space \(\mathcal {G}\) to \(\mathcal {H}\) are denoted by \(\text {dom}\, T\), \(\text {ker}\, T\), and \(\text {ran}\, T\), respectively. The resolvent set, the spectrum, the essential spectrum, the discrete spectrum, and the point spectrum of a self-adjoint operator T are denoted by \(\rho (T)\), \(\sigma (T)\), \(\sigma _{\text {ess}}(T)\), \(\sigma _{\text {disc}}(T)\), and \(\sigma _p (T)\). The eigenvalues of compact self-adjoint operators \(K \in \mathcal {L}(\mathcal {H})\) are denoted by \(\mu _n(K)\) and are ordered by their absolute values.
2 Proof of Theorem 1.1
In this section the main result of this paper will be proved. For this, some families of integral operators are used. Define for \(\lambda \in {\mathbb C}{\setminus } [0, \infty )\) the function \(L_{\lambda }\) by
and the operator \(\Psi _{\lambda }: L^2(\Sigma ) \rightarrow L^2({\mathbb R}^2)\) by
Moreover, for \(\lambda \in {\mathbb C}{\setminus } [0, \infty )\) we make use of the single layer potential \(SL(\lambda ): L^2(\Sigma ) \rightarrow H^1(\mathbb {R}^2)\) and the single layer boundary integral operator \(S(\lambda ): L^2(\Sigma ) \rightarrow L^2(\Sigma )\) associated with \(-\Delta - \lambda \) that are defined by
and
It is known that \(SL(\lambda )\) and \(S(\lambda )\) are bounded and \(\text {ran}\, S(\lambda ) \subseteq H^{1}(\Sigma )\); cf. [25, Theorem 6.12 and Theorem 7.2]. In particular, \(S(\lambda )\) gives rise to a compact operator in \(H^s(\Sigma )\) for every \(s \in [0,1]\). Furthermore, \(S(\lambda )\) is self-adjoint and positive for \(\lambda < 0\) (see Step 1 in the proof of Proposition 2.2). Some properties of \(\Psi _{\lambda }\) and \(S(\lambda )\) that are important in the proof of Theorem 1.1 are summarized in the following two propositions; cf. Appendix A for the proof of Propositions 2.1 and 2.2.
Proposition 2.1
Let \(\lambda \in {\mathbb C}{\setminus } [0, \infty )\) and let \(\Psi _{\lambda }\) be given by (2.2). Then
is bounded and the following is true:
-
(i)
\(\Psi _{\lambda }\) gives rise to a bijective mapping \(\Psi _{\lambda }: H^{1/2}(\Sigma ) \rightarrow \mathcal {H}_{\lambda }\), where
$$\begin{aligned} \begin{aligned} \mathcal {H}_{\lambda }:= \big \{ f \in H^1(\mathbb {R}^2 {\setminus } \Sigma ) \, | \, \partial _{\overline{z}}f_{+} \oplus \partial _{\overline{z}} f_{-} \in H^1({\mathbb R}^2), \left( - \Delta - \lambda \right) f_{\pm } = 0\, \textrm{on }\, \Omega _{\pm } \big \}. \end{aligned} \end{aligned}$$ -
(ii)
\(\Psi _{\lambda }^{*}: L^2({\mathbb R}^2) \rightarrow L^2(\Sigma )\) is a compact operator, \(\Psi _\lambda ^* = -2i \gamma _D \partial _{\overline{z}} (-\Delta -\overline{\lambda })^{-1}\), and \(\mathrm{ran\,}\Psi _{\lambda }^{*} \subseteq H^{1/2}(\Sigma )\).
-
(iii)
For all \(\varphi \in H^{1/2}(\Sigma )\) the jump relations
$$\begin{aligned} \begin{aligned} i ( \nu _1 + i \nu _2) \big ( \gamma _D^{+} ( \Psi _{\lambda } \varphi )_{+} - \gamma _D^{-} ( \Psi _{\lambda } \varphi )_{-} \big )&= \varphi ,\\ -i \big ( \gamma _D^{+} \partial _{\overline{z}} ( \Psi _{\lambda } \varphi )_{+} + \gamma _D^{-} \partial _{\overline{z}} ( \Psi _{\lambda } \varphi )_{-} \big )&= \lambda S(\lambda ) \varphi , \end{aligned} \end{aligned}$$hold.
For \(\lambda < 0\) denote by \(\mu _n(S(\lambda ))\) the discrete eigenvalues of the positive self-adjoint operator \(S(\lambda )\) ordered non-increasingly and with multiplicities taken into account.
Proposition 2.2
Let \(S(\lambda )\) be defined by (2.4) and let \(n \in \mathbb {N}\) be fixed. Then the following holds:
-
(i)
The function \((- \infty , 0) \ni \lambda \mapsto \lambda \mu _n( S(\lambda ) )\) is continuous, strictly monotonically increasing and
$$\begin{aligned} \lim _{\lambda \rightarrow 0^{-} }\lambda \mu _n( S(\lambda ) ) = 0 \quad \text {and} \quad \lim _{\lambda \rightarrow -\infty } \lambda \mu _n( S(\lambda ) ) = - \infty . \end{aligned}$$ -
(ii)
For \(a<0\) the unique solution \(\lambda _n(a) \in (- \infty , 0)\) of \(\lambda \mu _n(S(\lambda )) = a\) (see (i)) admits the asymptotic expansion \(\lambda _n(a) = - 4 a^2 + \mathcal {O}(1)\) for \(a \rightarrow - \infty \), where the dependence on n appears in the \(\mathcal {O}(1)\)-term.
Proof of Theorem 1.1
Step 1. We verify that \(T_{\alpha }\) is symmetric in \(L^2({\mathbb R}^2)\). Observe first that for \(f \in \mathrm{dom\,}T_\alpha \) we have \(\partial _{\overline{z}} f_{\pm } \in H^1(\Omega _{\pm })\) and \(\Delta f_\pm = 4 \partial _{z} \partial _{\overline{z}} f_{\pm } \in L^2(\Omega _{\pm })\), and hence \(T_\alpha \) is well-defined. Moreover, as \(C_0^{\infty }(\mathbb {R}^2 {{\setminus }} \Sigma ) \subseteq \mathrm{dom\,}T_{\alpha }\) it is also clear that \(\text {dom}\,T_\alpha \) is dense. In order to show that \(T_{\alpha }\) is symmetric, we note that integration by parts in \(\Omega _{\pm }\) yields for \(f,g \in \mathrm{dom\,}T_{\alpha }\)
Now, consider (2.6) for \(f=g\) and add the equations for \(\Omega _{+}\) and \(\Omega _{-}\). Then, using \(\gamma _D^{+} ( \partial _{\overline{z}} f_{+})=\gamma _D^{-} ( \partial _{\overline{z}} f_{-})\) and the transmission condition for \(f \in \mathrm{dom\,}T_{\alpha }\), one finds that
Since this holds for all \(f \in \mathrm{dom\,}T_{\alpha }\), we conclude that \(T_{\alpha }\) is symmetric.
Step 2. Proof of the Birman-Schwinger principle in (iii): To show the first implication, assume that \(\lambda \in {\mathbb C}{\setminus } [0, \infty )\) with \(1 \in \sigma _p ( \alpha \lambda S(\lambda ) )\) and choose \(\varphi \in {\mathrm{ker\,}\,}( I - \alpha \lambda S(\lambda )) {\setminus } \{0\}\). Then it follows from the mapping properties of \(S(\lambda )\) that \(\varphi = \alpha \lambda S(\lambda ) \varphi \in H^{1/2}(\Sigma )\) holds. Therefore, Proposition 2.1 (i) implies that \(f:= \Psi _{\lambda } \varphi \in \mathcal {H}_\lambda \) fulfils \(f \ne 0\), \(f \in H^1(\mathbb {R}^2 {\setminus } \Sigma )\), \(\partial _{\overline{z}}f_{+} \oplus \partial _{\overline{z}} f_{-} \in H^1({\mathbb R}^2)\) and, as \(\varphi \in {\mathrm{ker\,}\,}( 1 - \alpha \lambda S(\lambda )) {\setminus } \{0\}\), Proposition 2.1 (iii) implies
Hence, \(f \in \mathrm{dom\,}T_\alpha \). Moreover, as \(f \in \mathcal {H}_\lambda \), we conclude \(f \in {\mathrm{ker\,}\,}( T_{\alpha } - \lambda ) {\setminus } \{0\}\) and hence \(\lambda \in \sigma _p (T_{\alpha })\).
To show the second implication, we assume that \(\lambda \in \sigma _p (T_{\alpha })\) is given and we choose \(f \in {\mathrm{ker\,}\,}(T_{\alpha } - \lambda ) {\setminus } \{0\}\). Then, by Proposition 2.1 (i) there exists a unique \(\varphi \in H^{1/2}(\Sigma )\) such that \(f = \Psi _{\lambda } \varphi \). Moreover, using \(f \in \mathrm{dom\,}T_\alpha \) and Proposition 2.1 (iii) one finds that
Since \(\varphi \ne 0\), we conclude \(1 \in \sigma _p ( \alpha \lambda S(\lambda ))\).
Step 3. Next, we prove that \(T_{\alpha }\) is a self-adjoint operator and the resolvent formula in (iv). Let \(\lambda \in {\mathbb C}{\setminus } ( [0, \infty ) \cup \sigma _p (T_{\alpha }) )\) be fixed. First, we show that \(I - \alpha \lambda S(\lambda )\) gives rise to a bijective map in \(H^s(\Sigma )\) for every \(s \in [0,1]\). Recall that \(S(\lambda )\) is compact in \(H^s(\Sigma )\). Since \(I - \alpha \lambda S(\lambda )\) is injective for our choice of \(\lambda \) by the Birman-Schwinger principle in (iii), Fredholm’s alternative shows that \(I - \alpha \lambda S(\lambda )\) is indeed bijective.
Recall that \(T_\alpha \) is symmetric; cf. Step 1. Hence, to show that \(T_\alpha \) is self-adjoint, it suffices to verify that \(\text {ran}(T_\alpha -\lambda ) = L^2(\mathbb {R}^2)\) holds for \(\lambda \in {\mathbb C}{\setminus } ( [0, \infty ) \cup \sigma _p (T_{\alpha }) )\). Fix such a \(\lambda \), let \(f \in L^2(\mathbb {R}^2)\), and define
which is well-defined by the considerations above. Since \(\Psi _{\overline{\lambda }}^{*} f \in H^{1/2}(\Sigma )\) by Proposition 2.1 (ii) and \((I - \alpha \lambda S(\lambda ))^{-1}\) is bijective in \(H^{1/2}(\Sigma )\), we conclude with Proposition 2.1 (i) that \(\Psi _{\lambda } (I - \alpha \lambda S(\lambda ))^{-1} \Psi _{\overline{\lambda }}^{*} f \in \mathcal {H}_\lambda \subseteq H^1(\mathbb {R}^2 {\setminus } \Sigma )\). In particular, with \((- \Delta - \lambda )^{-1}f \in H^2(\mathbb {R}^2)\) this implies that \(g \in H^1(\mathbb {R}^2 {\setminus } \Sigma )\) and \(\partial _{\overline{z}}g_{+} \oplus \partial _{\overline{z}} g_{-} \in H^1({\mathbb R}^2)\). Moreover, with Proposition 2.1(ii)–(iii) we obtain that
and hence, \(g \in \mathrm{dom\,}T_{\alpha }\). As \(\Psi _{\lambda } (I - \alpha \lambda S(\lambda ))^{-1} \Psi _{\overline{\lambda }}^{*} f \in \mathcal {H}_\lambda \) by Proposition 2.1 (i), we conclude
i.e. \((T_{\alpha } - \lambda ) g = f\). Since \(f \in L^2({\mathbb R}^2)\) was arbitrary, we conclude that \(\mathrm{ran\,}(T_{\alpha } - \lambda ) = L^2({\mathbb R}^2)\) and that \(T_{\alpha }\) is self-adjoint. Moreover, the resolvent formula in item (iv) follows from (2.8).
Step 4. Next, we show \(\sigma _{\text {ess}}(T_{\alpha }) = [0, \infty )\). For this fix some \(\lambda \in {\mathbb C}{\setminus } \mathbb {R}\). Since \(\Psi _{\overline{\lambda }}^{*}: L^2({\mathbb R}^2) \rightarrow L^2(\Sigma )\) is compact by Proposition 2.1 (ii), the resolvent formula in (iv) implies that \((T_{\alpha } - \lambda )^{-1} - (-\Delta - \lambda )^{-1}\) is a compact operator in \(L^2({\mathbb R}^2)\). Consequently, Weyl’s Theorem [27, Theorem XIII.14] yields that \(\sigma _\text {ess}(T_\alpha ) = \sigma _\text {ess}(-\Delta )=[0,\infty )\).
Step 5. Proof of (i): Let \(\alpha \ge 0\). Then, (2.7) implies that \(T_\alpha \) is non-negative and hence, \(\sigma (T_\alpha ) \subset [0,\infty )\). Since the latter set coincides with \(\sigma _\text {ess}(T_\alpha )\), see Step 4, we conclude \(\sigma _\text {disc}(T_\alpha )=\emptyset \).
Step 6. Proof of (ii): Let \(\alpha <0\). Since \(\sigma _{\text {ess}}(T_{\alpha }) = [0, \infty )\), it follows from the Birman-Schwinger principle in (iii) that
holds. Note that by Proposition 2.2 the equation \(\lambda \mu _n(S(\lambda )) = \alpha ^{-1}\) has a unique solution \(\lambda _n\) for all \(n \in \mathbb {N}\). Moreover, for any \(n \in {\mathbb N}\) there cannot be infinitely many \(k \ne n\) with \(\lambda _n = \lambda _k\), since otherwise \(\alpha ^{-1} < 0\) would be an eigenvalue with infinite multiplicity of the self-adjoint and compact operator \(\lambda _n S(\lambda _n)\). Thus \(\sigma _{\text {disc}}(T_{\alpha })\) is indeed an infinite set. Furthermore, as \(S(\lambda )\) is a positive self-adjoint operator in \(L^2(\Sigma )\); cf. Step 1 in the proof of Proposition 2.2, we have by definition \(\mu _n(S(\lambda )) \ge \mu _{n+1}(S(\lambda ))\) implying \(\lambda \mu _n ( S(\lambda )) \le \lambda \mu _{n+1}(S(\lambda ))\). Therefore, the monotonicity of the map \(\lambda \mapsto \lambda \mu _n (S(\lambda ))\) from Proposition 2.2 yields \(\lambda _{n+1} \le \lambda _{n}\) for all \(n \in {\mathbb N}\). This shows that 0 cannot be an accumulation point of the sequence \((\lambda _n)_{n \in {\mathbb N}}\) and as \(\sigma _{\text {ess}}(T_{\alpha })\cap (-\infty ,0)=\emptyset \) the sequence \((\lambda _n)_{n \in {\mathbb N}}\) has no finite accumulation points, that is, \(\sigma _{\text {disc}}(T_{\alpha })\) must be unbounded from below.
It remains to prove the asymptotic expansion in item (ii). By the above considerations \(\lambda _n\) is determined as the unique solution of \(\lambda \mu _n(S(\lambda )) = \alpha ^{-1}\). Clearly, if \(\alpha \rightarrow 0^-\), then \(a:= \alpha ^{-1} \rightarrow -\infty \). Hence, it follows from Proposition 2.2 (ii) with \(a = \alpha ^{-1}\) that \(\lambda _n = - \frac{4}{\alpha ^2} + \mathcal {O}(1)\) for \(\alpha \rightarrow 0^-\) and that the dependence on n appears in the \(\mathcal {O}(1)\)-term. \(\square \)
3 Proof of Theorem 1.2
In this section we show that \(T_\alpha \) is the non-relativistic limit of a family of Dirac operators with electrostatic and Lorentz scalar \(\delta \)-shell potentials formally given by (1.8), whose interaction strengths are suitably scaled. First, we recall the rigorous definition of the operator \(A_{\eta , \tau }\) associated with (1.8), see [5,6,7] for details. Let
be the Pauli spin matrices and denote the \(2 \times 2\) identity matrix by \(I_2\). Furthermore, for \(x = (x_1,x_2) \in {\mathbb R}^2\) we will use the abbreviations
We define Dirac operators with electrostatic and Lorentz scalar \(\delta \)-shell interactions of strengths \(\eta , \tau \in {\mathbb R}\) in \(L^2({\mathbb R}^2)^2\) by
It is shown in [6, 7] that \(A_{\eta , \tau }\) is self-adjoint in \(L^2({\mathbb R}^2)^2\), whenever \(\eta ^2-\tau ^2 \ne 4 c^2\), and as in [5] one sees that these operators are the self-adjoint realisations of the formal differential expression (1.8). In the above definition we are using units such that \(\hbar = 1\) and consider the mass \(m=\frac{1}{2}\), but we keep the speed of light c as a parameter for the discussion of the non-relativistic limit \(c \rightarrow \infty \).
Throughout this section we make use of the self-adjoint free Dirac operator \(A_0\), which coincides with the operator\(A_{0,0}\) given in (3.3) and which is defined on \(H^1(\mathbb {R}^2)^2\). For \(\lambda \in \rho (A_0)=\mathbb {C} {\setminus }\big ((-\infty , -\frac{c^2}{2}] \cup [\frac{c^2}{2}, \infty )\big )\) the integral kernel of the resolvent of \(A_0\) is given by \(G_\lambda (x-y)\), where \(G_\lambda (x)\) is defined for \(x \in {\mathbb R}^2 {\setminus } \{0\}\) by
cf. [6, equation (3.2)]. With this function we define the two families of integral operators
where \(B(x,\varepsilon )\) is the ball of radius \(\varepsilon \) centered at x. Both operators \(\Phi _{\lambda }: L^2(\Sigma )^2 \rightarrow L^2({\mathbb R}^2)^2\) and \(\mathcal {C}_{\lambda }: L^2(\Sigma )^2 \rightarrow L^2(\Sigma )^2\) are well-defined and bounded; cf. [6, Proposition 3.3 and equation (3.7)].
In the following lemma, which is a preparation for the proof of Theorem 1.2, we will use the matrices
products of scalar operators and matrices are understood componentwise, e.g.
Lemma 3.1
Let \(\lambda \in {\mathbb C}{\setminus } {\mathbb R}\). Then there exists a constant \(K>0\), depending only on \(\lambda \) and \(\Sigma \), such that the estimates
are valid for all sufficiently large \(c>0\).
Proof
We use a similar strategy as in the proof of [4, Proposition 5.2]. In the following let \(\lambda \in {\mathbb C}{\setminus } {\mathbb R}\) be fixed. Then \(\lambda + \frac{c^2}{2} \in {\mathbb C}{\setminus } {\mathbb R}\) and hence all operators in (3.6a)–(3.6d) are well-defined. One verifies by direct calculation that for sufficiently large \(c>0\) and all \(t \in [0,1]\)
hold. With the well-known asymptotic expansions of the modified Bessel functions and \(K_1'(z) = -K_0(z) - \frac{1}{z} K_1(z)\), (see [1]) one shows that there exist constants \(\widehat{K}, \kappa , R > 0\), depending only on \(\lambda \), such that
and
hold for all \(x \in {\mathbb R}^2 {\setminus } \{0\}\), \(j \in \{0,1\}\), \(t \in [0,1]\), and sufficiently large \(c>0\).
Next, with \(G_{\lambda +c^2/2}\) defined by (3.4) we find
Let
be the integral kernel of the resolvent of the free Laplace operator; cf. [28, Chapter 7.5]. Then
holds, where the matrix-valued functions \(t_1, t_2\), and \(t_3\) are given by
With (3.7) and (3.8) applied with \(t=1\) one finds that there exist constants \(k_1, \kappa , R > 0\), depending only on \(\lambda \), such that for \(j \in \{1,3\}\) and sufficiently large \(c>0\) one has
To estimate \(t_2\), we use \(K_0' = -K_1\) and obtain with the fundamental theorem of calculus, (3.7), and (3.8)
with a constant \(k_2\) which depends only on \(\lambda \). Thus, if we define \(k_3 = 2 k_1 + \frac{k_2 R}{2 \pi }\), then
This estimation for the integral kernel yields with the Schur test; cf. [4, Proposition A.3] for a similar argument,
for all sufficiently large \(c>0\), which is the first claimed estimate (3.6a).
Next, we prove (3.6b). Recall that the integral kernel \(L_{\lambda }\) of \(\Psi _\lambda \) is given by (2.1). Using that \(\sigma _1 M_3 = M_2\), \(\sigma _2 M_3 = -i M_2\), and \(M_1 M_3 = 0\), we obtain with (3.10) the decomposition
with
Similar as above it can be shown that there exists a \(k_4>0\), depending only on \(\lambda \), such that for all \(j \in \{1,2,3\}\)
to see the estimate for \(\tau _2\) one has to use (3.9). With the help of the Schur test the estimate (3.6b) follows (see also [4, Proposition A.4] for a similar argument); the constant \(k_4\) depends in this case on \(\lambda \) and \(\Sigma \). The estimate in (3.6c) follows by taking adjoints.
It remains to prove (3.6d). Taking \(M_3 (\sigma \cdot x) M_3 = 0 \), which holds for any \(x \in \mathbb {R}^2\), and (3.11) into account we obtain that
holds for all \(x \in {\mathbb R}^2 {\setminus } \{0\}\). Using the dominated convergence theorem, one sees that
holds for all \(f \in L^2(\Sigma )^2\) and \(x \in \Sigma \), i.e. the integral does not have to be understood as principal value. Thus we obtain with the Schur test [23, III. Example 2.4] that
In this case, the constant K depends on \(\lambda \) and \(\Sigma \). This yields (3.6d) and finishes the proof of this lemma. \(\square \)
Now we are prepared to prove Theorem 1.2 and show that \(A_{-\alpha c^2/2, \alpha c^2/2}\) converges in the non-relativistic limit to \(T_\alpha \) defined in (1.6).
Proof of Theorem 1.2
Let \(\lambda \in {\mathbb C}{\setminus } {\mathbb R}\) be fixed. Then it follows from [7, Lemma 5.4, Proposition 5.5, Theorem 5.6, and Lemma 5.9] (see also [6, Theorem 4.6]) that the operator \(I_2 - \alpha c^2 M_3 \mathcal {C}_{\lambda + c^2/2}: L^2(\Sigma )^2 \rightarrow L^2(\Sigma )^2\) is boundedly invertible and the resolvent of \(A_{- \alpha c^2/2, \alpha c^2/2} - c^2/2\) is given by
Because of \(M_3 = M_3^2\) it follows from [26, Proposition 2.1.8] that
In particular, this yields that the operator \(I - \alpha c^2 M_3 \mathcal {C}_{\lambda + c^2/2} M_3\) is boundedly invertible in \(L^2(\Sigma )^2\) for all \(c>0\) and a direct calculation shows
Recall that for \(\lambda \in \mathbb {C} {\setminus } \mathbb {R}\) also \(I - \alpha \lambda S(\lambda )\) is boundedly invertible in \(L^2(\Sigma )\); cf. Theorem 1.1 (iv). Hence, we obtain from Lemma 3.1 and [23, IV. Theorem 1.16] that
holds for all sufficiently large \(c>0\) with a constant \(K>0\) which depends only on \(\lambda \), \(\alpha \), and \(\Sigma \).
To conclude, note that (3.12) and (3.13) yield
while Theorem 1.1 (iv) and \(M_2 M_3 M_2^{\top } = M_1\) show
Using Lemma 3.1 and (3.14) the last two displayed formulae finally lead to the claimed convergence result and it also follows that the order of convergence is \(\mathcal {O}(\frac{1}{c})\). \(\square \)
References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)
Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence (2005). (With an appendix by Pavel Exner)
Arrizabalaga, N., Mas, A., Vega, L.: Shell interactions for Dirac operators: on the point spectrum and the confinement. SIAM J. Math. Anal. 47, 1044–1069 (2015)
Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: On the spectral properties of Dirac operators with electrostatic \(\delta \)-shell interactions. J. Math. Pures Appl. 111, 47–78 (2018)
Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: On Dirac operators in \(\mathbb{R} ^3\) with electrostatic and Lorentz scalar \(\delta \)-shell interactions. Quant. Stud. 6, 295–314 (2019)
Behrndt, J., Holzmann, M., Ourmières-Bonafos, T., Pankrashkin, K.: Two-dimensional Dirac operators with singular interactions supported on closed curves. J. Funct. Anal. 279, 108700 (2020)
Behrndt, J., Holzmann, M., Stelzer, C., Stenzel, G.: Boundary triples and Weyl functions for Dirac operators with singular interactions. Preprint arXiv:2211.05191
Behrndt, J., Langer, M., Lotoreichik, V.: Schrödinger operators with \(\delta \) and \(\delta ^{\prime }\)-potentials supported on hypersurfaces. Ann. Henri Poincaré 14, 385–423 (2013)
Brasche, J., Exner, P., Kuperin, Y., Šeba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994)
Budyika, V., Malamud, M., Posilicano, A.: Nonrelativistic limit for \(2p \times 2p\)-Dirac operators with point interactions on a discrete set. Russ. J. Math. Phys. 24, 426–435 (2017)
Carlone, R., Malamud, M., Posilicano, A.: On the spectral theory of Gesztesy–Šeba realizations of 1-D Dirac operators with point interactions on a discrete set. J. Differ. Equ. 254, 3835–3902 (2013)
Exner, P.: Leaky quantum graphs: a review. In: Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, vol. 77, pp. 523–564. American Mathematical Society, Providence (2008)
Exner, P., Kovařík, H.: Quantum Waveguides, Theoretical and Mathematical Physics. Springer, New York (2015)
Exner, P., Yoshitomi, K.: Asymptotics of eigenvalues of the Schrödinger operator with a strong \(\delta \)-interaction on a loop. J. Geom. Phys. 4, 344–358 (2001)
Galkowski, J.: Resonances for thin barriers on the circle. J. Phys. A 49, 125205 (2016)
Galkowski, J.: The quantum Sabine law for resonances in transmission problems. Pure Appl. Anal. 1, 27–100 (2019)
Galkowski, J.: Distribution of resonances in scattering by thin barriers. Mem. Am. Math. Soc. 259, 1248 (2019)
Galkowski, J., Smith, H.: Restriction bounds for the free resolvent and resonances in lossy scattering. Int. Math. Res. Not. 16, 7473–7509 (2015)
Gesztesy, F., Šeba, P.: New analytically solvable models of relativistic point interactions. Lett. Math. Phys. 13, 345–358 (1987)
Gohberg, I., Goldberg, S., Kaashoek, M.: Classes of Linear Operators, vol. 1. Birkhäuser, Boston (1990)
Holzmann, M.: The nonrelativistic limit of Dirac operators with Lorentz scalar \(\delta \)-shell interactions. Proc. Appl. Math. Mech. 19, e201900126 (2019)
Holzmann, M., Unger, G.: Boundary integral formulations of eigenvalue problems for elliptic differential operators with singular interactions and their numerical approximation by boundary element methods. Oper. Matrices 14, 555–599 (2020)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition
Mantile, A., Posilicano, A., Sini, M.: Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces. J. Differ. Equ. 261, 1–55 (2016)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Palmer, T.: Banach Algebras and the General Theory of *-Algebras, vol. 1. Cambridge University Press, Cambridge (1994)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, Cambridge (1977)
Teschl, G.: Mathematical Methods in Quantum Mechanics. Graduate Studies in Mathematics, vol. 157, 2nd edn. American Mathematical Society, Providence (2014)
Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil I. Mathematische Leitfäden. B. G. Teubner, Stuttgart (2000)
Acknowledgements
We are indebted to the referee for a very careful reading of our manuscript and various helpful suggestions to improve the text. Jussi Behrndt and Markus Holzmann gratefully acknowledge financial support by the Austrian Science Fund (FWF): P33568-N. This publication is based upon work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu.
Funding
Open access funding provided by Austrian Science Fund (FWF).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Dyatlov.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Proof of Propositions 2.1 and 2.2
Proof of Propositions 2.1 and 2.2
Recall that for \(\lambda \in \mathbb {C} {\setminus } [0, \infty )\) the operators \(\Psi _\lambda \), \(SL(\lambda )\), and \(S(\lambda )\) are defined by (2.2), (2.3), and (2.4), respectively. First, we collect some properties of the single layer potential \(SL(\lambda )\) that are needed in the following. It is well-known that \(SL(\lambda ): H^{1/2}(\Sigma ) \rightarrow H^2(\mathbb {R}^2 {\setminus } \Sigma )\) gives rise to a bounded operator, that \((-\Delta - \lambda ) SL(\lambda ) \varphi =0\) in \(\mathbb {R}^2 {\setminus } \Sigma \), and that for \(\varphi \in H^{1/2}(\Sigma )\) the jump relations
hold; cf. [25] or [22, Section 3.3]. Furthermore, for the single layer boundary integral operator \(S(\lambda )\) from (2.4) we have \(S(\lambda ) = \gamma _D SL(\lambda )\) and for all \(\varphi \in L^2(\Sigma )\) the representations
hold (see [22, 25]); here \(\gamma _D:H^1(\mathbb R^2)\rightarrow L^2(\Sigma )\) and \(\gamma _D':L^2(\Sigma )\rightarrow H^{-1}(\mathbb R^2)\) is the anti-dual operator.
Proof of Proposition 2.1
First, we prove item (ii). For \(\lambda \in {\mathbb C}{\setminus } [0, \infty )\) define the operator
Since \((-\Delta -\overline{\lambda })^{-1}: L^2(\mathbb {R}^2) \rightarrow H^2(\mathbb {R}^2)\) and \(\gamma _D: H^1(\mathbb {R}^2) \rightarrow H^{1/2}(\Sigma )\) are bounded, we get that \(\widehat{\Psi }_{\lambda }: L^2({\mathbb R}^2) \rightarrow H^{1/2}(\Sigma )\) is well-defined and bounded. Furthermore, as \(H^{1/2}(\Sigma )\) is compactly embedded in \(L^2(\Sigma )\) by Rellich’s embedding theorem, the operator \(\widehat{\Psi }_{\lambda }: L^2({\mathbb R}^2) \rightarrow L^2(\Sigma )\) is compact. Note that \(\widehat{\Psi }_\lambda \) is an integral operator with integral kernel
where we used \(K_0'=-K_1\) in the second step and \(\overline{\sqrt{\lambda }} = - \sqrt{\overline{\lambda }}\) in the last step (recall that \(Im \sqrt{\omega } > 0\) for \(\omega \in \mathbb {C} {\setminus } [0, \infty )\)). Hence, we conclude that
is bounded and that all claims in item (ii) are true.
Next, we show (2.5). Let \(\varphi \in L^2(\Sigma )\) and \(f \in H^1(\mathbb {R}^2)\). Since \(\Delta = 4 \partial _{\overline{z}} \partial _z = 4 \partial _z \partial _{\overline{z}}\), we see that \(\partial _{\overline{z}} (-\Delta -\overline{\lambda })^{-1} f = (-\Delta -\overline{\lambda })^{-1} \partial _{\overline{z}} f\). Hence, item (ii) and (A.2) imply
Since \(H^1(\mathbb {R}^2)\) is dense in \(L^2(\mathbb {R}^2)\), we conclude that (2.5) is true. In particular, this and the properties of the single layer potential mentioned at the beginning of this appendix imply that
is bounded and for \(\varphi \in H^{1/2}(\Sigma )\) we have
Since \(SL(\lambda ) \varphi \in H^1({\mathbb R}^2)\) it follows that \( \partial _{ \overline{z}} \left( \Psi _{\lambda } \varphi \right) _{+} \oplus \partial _{ \overline{z}} \left( \Psi _{\lambda } \varphi \right) _{-} \in H^1({\mathbb R}^2)\) holds for any \(\varphi \in H^{1/2}(\Sigma )\).
Now, we show (iii). Let \(\varphi \in H^{1/2}(\Sigma )\). With (A.5) we see that
holds. Moreover, we obtain with \(SL(\lambda ) \varphi \in H^2(\mathbb {R}^2 {\setminus } \Sigma )\)
where \(\partial _t\) is the tangential derivative on \(\Sigma \). As \(SL(\lambda ) \varphi \in H^1(\mathbb {R}^2)\), one has the relation \(\partial _t\left( SL(\lambda ) \varphi \right) _{+} = \partial _t \left( SL(\lambda ) \varphi \right) _{-}\) and consequently with (A.1)
This finishes the proof of (iii).
It remains to prove item (i). By applying the Wirtinger derivative \(\partial _{z}\) to (A.5) one gets with (2.5) that
holds for all \(\varphi \in L^2(\Sigma )\) in the distributional sense. This, (A.4), (A.5), and the properties of \(SL(\lambda )\) described at the beginning of this appendix show that \(\Psi _{\lambda } \varphi \in \mathcal {H}_{\lambda }\) for all \(\varphi \in H^{1/2}(\Sigma )\) and therefore the mapping \(\Psi _{\lambda }: H^{1/2}(\Sigma ) \rightarrow \mathcal {H}_{\lambda }\) is well-defined. Moreover, it follows from (iii) that this mapping is injective. To prove that \(\Psi _{\lambda }: H^{1/2}(\Sigma ) \rightarrow \mathcal {H}_{\lambda }\) is surjective, let \(f \in \mathcal {H}_{\lambda }\) be fixed. Define \(\varphi = i (\nu _1 + i \nu _2) ( \gamma _D^+ f_{+} - \gamma _D^- f_{-}) \in H^{1/2}(\Sigma )\) and \(g = \Psi _{\lambda } \varphi \in \mathcal {H}_{\lambda }\). By (iii) we have that
This shows \(f - g \in H^1({\mathbb R}^2)\). Moreover, due to \(f, g \in \mathcal {H}_{\lambda }\) we have that \(\partial _{\overline{z}} (f - g) \in H^1({\mathbb R}^2)\), which implies \(f-g \in H^2(\mathbb {R}^2)\). Combining this with \(f, g \in \mathcal {H}_\lambda \) we find that \(f-g \in \text {ker}\left( - \Delta - \lambda \right) = \{ 0 \}\), i.e. \(f = g = \Psi _\lambda \varphi \). Thus \(\Psi _{\lambda }: H^{1/2}(\Sigma ) \rightarrow \mathcal {H}_{\lambda }\) is also surjective and all claims in assertion (i) are shown. \(\square \)
Proof of Proposition 2.2
The proof of item (i) is divided into 3 separate steps. In Step 1 we show that the map \((-\infty , 0) \ni \lambda \mapsto \mu _n(S(\lambda )) \in (0, \infty )\) is continuous and strictly monotonically increasing, and in Step 2 we verify that the same is true for the map \((-\infty , 0) \ni \lambda \mapsto \lambda \mu _n(S(\lambda )) \in (-\infty , 0)\). Using these results, we complete the proof of assertion (i) in Step 3.
Step 1. Let \(n \in \mathbb {N}\). We show that the map \((-\infty , 0) \ni \lambda \mapsto \mu _n(S(\lambda )) \in (0,\infty )\) is continuous and strictly monotonically increasing. To verify that \(\mu _n(S(\lambda ))>0\) for \(\lambda \in (-\infty ,0)\), it suffices to prove that \(S(\lambda )\) is a positive self-adjoint operator. From the definition of \(S(\lambda )\) in (2.4) it follows that \(S(\lambda )\) is self-adjoint. Next, let \(\varphi \in L^2(\Sigma )\) with \(\varphi \ne 0\) and set \(f:= SL(\lambda ) \varphi \). Using the properties of \(SL(\lambda )\) described at the beginning of this appendix one finds that \(f \ne 0\) and
Therefore, \(\mu _n(S(\lambda )) > 0\) must be true.
Next, we show that \((-\infty , 0) \ni \lambda \mapsto \mu _n(S(\lambda )) \in (0,\infty )\) is monotonically increasing and continuous. With (A.2) one sees that \(S(\cdot ): \mathbb {C} {\setminus } [0, \infty ) \rightarrow \mathcal {L}(L^2(\Sigma ))\) is holomorphic and that \(\frac{d}{d \lambda } S(\lambda ) = \gamma _D (-\Delta - \lambda )^{-2} \gamma _D'\) holds. In particular, for any \(\varphi \in L^2(\Sigma )\) the function \((- \infty , 0) \ni \lambda \mapsto (S(\lambda ) \varphi , \varphi )_{L^2(\Sigma )}\) is continuously differentiable and
is true. Thus, the min–max principle implies that the map \((- \infty , 0) \ni \lambda \mapsto \mu _n(S(\lambda ))\) is monotonically increasing for every \(n \in {\mathbb N}\). Furthermore, due to the holomorphy of \(S(\cdot ): \mathbb {C} {\setminus } [0, \infty ) \rightarrow \mathcal {L}(L^2(\Sigma ))\) and the estimate
(see [29, Satz 3.17]), we find that \((- \infty , 0) \ni \lambda \mapsto \mu _n(S(\lambda ))\) is continuous for \(n \in {\mathbb N}\).
It remains to show that the latter map is strictly monotonically increasing. Define for \(\alpha \in {\mathbb R}{\setminus } \{0\}\) the operator-valued function \(\mathcal {B}_1: {\mathbb C}{\setminus } [0,\infty ) \rightarrow \mathcal {L}(L^2(\Sigma ))\) by \(\mathcal {B}_1(\lambda ) = I - \alpha S(\lambda )\). By the properties of \(S(\lambda )\) it is easy to see that \(\mathcal {B}_1\) is holomorphic and \(\mathcal {B}_1(\lambda )\) is a Fredholm operator with index 0 for any fixed \(\lambda \), since \(S(\lambda )\) is compact in \(L^2(\Sigma )\). Moreover, by [18, Theorem 1.2] there exists a constant \(K>0\) such that
Hence, there exists \(\lambda _0 < 0\) such that \(\Vert S(\lambda ) \Vert < |\alpha |^{-1}\) is valid for all \(\lambda < \lambda _0\). This implies that \(\mathcal {B}_1(\lambda )\) is boundedly invertible for every \(\lambda < \lambda _0\). Therefore, by [20, Chapter XI., Corollary 8.4] the set
is at most countable and does not have an accumulation point in \({\mathbb C}{\setminus } [0, \infty )\). Now assume that \(\lambda _1< \lambda _2 < 0\) satisfy \(\mu _n(S(\lambda _1)) = \mu _n(S(\lambda _2)) =: \alpha ^{-1}\) for some \(n \in {\mathbb N}\). Then it follows from the monotonicity of \(\lambda \mapsto \mu _n(S(\lambda ))\) that \([\lambda _1, \lambda _2] \subseteq \mathcal {M}_{\alpha ,1} \), which is a contradiction to the fact that \(\mathcal {M}_{\alpha ,1}\) is at most countable. Therefore, the mapping \((- \infty ,0) \ni \lambda \mapsto \mu _n(S(\lambda ))\) is continuous and strictly monotonically increasing for \(n \in {\mathbb N}\).
Step 2. To show the continuity and strict monotonicity of the map \((- \infty , 0) \ni \lambda \mapsto \lambda \mu _n(S(\lambda ))\) for all \(n \in {\mathbb N}\), we note first that the continuity follows from the continuity of the map \(\lambda \mapsto \mu _n(S(\lambda ))\) shown in Step 1. In order to prove the monotonicity, we use again \(\frac{d}{d \lambda } S(\lambda ) = \gamma _D (-\Delta - \lambda )^{-2} \gamma _D'\) and compute for a fixed \(\varphi \in L^2(\Sigma )\) and \(\lambda \in (-\infty , 0)\) with the help of (2.5) and (A.2)
Thus, the min-max principle yields the monotonicity of the mapping \((- \infty , 0) \ni \lambda \mapsto \lambda \mu _n(S(\lambda ))\). To see the strict monotonicity, we use a similar strategy as in Step 1 and define for \(\alpha \in \mathbb {R} {\setminus } \{ 0 \}\) the holomorphic mapping \(\mathcal {B}_2: {\mathbb C}{\setminus } [0, \infty ) \rightarrow \mathcal {L}( L^2(\Sigma ))\) by \(\mathcal {B}_2(\lambda ) = I - \alpha \lambda S(\lambda )\). Again, \(\mathcal {B}_2(\lambda )\) is a Fredholm operator with index zero for any fixed \(\lambda \) and it follows from (A.6) that there exists \(\lambda _3 < 0\) such that \(\Vert \lambda S(\lambda ) \Vert < |\alpha |^{-1}\) holds for all \(\lambda \in (\lambda _3, 0)\). In particular, \(\mathcal {B}_2(\lambda )\) is boundedly invertible for all \(\lambda \in (\lambda _3, 0)\). It follows from [20, Chapter XI., Corollary 8.4] that the set
is at most countable and does not have an accumulation point in \({\mathbb C}{\setminus } [0, \infty )\). Now the same argument as in Step 1 shows that \((- \infty , 0) \ni \lambda \mapsto \lambda \mu _n(S(\lambda ))\) is strictly monotonously increasing.
Step 3. To study the limiting behaviour of \(\lambda \mu _n(S(\lambda ))\) for \(\lambda \rightarrow 0\), note that (A.6) implies \(\Vert \lambda S(\lambda )\Vert \rightarrow 0\) for \(\lambda \rightarrow 0^-\) and hence,
Next, we consider the limit of \(\lambda \mu _n(S(\lambda ))\) for \(\lambda \rightarrow -\infty \). For this purpose, results on Schrödinger operators with \(\delta \)-interactions will be used. Define for \(\alpha < 0\) the sesquilinear form
By [9, 14] the form \(\mathfrak h_{\delta , \alpha }\) is semi-bounded and closed, and one can show for the self-adjoint operator \(H_{\delta , \alpha }\), which is associated with \(\mathfrak h_{\delta , \alpha }\) by the first representation theorem, that \(\sigma _{\text {ess}}(H_{\delta , \alpha }) = [ 0, \infty )\), that its discrete spectrum \(\sigma _{\text {disc}}(H_{\delta , \alpha })\) is finite, and for \(\lambda \in (-\infty ,0)\) one has that
see for instance [9, Lemma 2.3 and Theorem 4.2] and [8, Theorems 3.5 and 3.14]. Recall that the eigenvalues \(\mu _n(S(\lambda ))\) are ordered non-increasingly with multiplicities taken into account. If we order the discrete eigenvalues of \(H_{\delta , \alpha }\) in an increasing way then the strict monotonicity of \(\lambda \mapsto \mu _n(S(\lambda ))\) implies that the k-th discrete eigenvalue \(E_k(\alpha )\) (if it exists) satisfies the equation \(-1 = \alpha \mu _k(S(E_k(\alpha )))\).
Let \(n \in {\mathbb N}\). Then by [14, Theorem 1] the operator \(H_{\delta , \alpha }\) has at least n negative discrete eigenvalues (counted with multiplicities) if \(-\alpha >0\) is sufficiently large, and the n-th discrete eigenvalue \(E_n(\alpha )\) of \(H_{\delta , \alpha }\) admits the asymptotic expansion
Here H is a fixed semibounded differential operator on \(\Sigma \) that is independent of \(\alpha \) and has purely discrete spectrum \(\mu _1(H)\le \mu _2(H)\le \dots \). Thus for \(\alpha \rightarrow -\infty \) we obtain with (A.8) that
This shows
and finishes the proof of item (i).
To show item (ii), we note first that by (A.7), (A.11), and the strict monotonicity and continuity of the mapping \(\lambda \mapsto \lambda \mu _n(S(\lambda ))\) it is clear that for any \(a<0\) there is a unique solution \(\lambda _n(a)\) of \(\lambda \mu _n(S(\lambda )) = a\). Let \(\mu _n(H)\) be as in (A.9), define the numbers \(k_{\pm } = \pm ( |\mu _n(H)|+1 )\) and let
with some functions \(f_{\pm }(a) = \mathcal {O}( a^{-3} )\) for large \(|a|>0\), where the latter representation holds due to a Taylor series expansion. Then one has
and it follows with (A.10) that
and
Since \(\lambda \mapsto \lambda \mu _n( S(\lambda ))\) is monotone we find
From (A.12) we obtain
with functions \(g_{\pm }(a) = \mathcal {O}( a^{-2} )\) for large \(|a|>0\) and hence (A.9) implies
for some constant \(C_1 >0\). Note that there exist positive constants \(C_2, C_3 >0\) such that \(C_2 \vert a \vert \le \vert \alpha _\pm \vert \le C_3 \vert a \vert \) holds for large \(|a|>0\). With this we conclude from (A.15) that
holds for some constant \(C_4 > 0\) and for large \(|a|>0\). Taking (A.14) and (A.16) into account, one concludes finally that
\(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Behrndt, J., Holzmann, M. & Stenzel, G. Schrödinger Operators with Oblique Transmission Conditions in \({\mathbb R}^2\). Commun. Math. Phys. 401, 3149–3167 (2023). https://doi.org/10.1007/s00220-023-04708-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-023-04708-7