Introduction

We consider the formal Dirac operators with singular potentials

$$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{\varvec{A},\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$
(3)

where \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) is a Dirac operator on \(\mathbb {R}^{n}\)

$$\begin{aligned} \mathfrak {D}_{\varvec{A},{\Phi },m}= & {} \alpha \cdot (D+\varvec{A})+\alpha _{n+1}m+{\Phi } I_{N}\nonumber \\= & {} \sum \limits _{j=1}^{n}\alpha _{j}(D_{x_{j}}+A_{j})+\alpha _{n+1}m+{\Phi }I_{N},D_{x_{j}}=-i\partial _{x_{j}}, \end{aligned}$$
(4)

with magnetic and electrostatic potentials \(\varvec{A=}(A_{1},...,A_{n}), \Phi\), and the variable mass m,  such that \(A_{j},\Phi ,m\in L^{\infty }(\mathbb {R}^{n}).\) In formula (2), \(\alpha _{j}\) are the \(N\times N\) Dirac matrices, that is

$$\begin{aligned} \alpha _{j}\alpha _{k}+\alpha _{k}\alpha _{j}=2\delta _{jk}I_{N}, \end{aligned}$$

\(I_{N}\) is the unit \(N\times N\) matrix, \(N=2^{\left[ \left( n+1\right) /2\right] }\) (see [18, 24]), and \(\Gamma \delta _{\Sigma }\) is a singular delta-type potential supported on a \(C^{2}-\)hypersurface \(\Sigma \subset \mathbb {R}^{n}\) periodic with respect to the action of a lattice \(\mathbb {G}\subset \mathbb {R}^{n}.\) More exactly, we assume that

$$\begin{aligned} \Sigma = {\bigcup \limits _{g\in \mathbb {G}}} \Sigma _{g} \end{aligned}$$
(5)

, where \(\Sigma _{g}=\Sigma _{0}+g,\) \(\Sigma _{0}\) is a closed \(C^{2}\)-hypersurface which is a boundary of the open bounded set \(\Omega _{0}.\) We assume that \(\Sigma _{g_{1}}\cap \Sigma _{g_{2}}=\varnothing\) if \(g_{1}\ne g_{2}.\) Let \(\Omega _{+}= {\bigcup \limits _{g\in \mathbb {G}}} \Omega _{g},\Omega _{g}=\Omega _{0}+g\), and \(\Omega _{-}=\mathbb {R}^{n} \diagdown \overline{\Omega }_{+},\) that is \(\Sigma\) is a common boundary of the sets \(\Omega _{+}\) and \(\Omega _{-}.\)

Such Dirac operators arise as approximation of Hamiltonians of interactions of relativistic quantum particles with potentials localized in thin tubular neighborhoods of the supports of singular potentials (see for instance [15, 30, 31]). In physical statements such problems describe the transitions of relativistic particles through obstacles generated by the potentials supported on the mentioned domains in \(\mathbb {R}^{n}\) (see for instance [9, 14, 16, 22, 23, 28]).

The formal Dirac operators with singular potentials are realized as unbounded operators \(\mathcal {D}\) in Hilbert spaces with domains described by interaction conditions on the sets carrying the singular potentials. Recently, many papers devoted to their spectral properties for the dimensions \(n=2,3\) have appeared; see, for instance, [4, 7, 8, 11,12,13, 15, 21, 29,30,31, 37, 38].

In the paper [39], it was considered the self-adjointness of the unbounded operators in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) associated with the operators \(D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}\) for \(\Sigma\) belonging to the class of so-called uniformly regular \(C^{2}-\)hypersurfaces which contain all closed \(C^{2}\)-hypersurfaces and a wide set of unbounded \(C^{2}-\)hypersurfaces, in particular, \(\mathbb {G-}\)periodic \(C^{2}-\)hypersurfaces described by formula (5).

Let \(H^{1}(\Omega_{\pm},\mathbb {C}^{N})\) be the Sobolev spaces of distributions on \(\Omega_{\pm}\) with values in \(\mathbb {C}^{N}\) and we set

$$\begin{aligned} H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})=H^{1}(\Omega _{+},\mathbb {C}^{N})\oplus H^{1}(\Omega _{-},\mathbb {C}^{N}). \end{aligned}$$

We associate with the formal Dirac operator \(D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}\) the unbounded in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) defined by the Dirac operator \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) with the domain

$$\begin{aligned}&H_{\mathfrak {B}_{\Sigma }}^{1}(\mathbb {R}^{n}\mathbb {\diagdown } \Sigma ,\mathbb {C}^{N}\mathbb {)}\\&=\left\{ u\in H^{1}(\mathbb {R}^{n}\mathbb {\diagdown }\Sigma ,\mathbb {C}^{N}\mathbb {)}:\mathfrak {B}_{\Sigma }u(s)=a_{+}\mathcal {(}s)\gamma ^{+}_{\Sigma }u(s)(s)+a_{-}\mathcal {(}s)\gamma _{\Sigma }^{-}u(s)=0,s\in \Sigma \right\} \nonumber \end{aligned}$$
(6)

where \(\gamma _{\Sigma }^{\pm }:H^{1}(\Omega _{\pm }\mathbb {C}^{N})\rightarrow H^{1/2}(\Sigma ,\mathbb {C}^{N})\) are the trace operators, and

$$\begin{aligned} a_{\pm }(s)=\frac{1}{2}\Gamma (s)\mp i\alpha \cdot \nu (s),\;\alpha \cdot \nu (s)=\sum _{j=1}^{n}\alpha _{j}\nu _{j}(s),s\in \Sigma , \end{aligned}$$
(7)

\(\nu (s)=\left( \nu _{1}(s)...,\nu _{n}(s)\right) ,s\in \Sigma\) is the field of unit normal vectors to \(\Sigma\) pointed into \(\Omega _{-}.\) We also associate the operator \(\mathbb{D}_{A,\Phi ,m,\mathfrak{B}}\) of the interaction (transmission) problem with the formal Dirac operator \(D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}\)

$$\begin{aligned} \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}u=\left\{ \begin{array}{c} \mathfrak {D}_{\varvec{A},\Phi ,m}u\text { on }\mathbb {R}^{n}\diagdown \Sigma \\ \mathfrak {B}_{\Sigma }u=0\text { on }\Sigma \end{array} \right. . \end{aligned}$$
(8)

acting from \(H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\) into \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}).\)

The following problems are considered in the paper.

  1. 1.

    We study the Dirac operators on \(n-\)dimensional torus \(\mathbb {T}\) with singular potentials \(\Gamma \delta _{\Sigma }\) where \(\Sigma\) is \(\left( n-1\right) -\)dimensional \(C^{2}-\)submanifold of \(\mathbb {T\,},\) \(\Gamma =\left( \Gamma _{ij}\right) _{i,j=1}^{N}\) is the matrix with elements \(\Gamma _{ij}\in C^{1}(\Sigma ).\) As above, we associate with the formal Dirac operator with singular potential an unbounded \(L^{2}(\mathbb {T},\mathbb {C}^{n})\) operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) and the interaction operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) bounded from \(H^{1}(\mathbb {T\diagdown }\Sigma ,\mathbb {C}^{N})\) into \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\). We study the Fredholm properties of \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }},\) the self-adjointness and discreetness of the spectrum of the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) on the torus \(\mathbb {T}.\)

  2. 2.

    We consider the Floquet theory for the formal Dirac operator (3) where \(\Sigma \subset \mathbb {R}^{n}\) is a \(\mathbb {G-}\)periodic \(C^{2} -\)hypersurface, \(\Gamma\) is a \(\mathbb {G-}\) periodic matrix, and the potentials \(\varvec{A}, \Phi , m\) are \(\mathbb {G-}\) periodic. We describe the band-gap structure of the spectrum for the self-adjoint operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}.\)

  3. 3.

    We consider the Fredholm property of \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) and essential spectrum of \(\mathcal {D} _{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) in the case if \(\Sigma\) is \(\mathbb {G-}\)periodic hypersurface in \(\mathbb {R}^{n}\) but the matrix \(\Gamma\), and potentials \(\varvec{A}, \Phi , m\) are not periodic. Our approach to the investigation of the Fredholm property of \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) and the essential spectrum of the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is based on the limit operator method (see [32,33,34]). We associate the sets of the limit operators with the operators \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) and \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\)

    $$\begin{aligned} \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}=\mathbb {D}_{\varvec{A}^{h},\Phi ^{h},m^{h},\mathfrak {B}_{\Sigma }^{h}},\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}=\mathcal {D}_{\varvec{A} ^{h},\Phi ^{h},m^{h},\mathfrak {B}_{\Sigma }^{h}} \end{aligned}$$

    defined by the sequences \(\mathbb {G}\ni h_{k}\rightarrow \infty ,\) where \(\varvec{A}^{h}(x),\Phi ^{h}(x),m^{h}(x)\) are the limits of the sequences \(\varvec{A}\left( x+h_{k}\right) ,\Phi (x+h_{k}),m(x+h_{k})\) in the sense of uniform convergence on compact sets in \(\mathbb {R}^{n},\) and

    $$\begin{aligned} \mathfrak {B}_{\Sigma }^{h}u=a_{+}^{h}\gamma _{\Sigma }^{+}u+a_{-}^{h} \gamma _{\Sigma }^{-}u, \end{aligned}$$

    where \(a_{\pm }^{h}(x)=\lim _{k\rightarrow \infty }a_{\pm }(x+h_{k})\) in the sense of uniform convergence on compact sets in \(\Sigma\). We denote by \(Lim(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\mathcal {)}\) \(Lim(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\mathcal {)}\) the set of all limit operators of \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }},\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}.\) Applying the limit operators approach, we obtain the following result.

Theorem 1

Let \(\Sigma\) be \(\mathbb {G-}\)periodic \(C^{2}\)-hypersurface, \(A_{j}, \Phi , m\in C_{b}^{1}(\Omega ),\Gamma =\left( \Gamma _{jl}\right) _{j,l=1}^{N}\) be an Hermitian matrix defined on \(\Sigma\) such that \(\Gamma _{jl}\in C_{b}^{1}(\Sigma ),k,l=1,...,N.\) We assume that the Lopatinsky-Shapiro condition

$$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}\right) \ne 0,\xi _{x}\in T_{x}^{*}(\Sigma ):\left| \xi _{x}\right| =1, \end{aligned}$$

holds at every point \(x\in \Sigma\) where \(T_{x}^{*}(\Sigma )\) is the cotangent space to \(\Sigma\) at the point x. Then:

(i) \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}:H^{1} (\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {R} ^{n},\mathbb {C}^{N})\) is a Fredholm operator if and only if all limit operators \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h} :H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\rightarrow L^{2} (\mathbb {R}^{n},\mathbb {C}^{N})\) are invertible;

(ii) The operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is closed and

$$\begin{aligned} sp_{ess}\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}={\bigcup \limits _{\mathcal {D}^{h}\in Lim(\mathcal {D} _{A,\Phi ,m,\mathfrak {B}_{\Sigma }})}}sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h} \end{aligned}$$

As an example, we consider the essential spectrum of operators which are perturbations of periodic operators \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) by slowly oscillating at infinity potentials.

Notations and auxiliary material

Notations

  • If \(\varvec{X,Y}\) are Banach spaces, then we denote by \(\mathcal {B}(\varvec{X,Y})\) the space of bounded linear operators acting from \(\varvec{X}\) into \(\varvec{Y}\) with the uniform operator topology, and by \(\mathcal {K}(\varvec{X,Y})\) the subspace of \(\mathcal {B}(\varvec{X,Y})\) of all compact operators. In the case \(\varvec{X}=\varvec{Y}\), we write shortly \(\mathcal {B}(\varvec{X})\) and \(\mathcal {K}(\varvec{X}).\)

  • An operator \(A\in \mathcal {B(}\varvec{X,Y})\) is called a Fredholm operator if \(ker A\), and \(coker A= \varvec{Y} /\text {Im}A\) are finite dimensional spaces. Let \(\mathcal {A}\) be a closed unbounded operator in a Hilbert space \(\mathcal {H}\) with a dense in \(\mathcal {H}\) domain \(dom\mathcal {A}.\) Then \(\mathcal {A}\) is called a Fredholm operator if \(ker \mathcal {A}=\left\{ u\in dom\mathcal {A}:\mathcal {A} u=0\right\}\) and \(coker A=\mathcal {H}/{\text {Im}}\mathcal {A}\) where \({\text {Im}}\mathcal {A}\mathcal {=}\left\{ \varvec{w}\in \mathcal {H}:\varvec{w}=\mathcal {A}u,u\in \mathcal {D}_{\mathcal {A}}\right\}\) are the finite-dimensional spaces. Note that \(\mathcal {A}\) is a Fredholm operator as the unbounded operator in \(\mathcal {H}\) if and only if \(\mathcal {A}:dom\mathcal {A}\rightarrow \mathcal {H}\) is a Fredholm operator as the bounded operator where \(dom\mathcal {A}\) is equipped by the graph norm

    $$\begin{aligned} \left\| u\right\| _{dom\mathcal {A}}=\left( \left\| u\right\| _{\mathcal {H}}^{2}+\left\| \mathcal {A}u\right\| _{\mathcal {H}} ^{2}\right) ^{1/2},u\in dom\mathcal {A} \end{aligned}$$

    (see for instance [1]).

  • The essential spectrum \(sp_{ess}\mathcal {A}\) of an unbounded operator \(\mathcal {A}\) is a set of \(\lambda \in \mathbb {C}\) such that \(\mathcal {A} -\lambda I\) is not the Fredholm operator as the unbounded operator, and the discrete spectrum \(sp_{dis}\mathcal {A}\) of \(\mathcal {A}\) is a set of isolated eigenvalues of finite multiplicity. It is well known that if \(\mathcal {A}\) is a self-adjoint operator, then \(\ sp_{dis}\mathcal {A}=sp\mathcal {A} \mathfrak {\diagdown }sp_{ess}\mathcal {A}\mathfrak {.}\)

  • We denote by \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) the Hilbert space of \(N-\)dimensional vector functions \(u(x)=(u^{1}(x),...,u^{N}(x)),x\in \mathbb {R}^{n}\) with the scalar product

    $$\begin{aligned} \left\langle u, v\right\rangle =\int _{\mathbb {R}^{n}}u(x)\cdot v(x)dx, \end{aligned}$$

    where \(u\cdot v=\sum _{j=1}^{n}u_{j}\bar{v}_{j}.\)

  • We denote by \(H^{s}(\mathbb {R}^{n},\mathbb {C}^{N})\) the Sobolev space of vector-valued distributions \(u\in \mathcal {D}^{\prime }(\mathbb {R}^{n},\mathbb {C}^{N})\) such that

    $$\begin{aligned} \left\| u\right\| _{H^{s}(\mathbb {R}^{n},\mathbb {C}^{N})}=\left( \int _{\mathbb {R}^{n}}(1+\left| \xi \right| ^{2})^{s}\left\| \varvec{\hat{u}}(\xi )\right\| _{\mathbb {C}^{N}}^{2}d\xi \right) ^{1/2}<\infty ,s\in \mathbb {R}, \end{aligned}$$

    where \(\varvec{\hat{u}}\) is the Fourier transform of u. If \(\Omega\) is a domain in \(\mathbb {R}^{n}\) then \(H^{s}(\Omega ,\mathbb {C}^{N})\) is the space of restrictions of \(u\in H^{s}(\mathbb {R}^{n},\mathbb {C}^{N})\) on \(\Omega\) with the norm

    $$\begin{aligned} \left\| u\right\| _{H^{s}(\Omega ,\mathbb {C}^{N})}=\inf _{lu\in H^{s}(\mathbb {R}^{n},\mathbb {C}^{N})}\left\| lu\right\| _{H^{s} (\mathbb {R}^{n},\mathbb {C}^{N})}, \end{aligned}$$

    where lu is an extension of u on \(\mathbb {R}^{n}.\) If \(\Sigma\) is a smooth enough hypersurface in \(\mathbb {R}^{n}\), we denote by \(H^{s-1/2}(\Sigma ,\mathbb {C}^{N})\) the space of restrictions on \(\Sigma\) the distributions in \(H^{s}(\mathbb {R}^{n},\mathbb {C}^{N}),s>1/2.\)

  • We denote by \(C_{b}(\mathbb {R}^{n})\) the class of bounded continuous functions on \(\mathbb {R}^{n},\) \(C_{b}^{m}(\mathbb {R}^{n})\) the class of functions a on \(\mathbb {R}^{n}\) such that \(\partial ^{\alpha }a\in C_{b}(\mathbb {R}^{n}\mathbb {)}\) for all multi-indices \(\alpha :\left| \alpha \right| \le m.\) We denote by \(C_{b}^{1}(\Sigma )\) the class of differentiable on \(\Sigma\) functions that are bounded with their first derivatives, and \(C_{b}^{\infty }(\mathbb {R}^{n})=\cap _{m\ge 0}C_{b} ^{m}(\mathbb {R}^{n}).\)

  • Let a \(C^{2}-\)hypersurface \(\Sigma \subset \mathbb {R}^{n},n\ge 2\) be the common boundary of the domains \(\Omega _{\pm }\). We say that \(\Sigma\) is uniformly regular (see for instance [3, 19]) if: (i) there exists \(r>0\) such that for every point \(x_{0}\in\) \(\Sigma\) there exists a ball \(B_{r}(x_{0})=\left\{ x\in \mathbb {R}^{n}:\left| x-x_{0}\right| <r\right\}\) and the diffeomorphism \(\varphi _{x_{0}} :B_{r}(x_{0})\rightarrow B_{1}(0)\) such that

    $$\begin{aligned} \varphi _{x_{0}}\left( B_{r}(x_{0})\cap \Omega _{\pm }\right)= & {} B_{1} (0)\cap \mathbb {R}_{\pm }^{n},\mathbb {R}_{\pm }^{n} \\= & {} \left\{ y=(y^{\prime },y_{n})\in \mathbb {R}_{y^{\prime }}^{n-1}\times \mathbb {R}_{y_{n}}:y_{n} \gtrless 0\right\} ,\\ \varphi _{x_{0}}\left( B_{r}(x_{0})\cap \Sigma \right)= & {} B_{1}(0)\cap \mathbb {R}_{y^{\prime }}^{n-1}; \end{aligned}$$

    (ii) Let \(\varphi _{x_{0}}^{i},\psi _{x_{0}}^{i},i=1,...,n\) be the coordinate functions of the mappings \(\varphi _{x_{0}},\varphi _{x_{0}}^{-1}.\) Then

    $$\begin{aligned} \sup _{x_{0}\in \Sigma }\sup _{\left| \alpha \right| \le 2,x\in B_{r}(x_{0})}\left| \partial ^{\alpha }\varphi _{x_{0}}^{i}(x)\right|&<\infty ,i=1,...,n;\\ \sup _{x_{0}\in \Sigma }\sup _{\left| \alpha \right| \le 2,y\in B_{1} (0)}\left| \partial ^{\alpha }\psi _{x_{0}}^{i}(y)\right|&<\infty ,i=1,...,n. \end{aligned}$$

    Note that each closed \(C^{2}-\)hypersurface is uniformly regular.

Auxiliary material

Dirac operators on \(\mathbb {R}^{n}\) with singular potentials ([39]).

  • Let

    $$\begin{aligned} \mathfrak {D}_{\varvec{A},\Phi ,m,\Gamma \delta }u(x)=\left( \mathfrak {D}_{\varvec{A},\Phi ,m}+\Gamma \delta _{\Sigma }\right) u(x),x\in \mathbb {R}^{n} \end{aligned}$$

    be the formal Dirac operator defined by formulas (3), (5). We assume that \(\Sigma\) is the uniformly regular \(C^{2}-\)hypersurface in \(\mathbb {R}^{n},A_{j},\Phi ,m\in L^{\infty }(\mathbb {R}^{n}),\) \(\Gamma =\left( \Gamma _{i,j}\right) _{i,j=1}^{N},\Gamma _{i,j}\in C_{b}^{1}(\Sigma ).\) We define the product \(\Gamma \delta _{\Sigma }u\) where u \(\in H^{1} (\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\) as a distribution in \(\mathcal {D}^{\prime }(\mathbb {R}^{n},\mathbb {C}^{N})=\mathcal {D}^{\prime }(\mathbb {R}^{n})\otimes \mathbb {C}^{N}\) acting on the test functions \({\varphi }\in C_{0}^{\infty }(\mathbb {R}^{n},\mathbb {C}^{N})\) as

    $$\begin{aligned} \left( \Gamma \delta _{\Sigma }u\right) (\varvec{\varphi })=\frac{1}{2} \int _{\Sigma }\Gamma (s)\left( \gamma _{\Sigma }^{+}u(s)+\gamma _{\Sigma }^{-}u(s)\right) \cdot \varphi (s)ds. \end{aligned}$$
    (9)

    Integrating by parts and taking into account (9), we obtain that

    $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}u=\mathfrak {D}_{\varvec{A},\Phi ,m}u-\left[ i\alpha \cdot \nu \left( \gamma _{\Sigma } ^{+}u-\gamma _{\Sigma }^{-}u\right) +\frac{1}{2}\Gamma \left( \gamma _{\Sigma }^{+}u+\gamma _{\Sigma }^{-}u\right) \right] \delta _{\Sigma }, \end{aligned}$$
    (10)

    where \(\gamma _{\Sigma }^{\pm }:H^{1}(\Omega _{\pm },\mathbb {C}^{N})\rightarrow H^{1/2}(\Omega _{\pm },\mathbb {C}^{N})\) are the trace operators, \(\nu (s)=(\nu _{1}(s),...,\nu _{n}(s))\) is the field of unit normal vectors pointed to \(\Omega _{-}\). Formula (10) yields that in the distribution sense

    $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}u=\mathfrak {D}_{\varvec{A},\Phi ,m}u-\left[ i\alpha \cdot \nu \left( \gamma _{\Sigma }^{+}u-\gamma _{\Sigma }^{-}u\right) +\frac{1}{2}\Gamma \left( \gamma _{\Sigma }^{+}u+\gamma _{\Sigma }^{-}u\right) \right] \delta _{\Sigma }, \end{aligned}$$
    (11)

    where \(\mathfrak {D}_{\varvec{A},\Phi ,m}u\) is the regular distribution given by the function \(\mathfrak {D}_{\varvec{A},\Phi ,m}u\in L^{2} (\mathbb {R}^{n},\mathbb {C}^{N}).\) Formula (11) yields that \(\mathfrak {D}_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}u\in L^{2} (\mathbb {R}^{n},\mathbb {C}^{N})\) if and only if

    $$\begin{aligned} -i\alpha \cdot \nu \left( \gamma _{\Sigma }^{+}u-\gamma _{\Sigma }^{-}u\right) +\frac{1}{2}\Gamma \left( \gamma _{\Sigma }^{+}u+\gamma _{\Sigma }^{-}u\right) =0\text { on }\Sigma . \end{aligned}$$
    (12)

    Condition (12) can be written in the form

    $${\mathfrak B}_\Sigma u=a_+\gamma_\Sigma^+u+a_-\gamma_\Sigma^-u=\mathbf0\;\mathrm{on}\;\mathrm\Sigma$$
    (13)

    where \(a_{\pm }\) are \(N\times N\) matrices:

    $$\begin{aligned} a_{\pm }=\frac{1}{2}\Gamma \mp i\alpha \cdot \nu \text { on } \Sigma . \end{aligned}$$
    (14)

    We associate with the formal Dirac operator \(\mathfrak {D}_{\varvec{A},\Phi ,\Gamma \delta _{\Sigma }}\) the unbounded in \(L^{2}(\mathbb {R} ^{n},\mathbb {C}^{N})\) operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) defined by the Dirac operator \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) with the domain

    $$\begin{aligned} dom\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}= & {} H_{\mathfrak {B}_{\Sigma }}^{1}(\mathbb {R}^{n} \diagdown \Sigma ,\mathbb {C}^{N})\\= & {} \left\{ u\in H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N}):\mathfrak {B}_{\Sigma }u\ {=0}\text { on }\Sigma \right\} ,\nonumber \end{aligned}$$
    (15)

    and the bounded operator of the interaction (transmission) problem

    $$\begin{aligned} \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}u=\left\{ \begin{array}{c} \mathfrak {D}_{\varvec{A},\Phi ,m}u\text { on }\mathbb {R}^{n}\diagdown \Sigma ,\\ \mathfrak {B}_{\Sigma }u=a_{+}\gamma _{\Sigma }^{+}u+a_{-}\gamma _{\Sigma }^{-}u\ {=0}\text { on }\Sigma \end{array}\right. \end{aligned}$$
    (16)

    acting from \(H^{1}(\mathbb {R}^{n}\mathbb {\diagdown }\Sigma ,\mathbb {C}^{N}\mathbb {)}\) into \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}).\)

  • We consider the parameter-dependent operator

    $$\begin{aligned}&\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}(i\mu )u=\left( \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}-i\mu I_{N}\right) u\\= & {} \left\{ \begin{array}{c} \mathfrak {D}_{\varvec{A},\Phi ,m}(i\mu )u=(\mathfrak {D}_{\varvec{A},\Phi ,m}-i\mu I_{N})u\text { on }\mathbb {R}^{n}\diagdown \Sigma ,\\ \mathfrak {B}_{\Sigma }u=a_{+}\gamma _{\Sigma }^{+}u+a_{-}\gamma _{\Sigma }^{-}u=0\, {on}\,\Sigma \end{array}\right. ,\mu \in \mathbb {R}\nonumber \end{aligned}$$
    (17)

    acting from \(H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\) into \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}).\)

  • Condition

    $$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}-i\mu I_{N}\right) \ne 0\text { for }(\xi _{x},\mu )\in T_{x}^{*}(\Sigma )\times \mathbb {R}:\left| \xi _{x}\right| ^{2}+\mu ^{2}=1 \end{aligned}$$
    (18)

    is called the local parameter-dependent Lopatinsky-Shapiro condition where \(T_{x}^{*}(\Sigma )\) is the cotangent space to \(\Sigma\) at the point \(x\in \Sigma ,\) and the condition

    $$\begin{aligned} \inf _{x\in \Sigma }\inf _{(\xi _{x},\mu )\in T_{x}^{*}(\Sigma )\times \mathbb {R}:\left| \xi _{x}\right| ^{2}+\mu ^{2}=1}\left| \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}-i\mu I_{N}\right) \right| >0 \end{aligned}$$
    (19)

    is called the uniform parameter-dependent Lopatinsky-Shapiro condition.

  • Note that if the matrix \(\Gamma (x)\) is Hermitian for every \(x\in \Gamma ,\) then condition (18) becomes the local Lopatinsky-Shapiro condition

    $$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}\right) \ne 0\text { for }\xi _{x}\in T_{x}^{*}(\Sigma ):\left| \xi _{x}\right| =1 \end{aligned}$$
    (20)

Theorem 2

Let \(\Sigma\) be the uniformly regular \(C^{2}-\)hypersurface in \(\mathbb {R}^{n}\), \(A_{j}\), \(\Phi\), \(m\in L^{\infty }(\mathbb {R}^{n}),\) \(\Gamma =\left( \Gamma _{i,j}\right) _{i,j=1}^{N},\Gamma _{i,j}\in C_{b}^{1}(\Sigma )\), and the uniform parameter-dependent Lopatinsky-Shapiro condition (19) hold. Then there exists \(\mu _{0}\in \mathbb {R}\) such that the operator

$$\begin{aligned} \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}(i\mu ):H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}) \end{aligned}$$

is invertible for every \(\mu \in \mathbb {R}:\left| \mu \right| >R.\)

Theorem 3

Let \(\Sigma\) be the uniformly regular \(C^{2}-\)hypersurface in \(\mathbb {R}^{n}\), \(A_{j}\), \(\Phi\), \(m\in L^{\infty }(\mathbb {R}^{n}),\) \(\Gamma =\left( \Gamma _{i,j}\right) _{i,j=1}^{N},\Gamma _{i,j}\in C_{b}^{1}(\Sigma )\), \(A_{j},\Phi ,m\) be real-valued functions, \(\Gamma (x)\) be Hermitian matrix for every \(x\in \Sigma ,\) and the uniform Lopatinsky-Shapiro condition

$$\begin{aligned} \inf _{x\in \Sigma }\inf _{\xi _{x}\in T_{x}^{*}(\Sigma ):\left| \xi _{x}\right| ^{2}=1}\left| \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}\right) \right| >0 \end{aligned}$$
(21)

hold. Then the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is self-adjoint.

Example 4

Let \(\Gamma (x)=\eta (x)I_{N}+\tau (x)\alpha _{n+1}\) where \(\eta (x),\tau (x)\in C_{b}^{1}(\Sigma )\) be real-valued functions. Then condition

$$\begin{aligned} \inf _{x\in \mathbb {R}^{n}}\left| \eta ^{2}(x)-\tau ^{2}(x)-4\right| >0 \end{aligned}$$
(22)

ensures the condition (21), and therefore the invertibility \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}(i\mu )\) for large enough \(\left| \mu \right| ,\) and the self-adjointness of \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}.\)

Note that the singular potential \(\Gamma \delta _{\Sigma }\) describes the electrostatic and Lorentz scalar shell interactions in \(\mathbb {R}^{n}\) (see [11,12,13].)

Band-dominated operators on \(\mathbb {R}^{n}\) and their local invertibility at infinity ([32,33,34] ,[26])

Let \(\psi \in C_{0}^{\infty }(\mathbb {R}^{n}),\psi (x)=1\) if \(\left| x\right| \le 1/2\) and \(\psi (x)=0\) if \(\left| x\right| \ge 1,\) \(\chi (x)=1-\psi (x),\) \(\psi _{R}(x)=\psi _{R}(x/R),\) \(\chi _{R}(x)=\chi (x/R).\)

Definition 5

We say that \(A\in \mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))\) is locally invertible at infinity if there exists \(R>0\) and the operators \(\mathcal {L}_{R},\mathcal {R}_{R}\in \mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))\) such that

$$\begin{aligned} \mathcal {L}_{R}A\chi _{R}I=\chi _{R}I,\chi _{R}A\mathcal {R}_{R}=\chi _{R}I. \end{aligned}$$

Definition 6

We say that the operator \(A \in \mathcal {B}\left( L^{2}(\mathbb {R}^{n},\mathbb {C}^{N} ) \right)\) belongs to the class \(\mathcal {A} (\mathbb {R}^{n},\mathbb {C}^{N})\) of band-dominated operators on \(\mathbb {R}^{n}\) if for every function \(\varphi \in C_{b}^{\infty }(\mathbb {R}^{n})\)

$$\begin{aligned} \lim _{t\rightarrow 0}\left\| \left[ \varphi _{t}I,A\right] \right\| _{\mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))}=\lim _{t\rightarrow 0}\left\| \left[ \varphi _{t}A-A\varphi _{t}I\right] \right\| _{\mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))}=0 \end{aligned}$$

where \(\varphi _{t}(x)=\varphi (t_{1}x_{1},...,t_{n}x_{n}),t=(t_{1},...,t_{n})\in \mathbb {R}^{n}.\)

Note that \(\mathcal {A}(\mathbb {R}^{n},\mathbb {C}^{N})\) is an inverse closed \(C^{*}-\)algebra.

We denote by \(V_{h},h\in \mathbb {G}\) the unitary in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) shift operator \(V_{h}u(x)=u(x-h).\)

Ler \(\mathbb {G}\) be the lattice in \(\mathbb {R}^{n},\) that is

$$\begin{aligned} \mathbb {G=}\left\{ g\in \mathbb {R}^{n}:g=\sum _{j=1}^{n}g_{j}a_{j},g_{j}\in \mathbb {Z}\right\} , \end{aligned}$$
(23)

where \(\left\{ a_{1},...,a_{n}\right\}\) is a linearly independent system of vectors in \(\mathbb {R}^{n}.\)

Definition 7

Let the sequence \(\mathbb {G}\ni h_{k}\rightarrow \infty .\) We say that the operator \(A^{h}\in \mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))\) is a limit operator of \(A\in \mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))\) if for every function \(\varphi \in C_{0}^{\infty }(\mathbb {R}^{n})\)

$$\begin{aligned}&\lim _{k\rightarrow \infty }\left\| \left( V_{-h_{k}}AV_{h_{k}}-A^{h}\right) \varphi I\right\| _{\mathcal {B}(L^{2}(\mathbb {R} ^{n},\mathbb {C}^{N})}\\= & {} \lim _{k\rightarrow \infty }\left\| \varphi \left( V_{-h_{k}}AV_{h_{k}}-A^{h}\right) \right\| _{\mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})}=0. \end{aligned}$$

We say that the operator \(A\in \mathcal {B}(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}))\) is rich if every sequence \(\mathbb {G}\ni h_{k}\rightarrow \infty\) has a subsequence \(h_{k_{l}}\) defining the limit operator.

Theorem 8

(see [26, 32, 33]). Let \(A\in \mathcal {A}(\mathbb {R}^{n},\mathbb {C}^{N})\) be a rich operator acting in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}).\) Then the following assertions are equivalent:

  1. (i)

    A is a locally invertible at infinity operator;

  2. (ii)

    The family Lim(A) of all limit operators is uniformly invertible in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) that is every limit operator \(A^{h}\) has inverse \(\left( A^{h}\right) ^{-1}\), and

    $$\begin{aligned} \sup _{A^{h}\in Lim(A)}\left\| \left( A^{h}\right) ^{-1}\right\| <\infty ; \end{aligned}$$
  3. (iii)

    Each limit operator \(A^{h}\in Lim(A)\) is invertible in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}).\)

Remark 9

The equivalence of conditions (i) and (ii) has been proved in [32, 33], but the question of the equivalence of conditions (ii) and (iii) has been open for a long time. The affirmative answer to this question has been obtained in [26].

Interaction problems for Dirac operators on the torus

Let \(\mathbb {G}\) be the lattice defined by (23). We consider the formal Dirac operator on \(\mathbb {R}^{n}\) given by formulas (3), (5) with the \(\mathbb {G-}\)periodic potentials \(A_{j},\Phi ,m,\) and the \(\mathbb {G-}\)periodic singular potentials \(\Gamma \delta _{\Sigma }.\) Let W be a fundamental domain for the action of the group \(\mathbb {G}\) on \(\mathbb {R}^{n},\) and \(\Omega _{0}\) be a domain such that \(\overline{\Omega _{0}}\subset int(W).\) We set

$$\begin{aligned} \Omega _{+}=\bigcup _{g\in G}\Omega _{g},\Omega _{-}=\mathbb {R}^{n}\diagdown \overline{\Omega }_{+}, \end{aligned}$$

and

$$\begin{aligned} \Sigma ={\bigcup \limits _{g\in \mathbb {G}}}\Sigma _{g},\; \Sigma _{g}=\partial \Omega _{g} \end{aligned}$$

is the periodic common boundary of the domains \(\Omega _{\pm }.\)

We associate with the periodic formal Dirac operator

$$\begin{aligned} D_{\varvec{A},\Phi ,m,\mathfrak {B},\Gamma \delta _{\Sigma }}= \mathfrak {D}_{\varvec{A} ,\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$

where \(A_{j},\Phi ,m\in C^{1}(\mathbb {R}^{n})\) are \(\mathbb {G-}\)periodic function on \(\mathbb {R}^{n},\Gamma =\left( \Gamma _{kl}\right) _{k.l=1}^{N}\) is a periodic matrix with \(\Gamma _{kl}\in C^{1}(\Sigma ),\)the interaction operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{\mathbb {T}}\) on the torus \(\mathbb {T=R}^{n}\diagup \mathbb {G}\)

$$\begin{aligned} \mathbb {D}_{\varvec{A},\Phi ,\Psi ,\mathfrak {B}_{\Sigma }}^{\mathbb {T}}u=\left\{ \begin{array} [c]{c} \mathfrak {D}_{\varvec{A},\Phi ,m}u\text { on }\mathbb {T}\diagdown \tilde{\Sigma }\\ \mathfrak {B}_{\tilde{\Sigma }}u\text { }=a_{+}\gamma _{\Sigma _{0}}^{+} u+a_{-}\gamma _{\Sigma _{0}}^{-}u=0\text { on }\tilde{\Sigma } \end{array}\right. , \end{aligned}$$
(24)

where \(a_{\pm }=\frac{\Gamma }{2}\mp i\alpha \cdot \nu ,\) \(\tilde{\Sigma }\) is a \(C^{2}\) manifold on \(\mathbb {T}\) of dimension \((n-1)\) which is the natural projection on \(\mathbb {T}\) by the hypersurface \(\Sigma \subset \mathbb {R}^{n},\) and \(\tilde{\Sigma }\) is the common boundary of the domain \(\tilde{\Omega }_{\pm }\subset \mathbb {T},\) which are the projections of \(\Omega _{\pm }\) on \(\mathbb {T},\) \(\nu (s)\) is the unit normal vector to \(\tilde{\Sigma }\) pointed to \(\tilde{\Omega }_{-}.\)

Let

$$\begin{aligned} H^{1}(\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C}^{N})=H^{1}(\tilde{\Omega }_{+},\mathbb {C}^{N})\oplus H^{1}(\tilde{\Omega }_{-},\mathbb {C}^{N}), \end{aligned}$$

\(H^{1}(\tilde{\Omega }_{\pm },\mathbb {C}^{N})\) are Sobolev spaces on domains \(\tilde{\Omega }_{\pm }\subset \mathbb {T}.\) We consider \(\mathbb {D} _{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{\mathbb {T}}\) as a bounded operator from \(H^{1}(\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C}^{N})\) into \(L^{2}(\mathbb {T},\mathbb {C}^{N})\). We denoted by \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{\mathbb {T}}\) the unbounded operator in \(L^{2}(\mathbb {T},\mathbb {C}^{N})\) generated by the Dirac operator \(\mathfrak {D}_{\varvec{A},\Phi ,\Psi }\) on the torus \(\mathbb {T}\) with the domain

$$\begin{aligned} dom\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{\mathbb {T}}=\left\{ u\in H^{1}(\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C} ^{N}):\mathfrak {B}_{\tilde{\Sigma }}u=0\text { on }\tilde{\Sigma }\right\} . \end{aligned}$$

Theorem 10

(i) Let \(\tilde{\Sigma }\subset \mathbb {T}\) be a \(C^{2}\)-submanifold of the dimension \((n-1)\), \(A_{j},\Phi ,m\in C^{1}(\mathbb {T})\), the matrix \(\Gamma =\left( \Gamma _{ij}\right) _{i,j=1}^{N}\) be defined on \(\tilde{\Sigma }\) and such that \(\Gamma _{ij}\in C^{1}(\tilde{\Sigma }).\) Moreover, let for every point \(x\in \tilde{\Sigma }\) the Lopatinsky-Shapiro condition

$$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}\right) \ne 0\text {, for each }\xi _{x}\in T_{x}^{*}(\tilde{\Sigma }):\left| \xi _{x}\right| =1 \end{aligned}$$
(25)

hold, where \(T_{x}^{*}(\tilde{\Sigma })\) is the cotangent space to the manifold \(\tilde{\Sigma }\) at the point x. Then, \(\mathbb {D}_{\varvec{A},\Phi ,\Psi ,\mathfrak {B}\tilde{\Sigma }}:H^{1}(\mathbb {T},\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {T},\mathbb {C}^{N})\) is the Fredholm operator.

(ii) Let in addition to the above conditions the matrix \(\Gamma (x)\) be Hermitian for each \(x\in \tilde{\Sigma }.\) Then \(ind(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}})=0,\) and the operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}-\lambda I:H^{1} (\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {T},\mathbb {C}^{N})\) is invertible for each \(\lambda \in \mathbb {C}\diagdown \Pi\) where \(\Pi\) is a discrete set in \(\mathbb {C}\) with a unique limit point \(\infty .\) The unbounded operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}\) is closed and has the discrete spectrum only.

Proof

(i) As in the paper [39], one can prove that the Lopatinsky-Shapiro condition (25) is sufficient for the local Fredholmness of the operator \(\mathbb {D}_{\varvec{A},\Phi ,\Psi ,\mathfrak {B}_{\tilde{\Sigma }}}\) at the point \(x\in \tilde{\Sigma }.\) Since the operator \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) is elliptic at every point \(x\in \mathbb {T}\) the local principle of the elliptic theory [1] yields that the operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}\) is Fredholm if condition (25) holds at every point \(x\in \tilde{\Sigma };\)

(ii) It follows from (i) the operator \(\mathbb {D}_{\varvec{A},\Phi ,\Psi ,\mathfrak {B}_{\tilde{\Sigma }}}-i\mu I_{N}\) is the Fredholm operator for every \(\mu \in \mathbb {C}\). Hence \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}-i\mu I_{N}\) is the analytical family of the Fredholm operators. Moreover, since \(\Gamma (x)\) is a Hermitian matrix for every \(x\in \mathbb {R}^{n}\) the parameter-dependent Lopatinsky-Shapiro condition holds for every \(\mu \in \mathbb {R}\). By Theorem 2, the operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}-i\mu I_{N}\) is invertible for \(\mu \in \mathbb {R}\) with \(\left| \mu \right|\) is large enough. Hence, by the Analytic Fredholm Theorem (see [10, 20]), the operator

$$\begin{aligned}&~~~~~~~\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}-\lambda I :H^{1}(\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {T},\mathbb {C}^{N})\\&\text {is invertible for each }\lambda \in \mathbb {C}\diagdown \Pi \nonumber \end{aligned}$$
(26)

where \(\Pi\) is a discrete set with a possible limit point \(\infty .\)

Moreover, \(ind\left( \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}\right) =0.\) The Lopatinsky-Shapiro condition (25) yields the a priori estimate

$$\begin{aligned} \left\| u\right\| _{H^{1}(\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C}^{N})}\le C\left( \left\| \mathfrak {D}_{\varvec{A},\Phi ,m}u\right\| _{L^{2}(\mathbb {T},\mathbb {C}^{N})}+\left\| u\right\| _{L^{2}(\mathbb {T},\mathbb {C}^{N})}\right) \end{aligned}$$
(27)

for every \(u\in H^{1}(\mathbb {T}\diagdown \tilde{\Sigma },\mathbb {C}^{N})\) with a constant \(C>0\) independent of u. The a priori estimate (27) implies the closedness of \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}.\) Moreover, applying property (26) we obtain that \(sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}\) is discrete.

Theorem 11

Let conditions (i) of Theorem 10 hold. Moreover, \(A_{j}\Phi ,m\) are real-valued functions, and the matrix \(\Gamma (x)\) is Hermitian for every \(x\in \tilde{\Sigma }.\) Then the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{\mathbb {T}}\) is self-adjoint in \(L^{2}(\mathbb {T},\mathbb {C}^{n}).\)

Proof

We turn to the paper [39] where the similar result was obtined for the unbounded in \(L^{2}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\) opertator \(\mathcal {D}_{\mathbf {A}\Phi ,m,\mathfrak {B}_{\Sigma }}.\)

\(1^{0}.\) Let \(u,v\in dom\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}.\) Then integrating by parts we obtain that

$$\begin{aligned}&\left\langle \mathfrak {D}_{\varvec{A,}\Phi ,m}u,v\right\rangle _{L^{2}(\mathbb {T},\mathbb {C}^{N})}-\left\langle u, \mathfrak {D}_{\varvec{A,}\Phi ,m}v\right\rangle _{L^{2}(\mathbb {T},\mathbb {C}^{N})}\nonumber \\= & {} -\frac{1}{4}\left\langle \Gamma \left( \gamma _{\Sigma _{0}}^{+}u+\gamma _{\Sigma _{0}}^{-}u\right) ,\gamma _{\Sigma _{0}}^{+}v-\gamma _{\Sigma _{0}}^{-}v\right\rangle _{L^{2}(\tilde{\Sigma },\mathbb {C}^{N})}\\&+\frac{1}{4}\left\langle \gamma _{\Sigma _{0}}^{+}u+\gamma _{\Sigma _{0}}^{-}u,\Gamma (\gamma _{\Sigma }^{+}v-\gamma _{\Sigma }^{-}v\right\rangle _{L^{2}(\tilde{\Sigma },\mathbb {C}^{N})}.\nonumber \end{aligned}$$
(28)

Since \(\Gamma\) is an Hermitian matrix we obtain that \(\mathfrak {D}_{\varvec{A,}\Phi ,m}\) is a symmetric operator.

\(2^{0}.\) Let

$$\begin{aligned} \mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}(i\mu )u=\left\{ \begin{array}{c} \left( \mathfrak {D}_{\varvec{A},\Phi ,m}-i\mu I_{N}\right) u\text { on }\mathbb {T}\diagdown \tilde{\Sigma }\\ \mathfrak {B}_{\tilde{\Sigma }}u\text { }=0\text { on }\tilde{\Sigma } \end{array}\right. \end{aligned}$$
(29)

be the operator depending on the parameter \(\mu \in \mathbb {R}\) acting from \(H^{1}(\mathbb {T\diagdown }\tilde{\Sigma },\mathbb {C}^{N})\) into \(L^{2}(\mathbb {T},\mathbb {C}^{N}),\) and let the Lopatinsky-Shapiro condition (25) holds at every point \(\ x\in \tilde{\Sigma }.\) Then since \(\mathfrak {D}_{\varvec{A},\Phi ,\Psi }-i\mu I_{N}\) is the elliptic with parameter operator on the torus \(\mathbb {T}\), and the Lopatinsky-Shapiro condition (25) yields the parameter-dependent Lopatinsky-Shapiro condition

$$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}-i\mu I_{N}\right) \ne 0,\xi _{x}\in T_{x}^{*}(\tilde{\Sigma }):\left| \xi _{x}\right| ^{2}+\mu ^{2}=1 \end{aligned}$$
(30a)

since \(\Gamma\) is a Hermitian matrix. Condition (30a) yields that the interaction (transmission) operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma _{0}}}^{\mathbb {T}}(i\mu )\) is invertible for the large value of \(\left| \mu \right|\) (see [1, 2]). Moreover, the invertibility of \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}^{\mathbb {T}}(i\mu )\) for large \(\left| \mu \right|\) implies that \(Range\left( \mathfrak {D}_{\varvec{A},\Phi ,m}-i\mu I_{N}\right) =L^{2}(\mathbb {T},\mathbb {C}^{N})\) for all \(\mu \in \mathbb {R}\) with large enough \(\left| \mu \right| .\) Hence, the deficiency indices of \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}\) are equal to zero, and the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}\) is self-adjoint.

Floquet theory of interaction problems on periodic hypersurfaces

Let \(\mathbb {G}\) be the lattice (23), and \(\mathbb {G}^{*}\) be the reciprocal lattice

$$\begin{aligned} \mathbb {G}^{*}= & {} \left\{ k\in \mathbb {R}^{n}:k=\sum _{j=1}^{n}k_{j}b_{j}.k_{j}\in \mathbb {R}\right\} ,\\ a_{j}\cdot b_{l}= & {} 2\pi \delta _{jl},j,l=1,...,n. \end{aligned}$$

We fix a connected fundamental domain \(W_{0}\subset \mathbb {R}^{n}\) (Wigner–Seitz cell) of the lattice \(\mathbb {G}\) in \(\mathbb {R}^{n}\), i.e., a set

$$\begin{aligned} W_{0}=\left\{ x\in \mathbb {R}^{n}:x={\sum \limits _{j=1}^{n}} t_{j}a_{j},t_{j}\in \left[ 0,1\right) \right\} . \end{aligned}$$

such that \(\mathbb {R}^{n}=\sum _{g\in \mathbb {G}}W_{g},W_{g}=W_{0}+g.\) We will also fix a connected fundamental domain \(B_{0}\) (Brillouin zone) of the reciprocal lattice \(\mathbb {G}^{*}\)

$$\begin{aligned} B_{0}=\left\{ x\in \mathbb {R}^{n}:x= {\sum \limits _{j=1}^{n}} t_{j}b_{j},t_{j}\in \left[ 0,1\right) \right\} . \end{aligned}$$
(31)

We also introduce two tori \(\mathbb {T=R}^{n}\diagup \mathbb {G}\) and \(\mathbb {T}^{*}\mathbb {=R}^{n}\diagup \mathbb {G}^{*}\) which can be identified naturally with fundamental domains \(W_{0}\), \(B_{0},\) respectively. Let as above

$$\begin{aligned} \Omega _{+}= {\sum \limits _{g\in \mathbb {G}}} \Omega _{g},\text { where }\bar{\Omega }_{0}\subset int(W_{0}),\Omega _{g} =\Omega _{0}+g,g\in \mathbb {G},\Omega _{-}=\mathbb {R}^{n}\diagdown \overline{\Omega _{+}}, \end{aligned}$$
(32)

and

$$\begin{aligned} \Sigma = {\bigcup \limits _{g\in \mathbb {G}}} \Sigma _{g},\;\Sigma _{0}=\partial \Omega _{0},\;\Sigma _{g}=\Sigma _{0}+g \end{aligned}$$
(33)

is the common boundary of the domains \(\Omega _{\pm }.\)

We consider here the periodic formal Dirac operator on \(\mathbb {R}^{n}\) with singular potentials of the form

$$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{\varvec{A},\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$
(34)

where \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) is a Dirac operator on \(\mathbb {R}^{n}\) given by formula (4), \(A_{j},\Phi ,m\) \(\in C^{1}(\mathbb {R}^{n})\) are real-valued \(\mathbb {G-}\)periodic functions, and \(\Gamma =\left( \Gamma _{i,j}\right) _{i,j=1}^{N}\) is a \(\mathbb {G}\)-periodic Hermitian matrix with \(\Gamma _{i,j}\in C^{1}(\Sigma ).\)

Let \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) be the unbounded operator in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) generated by the Dirac operator \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) with the domain

$$\begin{aligned} dom\left( \mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\right) =\left\{ u\in H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N}):\mathfrak {B}_{\Sigma }u=0\right\} \end{aligned}$$

associated with the formal Dirac operator \(D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}.\)

We assume that the local Lopatinsky-Shapiro condition

$$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}\right) \ne 0,\xi _{x}\in T_{x}^{*}(\Sigma ):\left| \xi _{x}\right| =1 \end{aligned}$$
(35)

is satisfied at every point \(x\in \Sigma .\)

By Theorem 3\(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is a self-adjoint operator in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}).\) Since the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is invariant with respect to the shifts \(V_{g},g\in \mathbb {G}\)

$$\begin{aligned} sp_{ess}\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}=sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}. \end{aligned}$$

We consider the band-gap structure of the spectrum of \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) applying the Floquet transform (see for instance [25, 40]).

Let \(f\in S(\mathbb {R}^{n},\mathbb {C}^{N})=S(\mathbb {R}^{n})\otimes \mathbb {C}^{N},\) where \(S(\mathbb {R}^{n})\) is the Schwartz space. The Floquet transform of f is defined as

$$\begin{aligned} \left( \mathcal {F}_{\mathbb {G}}f\right) (x,k)=\sum _{\gamma \in \mathbb {G}}f(x-\gamma )e^{-ik\cdot (x-\gamma )},k\in B_{0}, \end{aligned}$$

and the inverse Floquet transform is

$$\begin{aligned} \left( \mathcal {F}_{\mathbb {G}}^{-1}v\right) (x)=\int _{B_{0}}v(x,k)e^{ix\cdot k}\frac{dk}{vol(B_{0})} \end{aligned}$$

The operator \(\mathcal {F}_{\mathbb {G}}\) is continued from the space \(S(\mathbb {R}^{n},\mathbb {C}^{N})\) to the unitary operator acting from \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) into the space \(L^{2}(\mathbb {T\times T}^{*},\mathbb {C}^{N}).\) Applying the Floquet transform, we obtain that

$$\begin{aligned} \mathcal {F}_{\mathbb {G}}\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B} _{\Sigma }}\mathcal {F}_{\mathbb {G}}^{*}=\int _{k\in \mathbb {T}^{*}} ^{\oplus }\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}(k)\frac{dk}{vol(B_{0})} \end{aligned}$$
(36)

where \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}(k),k\in \mathbb {T}^{*}\) is the unbounded operator in \(L^{2}(\mathbb {T},\mathbb {C}^{N})\) generated by the Dirac operator

$$\begin{aligned} \mathfrak {D}_{\varvec{A},\Phi ,m}(k)=\mathfrak {D}_{\varvec{A}+k,\Phi ,m}=\alpha \cdot (D+A+k)+m\alpha _{n+1}+\Phi I_{N}\text { on }\mathbb {T} \end{aligned}$$

with domain

$$\begin{aligned} dom(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}(k))=\left\{ u\in H^{1}(\mathbb {T}\setminus \tilde{\Sigma },\mathbb {C}^{N}):\mathfrak {B}_{\tilde{\Sigma }}u=0\right\} . \end{aligned}$$

As follows from Theorem 11 the operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}(k)\) has real discrete spectrum

$$\begin{aligned} sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\tilde{\Sigma }}}(k)=\left\{ \lambda _{j}(k)\right\} _{j=-\infty }^{\infty },k\in \mathbb {T}^{*} \end{aligned}$$

where \(\lambda _{j}(k)<\lambda _{j+1}(k)\) for every \(j\in \mathbb {Z}\) and \(\lambda _{j}(k)\) are continuous real-valued functions on the torus \(\mathbb {T}^{*}.\)

The decomposition of \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) in the direct integral (36) yields that

$$\begin{aligned} sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}=sp_{ess} \mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}= {\bigcup \limits _{j\in \mathbb {Z}}} \left[ a_{j},b_{j}\right] \end{aligned}$$
(37)

where

$$\begin{aligned} \left[ a_{j},b_{j}\right] =\left\{ \lambda \in \mathbb {R}:\lambda =\lambda _{j}(k),k\in \mathbb {T}^{*}\right\} . \end{aligned}$$
(38)

Fredholm theory and essential spectrum of interaction problems on periodic hypersurfaces in \(\mathbb {R}^{n}\)

We consider the interaction problem on the domains \(\Omega _{\pm }\) with the common boundary \(\Sigma\) described in (32) and (33). Hence, the domains \(\Omega _{\pm }\) and the hypersurface \(\Sigma\) are invariant with respect to the shifts on the vectors \(h\in \mathbb {G}.\) We do not assume the periodicity of the potentials and the matrix \(\Gamma\) with respect to the action of \(\mathbb {G}.\)

We assume that

$$\begin{aligned} A_{j},\Phi ,m\in C_{b}^{1}(\mathbb {R}^{n}),\Gamma _{ij}\in C_{b}^{1}(\Sigma ). \end{aligned}$$
(39)

Our approach is based on the limit operators method and Theorem 8. We introduce the limit operators defined by the sequence \(\mathbb {G}\ni h_{k}\rightarrow \infty .\) We set

$$\begin{aligned} \varvec{A}^{h}(x)=\lim _{k\rightarrow \infty }\varvec{A}(x+h_{k}),\Phi ^{h}(x)=\lim _{k\rightarrow \infty }\Phi (x+h_{k}),m^{h}(x)=\lim _{k\rightarrow \infty }m(x+h_{k}) \end{aligned}$$

where the limits are understood in the sense of the uniform converges on compact sets in \(\mathbb {R}^{n},\) and

$$\begin{aligned} \Gamma ^{h}(x)=\lim _{k\rightarrow \infty }\Gamma (x+h_{k}) \end{aligned}$$

is understood in the sense of the converges on the finite unions \(\cup _{\left| g\right| \le l}\Sigma _{g},l\in \mathbb {N}.\)

We use the notations \(\mathbb {X}=H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\) and \(\mathbb {Y}=L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}),\) and

$$\begin{aligned} \mathbb {D}_{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}u=(\mathfrak {D}_{\varvec{A,}\Phi ,m}u,\mathfrak {B}_{\Sigma }u=0) \end{aligned}$$

is a bounded operator from \(\mathbb {X}\) into \(\mathbb {Y}.\) We introduce the limit operator \(\mathbb {D}_{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\) defined by the sequence \(\mathbb {G}\ni h_{k}\rightarrow \infty\) as follows:

$$\begin{aligned} \mathbb {D}_{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}u=\mathbb {D} _{\varvec{A}^{h}\varvec{,}\Phi ^{h},m^{h},\mathfrak {B}_{\Sigma }^{h} }u=(\mathfrak {D}_{\varvec{A}^{h}\varvec{,}\Phi ^{h},m^{h} }u,\mathfrak {B}_{\Sigma }^{h}u=0) \end{aligned}$$

where \(\mathfrak {B}_{\Sigma }^{h}u=a_{+}^{h}\gamma _{\Sigma }^{+}u+a_{-}^{h}\gamma _{\Sigma }^{-}u,\) \(a^{\pm }=\frac{\Gamma ^{h}}{2}\mp i\alpha \cdot \nu .\)

One can see that for every \(\varphi \in C_{0}^{\infty }(\mathbb {R}^{n})\)

$$\begin{aligned}&\lim _{k\rightarrow \infty }\left\| \left( V_{-h_{k}}\mathbb {D} _{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}V_{-h_{k}}-\mathbb {D} _{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\right) \varphi I\right\| _{\mathcal {B}\left( \mathbb {X},\mathbb {Y}\right) }\\&=\lim _{k\rightarrow \infty }\left\| \varphi \left( V_{-h_{k}} \mathbb {D}_{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}V_{-h_{k}} -\mathbb {D}_{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\right) \right\| _{\mathcal {B}\left( \mathbb {X},\mathbb {Y}\right) }=0. \end{aligned}$$

The Arcela-Ascoli Theorem implies that every sequence \(\mathbb {G}\ni h_{k}\rightarrow \infty\) has a subsequence defining the limit operator \(\mathbb {D}_{\varvec{A,}\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}.\)

Definition 12

(i) We say that: (a) the operator

$$\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}:\mathbb {X}\rightarrow \mathbb {Y}$$

is the locally Fredholm if for every \(R>0\) there exist operators \(\mathcal {L}_{R},\mathcal {R}_{R}\in \mathcal {B(}\mathbb {Y},\mathbb {X})\) such that

$$\begin{aligned} \mathcal {L}_{R}\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\psi _{R}I_{\mathbb {X}}&=\psi _{R}I_{\mathbb {X}}+\mathcal {T}_{R}^{1},\\ \psi _{R}\mathbb {D}_{\varvec{A},\Phi , m,\mathfrak {B}_{\Sigma }}\mathcal {R}_{R}&=\psi _{R}I_{\mathbb {Y}}+\mathcal {T}_{Y}^{2}\nonumber \end{aligned}$$
(40)

where \(\mathcal {T}_{R}^{1}\in \mathcal {K(}\mathbb {X}),\mathcal {T}_{R}^{2}\in \mathcal {K}(\mathbb {Y});\)

(ii) The operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}:\mathbb {X}\rightarrow \mathbb {Y}\) is locally invertible at infinity if there exists \(R>0\) and the operators \(\mathcal {L}_{R}^{\prime },\mathcal {R}_{R}^{\prime }\in \mathcal {B(}\mathbb {Y},\mathbb {X)}\) such that

$$\begin{aligned} \mathcal {L}_{R}^{\prime }\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\chi _{R}I_{\mathbb {X}}&=\chi _{R}I_{\mathbb {X}},\\ \chi _{R}\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\mathcal {R}_{R}^{\prime }&=\chi _{R}I_{\mathbb {Y}}.\nonumber \end{aligned}$$
(41)

We will use the following simple statement.

Proposition 13

The operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}:\mathbb {X\rightarrow Y}\) is Fredholm if and only if:

(i) \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is locally Fredholm; (ii) \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is locally invertible at infinity.

Theorem 14

Let conditions (39) hold, and the Lopatinsky-Shapiro condition

$$\begin{aligned} \det \left( \alpha \cdot \xi _{x}+\frac{\Gamma (x)}{2}\right) \ne 0,\xi _{x}\in T_{x}^{*}(\Sigma _{0}):\left| \xi _{x}\right| =1 \end{aligned}$$
(42a)

be satisfied at every point \(x\in \Sigma .\) Then the operator

$$\begin{aligned} \mathbb {D}_{\varvec{A}, \Phi , m,\mathfrak {B}_{\Sigma }}:H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {R}^{n},\mathbb {C}^{N}) \end{aligned}$$

is Fredholm if and only if all limit operators \(\mathbb {D}_{\varvec{A}, \Phi , m,\mathfrak {B}_{\Sigma }}^{h}\) are invertible.

Proof

The ellipticity of \(\mathfrak {D}_{\varvec{A},\Phi ,m}\) and the local Lopatinsky-Shapiro condition imply that the operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is locally Fredholm. Hence, Proposition 13 yields that \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is the Fredholm operator if and only if \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is locally invertible at infinity. We reduce the study of local invertibility at infinity of \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) to the application of Proposition 13.

We introduce the operator

$$\begin{aligned} \mathbb {D}_{\mathfrak {B}_{\Sigma }}^{0}(i\mu )u =\left\{ \begin{array}{c}(\alpha \cdot D_{x}-i\mu I_{N})u\text { on }\mathbb {R}^{n}\diagdown \Sigma ,\\ \mathfrak {B}_{\Sigma }u=a_{+}^{0}\gamma _{\Sigma }^{+}u+a_{-}^{0}u\gamma _{\Sigma }^{-}=0\text { on }\Sigma \end{array} \right. \end{aligned}$$

where \(a_{\pm }^{0}=\frac{1}{2}\alpha _{n+1}\mp i\alpha \cdot \nu\) acting from \(\mathbb {X}\) into \(\mathbb {Y}.\) Then according to Example 4 the operator

$$\begin{aligned} \mathbb {D}_{\mathfrak {B}_{\Sigma }}^{0}(i\mu ):H^{1}(\mathbb {R}^{n} \diagdown \Sigma ,\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {R}^{n},\mathbb {C} ^{N}). \end{aligned}$$

is invertible for \(\left| \mu \right|\) large enough. We fix such \(\mu .\) Let \(\Xi (i\mu )=\left( \mathbb {D}_{\mathfrak {B}_{\Sigma }}^{0} (i\mu )\right) ^{-1}\). We introduce the bounded in \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) operator

$$\begin{aligned} \mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}=\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\Xi (i\mu ). \end{aligned}$$
(43)

It is easy to prove that

$$\begin{aligned} \lim _{R\rightarrow \infty }\left\| \left[ \chi _{R}I,\Xi (i\mu )\right] \right\| _{\mathcal {B}\left( L^{2}(\mathbb {R}^{n},\mathbb {C}^{N} ),H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\right) }=0. \end{aligned}$$
(44)

Formula (44) implies that the operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is locally invertible at infinity if and only if the operator \(\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B} _{\varvec{\Sigma }}}\) is locally invertible at infinity. One can prove that the operator \(\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) belongs to the algebra \(\mathcal {A}\left( \mathbb {R}^{n},\mathbb {C}^{N}\right) .\) Hence, \(\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) is locally invertible at infinity if and only if all limit operators \(\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\) are invertible. Formula

$$\begin{aligned} V_{-h}\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}V_{h}= & {} \left( V_{-h}\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}V_{h}\right) \left( V_{-h}\Xi (i\mu )V_{h}\right) \\= & {} \left( V_{-h}\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}V_{h}\right) \Xi (i\mu ),h\in \mathbb {G}\nonumber \end{aligned}$$
(45)

implies that \(\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}=\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\Xi (i\mu ).\)

Hence, \(\mathbb {\tilde{D}}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}:L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\rightarrow L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) is invertible if and only if \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\): \(H^{1}(\mathbb {R}^{n}\diagdown \Sigma ,\mathbb {C}^{N})\) \(\rightarrow\) \(L^{2}(\mathbb {R}^{n},\mathbb {C}^{N})\) is invertible, and by Theorem 8 the operator \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\) is locally invertible at infinity if and only if all limit operators \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\) are invertible. Hence, the Theorem has been proved.

Corollary 15

Let conditions of Theorem 14 hold. Then

$$\begin{aligned} sp_{ess}\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}={\bigcup \limits _{h}} sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h} \end{aligned}$$
(46)

where \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\) are unbounded operators associated with the operators \(\mathbb {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}^{h}\) and the union is taken with respect to all such limit operators.

Example 16

We consider an operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}\) where \(\Sigma\) is the above defined \(\mathbb {G}-\)periodic hypersurface in \(\mathbb {R}^{n},\) We assume that the real-valued potentials \(\varvec{A},\Phi\) have the form: \(\varvec{A}=\varvec{A}^{0}+\varvec{A}^{\prime },\Phi =\Phi ^{0}+\Phi ^{\prime }\), where \(\varvec{A}^{0}\) is a \(\mathbb {G-}\)periodic magnetic potential, \(\Phi ^{0}\) is a \(\mathbb {G-}\)periodic electrostatic potential, \(m\in \mathbb {R}\) is the mass of the particle, and \(\Gamma\) is a Hermitian \(\mathbb {G-}\)periodic matrix on \(\Sigma\) such that the local Lopatinsky-Shapiro condition is satisfied at every point \(x\in \Sigma .\) We assume that the perturbations \(\varvec{A}^{\prime }\) and \(\Phi ^{\prime }\) are slowly oscillating at infinity, such that their partial derivatives tend to zero at infinity. In this case the limit operators \(\mathcal {D}_{\varvec{A}^{h},\Phi ^{h},m,\mathfrak {B}_{\Sigma }}\) are such that \(\varvec{A}^{h}=\varvec{A}^{0}+\varvec{A}_{h}^{\prime },\Phi ^{h}=\Phi ^{0}+\Phi _{h}^{\prime }\) where \(\varvec{A}_{h}^{\prime }\in \mathbb {R}^{n},\Phi _{h}^{\prime }\in \mathbb {R}\). Then

$$\begin{aligned} sp\mathcal {D}_{\varvec{A}^{h},\Phi ^{h},m,\mathfrak {B}_{\Sigma }}=sp(\mathcal {D}_{\varvec{A}^{0},\Phi ^{0},m,\mathfrak {B}_{\Sigma }} +\Phi _{h}^{\prime }I)= {\sum \limits _{j\in \mathbb {Z}}} \left[ a_{j}+\Phi _{h}^{\prime },b_{j}+\Phi _{h}^{\prime }\right] . \end{aligned}$$

Applying formula (46), we obtain that

$$\begin{aligned} sp\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}={\sum \limits _{j\in \mathbb {Z}}} \left[ a_{j}+\mathfrak {m(}\Phi ^{\prime }),b_{j}+\mathfrak {M(}\Phi ^{\prime })\right] , \end{aligned}$$

where \(\mathfrak {m(}\Phi ^{\prime })=\liminf _{x\rightarrow \infty }\Phi ^{\prime }(x)\), \(\mathfrak {M(}\Phi ^{\prime })=\limsup _{x\rightarrow \infty }\Phi ^{\prime }(x).\)

Hence, if

$$\begin{aligned} a_{j+1}-b_{j}<\mathfrak {M(}\Phi ^{\prime })-\mathfrak {m(}\Phi ^{\prime }) \end{aligned}$$

the gap \((b_{j},a_{j+1})\) in the spectrum of operator \(\mathcal {D}_{\varvec{A}^{0},\Phi ^{0},m,\mathfrak {B}_{\Sigma }}\)disappears in the spectrum of perturbed operator \(\mathcal {D}_{\varvec{A},\Phi ,m,\mathfrak {B}_{\Sigma }}.\)