Construction of self-adjoint differential operators with prescribed spectral properties

In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schr\"odinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schr\"odinger operator with prescribed essential spectrum.

eigenvalues of is denoted by disc ( ); the complement of the discrete spectrum in ( ) is called the essential spectrum of and the notation ess ( ) is used for this set. It is clear that ( ) = disc ( )∪ ess ( ) and that ess ( ) consists of all those spectral points which are in the continuous spectrum, all eigenvalues embedded in the continuous spectrum and all isolated eigenvalues of infinite multiplicity. For the intuition it may be helpful to keep in mind that essential spectrum can only appear in an infinite dimensional Hilbert space, whereas the spectrum of any matrix is necessarily discrete and hence is always present (and the only type of spectrum) of self-adjoint operators in finite dimensional Hilbert spaces. We refer the reader to the monographs [2,17,31,59,81,83,84] for more details on the spectrum of self-adjoint operators.
The main objective of this expository paper is to illustrate and (partly) review how one can explicitely construct rough domains, singular potentials, or nonstandard boundary conditions with the property that the essential spectrum of a Schrödinger operator coincides with a predefined subset of the real line. The closely related problem of constructing Schrödinger operators with predefined discrete spectrum is also briefly discussed. Very roughly speaking, the results in Section 2 are contained in the well-known papers [8,29,53,54], whereas the main results Theorem 3.3 and Theorem 4.3 in the later sections seem to be new.
More precisely, in Section 2 we treat Laplace operators subject to Neumann boundary conditions (Neumann Laplacians) on bounded domains. It is often believed that self-adjoint Laplace-type operators on bounded domains always have purely discrete spectrum (or, equivalenty, a compact resolvent). This is indeed true for Laplace operators subject to Dirichlet boundary conditions (Dirichlet Laplacian), but, in general, not true for Neumann Laplacians. In fact, the discreteness of the spectrum of the Neumann Laplacian is equivalent to the compactness of the embedding 1 (Ω) ↪ 2 (Ω), and for this a necessary and sufficient criterion was obtained by C. J. Amick [8]; cf. Theorem 2.1. The standard example of a bounded domain for which essential spectrum for Neumann Laplacian appears is a so-called called "rooms-and-passages" domain: a chain of bounded domains ("rooms") connected through narrow rectangles ("passages"), see Figure 1. Roomsand-passages domains are widely used in spectral theory and in the theory of Sobolev spaces in order to demonstrate various peculiar effects (see, e.g., [8,35,45]). Some spectral properties of such domains were investigated in [26]. We also refer to the comprehensive monograph of V. G. Maz'ya [71] (see also earlier contributions [66,67,69,70]), where rooms-and-passages together with many other tricky domains were treated. In the paper [54] R. Hempel, L. Seco, and B. Simon constructed a rooms-and-passages domain such that the spectrum of the Neumann Laplacian coincides with a prescribed closed set ⊂ [0, ∞) with 0 ∈ . We review and prove their result in Theorem 2.4; here also the continuous dependence of the eigenvalues of Neumann Laplacians on varying domains discussed in Appendix A plays an important role. We also briefly recall another type of bounded domains -so-called "comb-like" domains -which allow to control the essential spectrum in the case 0 ∉ . Rooms-and-passages domains can also be used in a convenient way to control the discrete spectrum within compact intervals. We demonstrate this in Theorem 2.6, where we establish a slightly weaker version of the following result by Y. Colin de Verdière [29]: for arbitrary numbers 0 = 1 < 2 < ⋯ < there exists a bounded domain Ω ⊂ ℝ such that the spectrum of the Neumann Laplacian on Ω is purely discrete and its first eigenvalues coincide with the above numbers. One of the main ingredients in our proof is a multidimensional version of the intermediate value theorem by R. Hempel, T. Kriecherbauer, and P. Plankensteiner in [53]. In fact, our Theorem 2.6 is also contained in a more general result established in [53], where a domain was constructed in such a way that the essential spectrum and a part of the discrete spectrum of the Neumann Laplacian coincides with prescribed sets.
In Section 3 we show that similar tools and techniques can be used for a class of singular Schrödinger operators describing the motion of quantum particles in potentials being supported at a discrete set. These operators are known as solvable models of quantum mechanics [3]. Namely, we will treat differential operators defined by the formal expression where ′ is the distributional derivative of the delta-function supported at , ⟨ , ′ ⟩ denotes its action on the test function and ∈ ℝ ∪ {∞}. Such operators are called Schrödinger operator with ′ -interactions (or point dipole interactions) and have been studied (also in the multidimensional setting) in numerous papers; here we only refer the reader to [6, 11-14, 16, 24, 25, 27, 28, 34, 36-44, 48, 52, 57, 61-65, 75, 89] and the references therein. We will show in Theorem 3.1 (see also Theorem 3.3) that the points and coefficients can be chosen in such a way that the essential spectrum of the above operator coincides with a predefined closed set. In our proof we make use of well-known convergence results for quadratic forms, which we briefly recall in Appendix B. Some of our arguments are also based and related to results in the paper [63] by A. Kostenko and M. M. Malamud. Finally, in Section 4 we consider a slightly more abstract problem which can also be viewed as a generalization of some of the above problems: for a given densely defined symmetric operator with infinite deficiency indices, that is, admits a self-adjoint extension and dom( )∕dom( ) is infinite dimensional, and under the assumption that there exists a selfadjoint extension with discrete spectrum (or, equivalently, compact resolvent), we construct a self-adjoint extensions of with prescribed essential spectrum (possibly unbounded from below and above). Here the prescribed essential spectrum is generated via a perturbation argument and a self-adjoint operator Ξ that acts in an infinite dimensional boundary space and plays the role of a parameter in a boundary condition. Our result is also related to the series of papers [4,5,[19][20][21][22][23] by S. Albeverio, J. Brasche, M. M. Malamud, H. Neidhardt, and J. Weidmann in which the existence of self-adjoint extensions with prescribed point spectrum, absolutely continuous spectrum, and singular continuous spectrum in spectral gaps of a fixed underlying symmetric operator was discussed.

ESSENTIAL AND DISCRETE SPECTRA OF NEUMANN LAPLACIANS
The main objective of this section is to highlight some spectral properties of the Neumann Laplacian on bounded domains. In the following let Ω be a bounded domain in ℝ and assume that ≥ 2. As usual the Hilbert space of (equivalence classes of) square integrable complex functions on Ω is denoted by 2 (Ω), and 1 (Ω) denotes the first order Sobolev space consisting of functions in 2 (Ω) that admit weak derivatives (of first order) in 2 (Ω). An efficient method to introduce the Neumann Laplacian in a mathematically rigorous way is to consider the sesquilinear form Ω defined by hold. The operator Ω is called the Neumann Laplacian on Ω. Let 2 (Ω) denote the second order Sobolev space consisting of functions in 2 (Ω) that admit second order weak derivation in 2 (Ω). The corresponding local 2 -space consisting of functions in 2 (Ω) that belong to 2 ( ) for any compact subset of Ω is denoted by 2 loc (Ω). Then one can show that where denotes the normal derivative on Ω.
Typically the boundary condition ↾ Ω = 0 is referred to as Neumann boundary condition, which also justifies the terminology Neumann Laplacian. However, note that some regularity for the boundary of the domain has to be required in order to be able to deal with a normal derivative. For completeness, we note that the assumption of a 2 -boundary above is not optimal (but almost) for 2 -regularity of the domain of the Neumann Laplacian.
The rest of this section deals with some spectral properties of Neumann Laplacians. First of all we discuss in a preliminary situation that the Neumann Laplacian may have essential spectrum; since the domain Ω is bounded this may be a bit surprising at first sight. In this context we then recall a well-known result due to R. Hempel, L. Seco, and B. Simon from [54] how to explicitely construct a bounded rooms-and-passages-type domain (with nonsmooth boundary) such that the essential spectrum of the Neumann Laplacian coincides with a prescribed closed set. Another related topic is to construct Neumann Laplacians on appropriate domains such that finitely many discrete eigenvalues coincide with a given set of points. Here we recall a famous result due to Y. Colin de Verdière from [29], and supplement this theorem with a similar result which is proved with a simple rooms-and-passages-type strategy. Actually, Theorem 2.6 is also a special variant of a more general result by R. Hempel, T. Kriecherbauer, and P. Plankensteiner in [53].

Neumann Laplacians may have nonempty essential spectrum
Let Ω be the self-adjoint Neumann Laplacian in 2 (Ω) and denote by ess ( Ω ) the essential spectrum of Ω . It is wellknown that ess ( Remark 2.2. Necessary and sufficient conditions for ess ( Ω ) = ∅ have been also obtained in [68]. These conditions are formulated in terms of capacities.
In [8] an example of a bounded domain Ω ⊂ ℝ 2 consisting of countably many rooms and passages with Γ Ω > 0 was constructed (see Figure 1).
For the convenience of the reader we wish to recall this construction in the following. Note that we impose slightly different assumptions on the rooms and passages as compared to [8] In the next step define the sequence ( ) ∈ℕ by and define the rooms and passages by respectively. Finally, the union of and leads to the desired rooms-and-passages domain . (2.14) From (2.5), (2.7), and (2.9) it is clear that Ω is bounded. Using (2.6), (2.9), (2.10) and taking into account that > 3 we obtain the estimate Hence the thickness of the passage is not larger than the sides of the adjacent rooms and +1 , which also shows that Ω is indeed an open set. It will now be illustrated that for this particular domain Ω the quantity Γ Ω in (2.4) is positive, so that the embedding in (2.3) is not compact. In particular, the essential spectrum of the Neumann Laplacian in 2 (Ω) is not empty. For this purpose consider the piecewise linear functions , = 2, 3, … , defined by Note that −1 − + = −̂− 1 by (2.11). It is easy to see that the function belongs to 1 (Ω). Next we evaluate its 2 -norm. One computes (2.16) We also have (cf. (2.6))̂− Moreover, it is clear that for any > 0 there exists ( ) ∈ ℕ such that supp ( ) ⊂ Ω for all ≥ ( ). This, (2.18), (2.19), and (2.8) yield Γ Ω > 0 (recall that ≥ 3). As an immediate consequence we conclude the following corollary.

Corollary 2.3.
Let Ω be the bounded rooms-and-passages domain in (2.14) and let Ω be the self-adjoint Neumann Laplacian in 2 (Ω). Then The natural question that arises in the context of Corollary 2.3 is what form the essential spectrum of the Neumann Laplacian Ω may have. This topic is discussed in the next subsection.

Neumann Laplacian with prescribed essential spectrum
In the celebrated paper [54] R. Hempel, L. Seco, and B. Simon have shown, using rooms-and-passages-type domains of a similar form as above, that the essential spectrum of the Neumann Laplacian can be a rather arbitrary closed subset of the real line. Below we briefly describe their construction. We fix sequences and of course one then has (2.8). Let the domain Ω ⊂ ℝ 2 consist of countably many rooms and passages . Furthermore, in each room we insert an additional "wall" (see Figure 2), and the resulting modified room is then denoted by = ⧵ , so that where it is assumed that the sequence ( ) ∈ℕ satisfies 0 < ≤ . Proof. This sketch of the proof from [54] consists of three steps. In the first step, where we skip a perturbation argument, it is shown that the original problem to ensure (2.24) for the Neumann Laplacian Ω on the domain Ω can be reduced to show the same property for a "decoupled" Neumann Laplacian dec . The spectrum of this operator can be described explicitly, which is done in the second step. Finally, in a third step the parameters are adjusted in such a way that (2.24) holds.
Step  as an orthogonal sum of the self-adjoint operators and DN in the space Then one can show that the resolvent difference ( dec + I ) −1 − ( Ω + I ) −1 of the Neumann Laplacian Ω and the decoupled operator dec is a compact operator provided → 0 sufficiently fast as → ∞. We fix such a sequence ( ) ∈ℕ ; then from Weyl's theorem (see, e.g., [83, Theorem XIII.14]) one concludes and hence it remains to show that ess ( dec ) = .
Step 2. First we shall explain how the eigenvalues of the Neumann Laplacian on depend on the size of the wall inside . In this step of the proof the value = 0 is also allowed (in this case the room decouples: it becomes a union of two disjoint rectangles). We denote the eigenvalues of (counted with multiplicities) and ordered as a nondecreasing sequence by ( which means ≤˜in the sense of ordering of forms. Then it follows from the min-max principle (see, e.g., [31,Section 4.5]) that for each ∈ ℕ the function is nondecreasing and by Theorem A.1 this function is also continuous. In the present situation it is clear that and due to the monotonicity of the function (2.27) one also has ∈ℕ denote the eigenvalues (counted with multiplicities) of DN ordered as a nondecreasing sequence then one verifies that the first eigenvalue 1 (̊) is given by for all ∈ ℕ. From the orthogonal sum structure in (2.26) it is clear that where the symbol denotes the set of accumulation points of a sequence. Observe that the eigenvalues ( (̊)) ∈ℕ do not have any finite accumulation point (the smallest eigenvalue satisfies (2.31) and lim →∞̂= 0 by assumption) and hence we obtain , ∈ℕ ) .
Step 3. Now we complete the proof by adjusting the parameters in the above construction. Since by assumption is a closed subset of [0, ∞) one can always find a sequence ( ) ∈ℕ such that Next, for each ∈ ℕ we fix a number > 0 such that In addition, we assume that the numbers are chosen small enough so that We also fix a sequence of positive numbers Using the continuity of the function (2.27) and taking into account (2.29) and (2.33) it is clear that there exists such that the second eigenvalue 2 ( ) of the Neumann Laplacian satisfies for all ∈ ℕ. Furthermore, by construction we also have 1 ( (2.30)), and hence we conclude together with lim →∞ = 0, (2.28), (2.32), (2.34), and 0 ∈ that This implies ess ( dec ) = and completes the proof. □ Besides (2.24) it is also shown in [54] that the absolutely continuous spectrum ac ( of Ω is empty. The argument is as follows: It is verified that the difference = [0, ∞). In [54] R. Hempel, L. Seco, and B. Simon also constructed a domain for which (2.24) holds without the restriction 0 ∈ . For this purpose so-called comb-like domains are used, see Figure 3. To construct the comb one attaches a sequence of "teeth" ( ) ∈ℕ to a fixed rectangle ; the tooth is obtained from a rectangle by removing an internal wall . The teeth have bounded lengths, shrinking widths, and are stacked together without gaps.
The analysis is similar to the rooms-and-passages case. One can prove that the Neumann Laplacian Ω on such a comblike domain Ω is a compact perturbation of the decoupled operator where is the Neumann Laplacian on and DN is the Laplace operator on the tooth subject to Neumann boundary conditions on ⧵ and Dirichlet boundary conditions on ∩ . The walls are adjusted in such a way that the lowest eigenvalue of DN coincides with a predefined number , while the next eigenvalues tend to ∞ as → ∞ and do not contribute to essential spectrum.
The important and somewhat surprising element of the rooms-and-passage and comb-like domain constructions is the form of the decoupled operators with mixed Dirichlet and Neumann boundary conditions. The fact that one chooses Dirichlet conditions on the common part of the boundaries of the passages and modified rooms , and similarly on the common part of the boundaries of the modified teeth and the rectangle is due to the following well-known effect (see, e.g., [9,10,58]): the spectrum of the Neumann Laplacian on ∪ , where is a fixed domain and is an attached "handle" of fixed length and width , converges to the direct sum of the Neumann Laplacian on and the one-dimensional Dirichlet Laplacian on (0, ) as → 0.
Finally, we note that A. A. Kiselev and B. S. Pavlov [60] obtained Theorem 2.4 for (a kind of) Neumann Laplacian on a bounded set consisting of an array of two-dimensional domains connected by intervals.

Neumann Laplacian with prescribed discrete spectrum
In this section we are interested in the discrete spectrum of Neumann Laplacians. First we recall a result by Y. Colin de Verdière from [29].
where ( (Ω)) ∈ℕ denotes the sequence of the eigenvalues of Ω numbered in increasing order with multiplicities taken into account.
In fact, the main result in [29] concerns Riemannian manifolds: for arbitrary , = 1, … , , satisfying (2.35) and an arbitrary compact connected manifold one can construct a Riemannian metric on this manifold in such a way that the first eigenvalues of the corresponding Laplace-Beltrami operator coincide with the numbers , = 1, … , . The idea of the proof of this theorem in [29] is to first construct a suitable differential operator Γ on a metric graph Γ such that the first eigenvalues of Γ coincide with , = 1, … , . Then the graph Γ is "blown" up to a tubular thin domain Ω in such a way that the first eigenvalues of the Neumann Laplacian Ω on Ω are asymptotically close to the first eigenvalues of Γ provided the cross-section of Ω tends to zero. For dimensions ≥ 3 the above theorem is extended in [29] by allowing nonsimple eigenvalues. More precisely, one can construct a domain in ℝ , ≥ 3, such that the first eigenvalues of the Neumann Laplacian on this domain coincide with the predefined numbers A similar theorem for the Dirichlet Laplacian (at least for = 2) can not be expected. In fact, by a well-known result of L. E. Payne, G. Pólya, and H. F. Weinberger from [78] the ratio between the -th and the ( − 1)-th Dirichlet eigenvalue (for compact domains in ℝ 2 ) is bounded from above (by a domain independent constant) and hence the eigenvalues can not be placed arbitrarily on (0, ∞).
Theorem 2.5 was later improved by R. Hempel, T. Kriecherbauer, and P. Plankensteiner in [53], where a bounded domain Ω was constructed such that the essential spectrum and a finite part of the discrete spectrum of Ω coincide with predefined sets. In their construction comb-like domains were used; see Figure 3.
The next theorem may be viewed as a variant of Theorem 2.5, although it is actually a slightly weaker version. In fact, we present this result here since it can be proved with a similar rooms-and-passages domain strategy as Theorem 2.4. An important ingredient is a multidimensional version of the intermediate value theorem from [53]; cf. Lemma 2.7 below. Theorem 2.6. Let ∈ ℕ ⧵ {1} and assume that are fixed numbers. Then there exists a bounded domain Ω ⊂ ℝ such that the spectrum of the Neumann Laplacian Ω on Ω is purely discrete and where ( (Ω)) ∈ℕ denotes the sequence of the eigenvalues of Ω numbered in increasing order with multiplicities taken into account.
Proof. We consider the case = 2; the construction and the arguments in dimensions are zero, the next eigenvalues are 1 , … , , and all further eigenvalues are contained in (sup , ∞). Afterwards we connect the rooms by small windows of length (see Figure 4); the resulting domain is denoted by Ω. If is small the spectrum changes slightly. Namely, • the eigenvalues +1 (Ω), … , 2 (Ω) remain in small neighborhoods of 1 , … , , respectively, moreover they still belong to , • the rest of the spectrum remains in (sup , ∞).
It remains to establish the coincidence of + (Ω) and as = 1, … , (so far they are only close as > 0 is sufficiently small), which is done by using a multidimensional version of the mean value theorem from [53]. Roughly speaking, the shift of the eigenvalues that appears after inserting the small windows can be compensated by varying the constants appropriately. In the following we implement this strategy.
Step 1. Fix some positive numbers 1 , … , and let 1 , … , , and be nonnegative numbers satisfying ] . (2.36) In the following we shall use the notation ∶= where the walls are ]} and the windows are . Then the domain Ω , that we will use is defined as .
The Neumann Laplacian on Ω , will be denoted by Ω , and it is important to observe that for = 0 this self-adjoint operator in 2 (Ω , ) decouples in a finite orthogonal sum of Neumann Laplacians on the rooms , that is, one has with respect to the corresponding space decomposition Note that 2 ( Ω ,0 ) = 2 ( Ω , ) and hence Ω , and Ω ,0 act in the same space for all in (2.36). It is clear from (2.37) that the spectrum of the decoupled operator Ω ,0 is the union of the spectra of the Neumann Laplacians , = 1, … , . We recall (see Step 2 of the proof of Theorem 2.4) that the functions are continuous and nondecreasing. Moreover, one has Step 2. Now we approach the main part of the proof, where the parameters will be properly adjusted. Let 0 < 1 < ⋯ < be as in the assumptions of the theorem and fix an open interval  such that Assume that the numbers 1 , … , satisfy Furthermore, let us choose a constant > 0 such that the intervals are pairwise disjoint and holds. Next, we introduce the sets Using the continuity and monotonicity of the function in (2.38) and taking into account (2.39), (2.41)-(2.43) we conclude that each is a nonempty compact interval. We set It is clear from (2.39), (2.41), and (2.43) that ± > 0. From now on we assume that As usual we denote the eigenvalues of the decoupled operator Ω ,0 in (2.37) by ( ( Ω ,0 )) ∈ℕ , counted with multiplicities and ordered as a nondecreasing sequence. It follows from 1 ( ) = 0 for = 1, … , , that (  We introduce the functions 0 ∶  → ℝ by It is important to note that due to (2.45) the value + ( Ω ,0 ) of the function 0 depends only on the -th variable . Using this and taking into account that the mapping (2.38) is nondecreasing we get Step 3. Let ( (Ω , ) ) ∈ℕ be the eigenvalues of the Neumann Laplacian Ω , on Ω , counted with multiplicities and ordered as a nondecreasing sequence.
For ≥ 0 we introduce the functions ∶  → ℝ by  Then for any ∈ ∏
We apply this lemma to the function = ( 1 , … ,

SINGULAR SCHRÖDINGER OPERATORS WITH ′ -INTERACTIONS
In this section we show that the methods used to control the spectrum of the Neumann Laplacian in the previous section can also be applied to singular Schrödinger operators describing the motion of quantum particles in potentials being supported at a discrete (finite or infinite) set of points. These operators are often referred to as solvable models in quantum mechanics, since their mathematical and physical quantities (e.g., their spectrum) can be determined explicitly. We refer to the monograph [3] for an introduction to this topic. We also note that in the mathematical literature such operators are often called Schrödinger operators with point interactions.
The classical example of a Schrödinger operator with -interactions is the following formal expression where are Dirac delta-functions supported at the points ∈ ℝ and ∈ ℝ ∪ {∞}. In the present paper we treat the closely related model of a Schrödinger operator with ′ -interactions (or point dipole interactions) defined by the formal expression where ′ is the distributional derivative of the delta-function supported at ∈ ℝ, ⟨ , ′ ⟩ denotes its action on the test function , and ∈ ℝ ∪ {∞}. The above formal expression can be realized as a self-adjoint operator in 2 with the action − d 2 d 2 and domain consisting of local 2 -functions that satisfy (the case = ∞ stands for a decoupling with Neumann conditions at ± 0). The existence of this model was pointed out by A. Grossmann, R. Høegh-Krohn, M. Mebkhout in [52], the first rigorous mathematical treatment of ′ -interactions is due to F. Gesztesy and H. Holden in [48]. Among the numerous subsequent contributions we mention the more recent papers [62,63] by A. Kostenko and M. M. Malamud, in which also the more elaborate case | | − −1 | | → 0 as | | → ∞ was treated. In these papers self-adjointness, lower semiboundedness and spectral properties of the underlying operators have been studied in detail.
Our goal and strategy is similar to [54] in the context of the Neumann Laplacian: We wish to construct an operator of the form (3.1) with predefined essential spectrum; cf. Theorem 2.4. At this point we present the main result of this section on a formal level without giving a precise definition of the underlying operator. This will be done during its proof; cf. Theorem 3.3 for a more precise formulation of Theorem 3.1. Proof. For the construction of the self-adjoint Schrödinger operator with ′ -interactions we use a similar idea as in the construction of the rooms-and-passages domains in the previous section. Here we split the sequence of points , and instead of we denote the interaction strengths at the points by and at the points by . Instead of we shall write , for the corresponding Schrödinger operator with ′ -interactions, see Step 3 of the proof. The intervals ) will play the role of the rooms, the interactions at the points will play the role of the passages, and the interactions at the points will play the role of the additional walls inside the rooms.
As in the proof of Theorem 2.4 we fix a sequence ( ) ∈ℕ such that and for each ∈ ℕ we choose numbers > 0 that satisfy Moreover, we can assume that the are chosen sufficiently small, so that The proof consists of four steps. In the first step we discuss the spectral properties of the Schrödinger operator , on the interval  with a ′ -interaction of strength > 0 at and Neumann boundary conditions at the endpoints of  . In the second step we consider the direct sum of these operators: Note that the Neumann conditions at ± 0 can be regarded as a ′ -interaction with infinite strength. Thus ∞, corresponds to the Schrödinger operator on ( , ) with ′ -interactions of strengths at the points and ′ -interactions of strengths ∞ at the points . The interaction strengths will be adjusted in such a way that the essential spectrum of ∞, coincides with . In fact, ess is the union of the point 0 and all accumulation points of a sequence formed by the second eigenvalues of , . In the third step we perturb the decoupled operator ∞, linking the intervals  +1 and  by a ′ -interaction of a sufficiently large strength > 0 for all ∈ ℕ; the corresponding operator is denoted by , . We will prove in the last step that the essential spectra of , and ∞, coincide if the interaction strengths tend to ∞ for → ∞ sufficiently fast.
Step 1. Let ∈ (0, ∞] and let , be the sesquilinear form in 2 (  ) defined by for = ∞ we use the convention ∞ −1 = 0. The form , is densely defined, nonnegative, and closed in 2 (  ) . Hence by the first representation theorem there is a unique nonnegative self-adjoint operator ) .
Integration by parts shows that dom ( consists of all those functions ∈ 2 (  ⧵ { }) that satisfy the ′ -jump condition

at the point and Neumann boundary conditions
at the endpoints of  . Furthermore, the action of , is given by The spectrum of the self-adjoint operator , is purely discrete. We use the notation ∈ℕ for the corresponding eigenvalues counted with multiplicities and ordered as a nondecreasing sequence. Some properties of these eigenvalues are collected in the next lemma. Here we will also make use of the Neumann Laplacian on  , defined as usual via the form  | | where we have used > 0 in the third estimate. In the same way we get | | . Hence (3.9) leads to the estimate and from this we conclude It is well-known (see, e.g., [59,81]  Since the spectra of the operators , and 0, are purely discrete Theorem B.2 shows (3.8). □ Now we return to the spectral properties of the self-adjoint operators , . In particular, the eigenvalues of the Neumann Laplacian 0, on  and the direct sum of the Neumann Laplacians ∞, on ( −1 , can be easily calculated. For our purposes it suffices to note that (3.14) It follows from (3.7), (3.12), and (3.14) that for any ∈ (0, ∞] we have In a more explicit form ∞, is given by where ∶= ↾  stands for the restriction of the function onto the interval  . Note that the corresponding sesquilinear form ∞, associated with ∞, is It is clear that the operator ∞, in (3.17) is self-adjoint and nonnegative in 2 ( , ). Furthermore, it is not difficult to check that holds. Taking into account that 0 ∈ and using (3.2), (3.5), (3.15), (3.16), we arrive at Step 3. In this step we perturb the decoupled operator ∞, linking the intervals  +1 and  by a ′ -interaction of sufficiently large strength > 0 for all ∈ ℕ. The corresponding self-adjoint operator will be denoted by , . More precisely, for > 0, ∈ ℕ, we consider the sesquilinear form , . This form is nonnegative and densely defined in 2 ( , ). Moreover, the form is closed by [63,Lemma 2.6] and the corresponding nonnegative self-adjoint operator , is given by . (3.21) In fact, the inclusion dom ( follows directly from the definition of the above form domains. To prove the reverse inclusion we have to show that . Using the standard trace estimates (see, e.g. [18,Lemma 1.3.8]) and taking into account that sup ∈ℕ < − and > 0 we obtain (3.23) and thus (3.22) holds. We have shown (3.21). Now let , ∈ 2 ( , ) be arbitrary and consider the functions ) .

Using (3.21) and the fact that
) .

(3.24)
Next we introduce the operators Γ , Γ ∞ ∶ 2 ( , ) → 2 (ℕ) defined by on their natural domains Note that dom ( Γ ) coincides with the whole 2 ( , ); this follows immediately from the ran . Let us prove that the operator Γ is compact. For this purpose we introduce the finite rank operators Let ∈ 2 ( , ) = dom ( Γ ) and = ( , + I ) −1 . Using the same arguments as in the proof of (3.23) and (3.10) we obtain and hence it follows from (3.20) that ‖ ‖ Γ − Γ ‖ ‖ 2 (ℕ) → 0 as → ∞. Since Γ are finite rank operators we conclude that the operator Γ is compact. Furthermore, it is easy to see that Γ ∞ is a bounded operator defined on 2 ( , ). Indeed, for ∈ 2 ( , ) and = ( ∞, + I ) −1 one verifies in the same way as in (3.23) and (3.25) that

Now (3.24) can be rewritten in the form
and hence we have , = −(Γ ∞ ) * Γ . Since Γ is compact and Γ ∞ is bounded (thus (Γ ∞ ) * is also bounded) we conclude that , is compact. □ For the convenience of the reader we now formulate Theorem 3.1 in a more precise form. At the end of this section we note that there exist many other methods for the construction of Schrödinger operators with predefined spectral properties. For example, F. Gesztesy, W. Karwowski, and Z. Zhao constructed in [49,50] a smooth potential (which is a limit of suitably chosen -soliton solutions of the Korteweg-de Vries equation as → ∞) such that the Schrödinger operator = − d 2 d 2 + has purely absolutely continuous spectrum ℝ + and a prescribed sequence of points in ℝ − is contained in the set of eigenvalues of .

ESSENTIAL SPECTRA OF SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS
The aim of this slightly more abstract section is to discuss some possible spectral properties of self-adjoint extensions of a given symmetric operator in a separable Hilbert space. In a similar context the existence of self-adjoint extensions with prescribed point spectrum, absolutely continuous spectrum, and singular continuous spectrum in spectral gaps of a fixed underlying symmetric operator was discussed in [4,5,[19][20][21][22][23], see also [74] for a related result on prescribed eigenvalue asymptotics or, e.g., the earlier contributions [7,46,47,55]. Our main observation here is the fact that for a symmetric operator with infinite deficiency indices one can construct self-adjoint extensions with prescribed essential spectrum; cf. Theorem 4.3 below.
In the following let  be a separable (infinite dimensional) complex Hilbert space with scalar product (⋅, ⋅). Recall that a linear operator in  is said to be symmetric if ( , ) = ( , ), , ∈ dom( ).
We point out that a symmetric operator is in general not self-adjoint. More precisely, if the domain dom( ) of is dense in  then the adjoint * of the operator is given by * ℎ = , dom( * ) = { ℎ ∈  ∶ ∃ ∈  such that ( , ℎ) = ( , ) for all ∈ dom( ) } , and the fact that is symmetric is equivalent to the inclusion ⊂ * in the sense that we have dom( ) ⊂ dom( * ) and * = for all ∈ dom( ). However, this is obviously a weaker property than the more natural physical property of self-adjointness, that is, = * . A symmetric operator is not necessarily closed (although closable) and the spectrum of a symmetric operator which is not self-adjoint may cover the whole complex plane (or at least the upper or lower complex halfplane). We also point out that the closure of a symmetric operator need not be self-adjoint and it need not have self-adjoint extensions in . A symmetric operator is called essentially self-adjoint if its closure is self-adjoint, that is, = * -however, we shall not deal with essential self-adjoint operators here. We emphasize that from a spectral theoretic point of view a symmetric operator (or an essentially self-adjoint operator) which is not self-adjoint is not suitable as an observable in the description of a physical quantum system.
It is an important issue to understand in which situations a symmetric operator admits self-adjoint extensions and how these self-adjoint extensions can be described. These questions have already been discussed in the classical contribution [76] by J. von Neumann. For completeness we recall that a self-adjoint operator in  is an extension of a densely defined symmetric operator if ⊂ ; since is self-adjoint this is equivalent to ⊂ * . We start by recalling the so-called first von Neumann formula in the next theorem. (4.1) Note that * = for all ∈ ker( * − ) and similarly * − = − − for all − ∈ ker( * + ). The spaces ker( * − ) and ker( * + ) are usually called defect spaces of and their dimensions are the deficiency indices of . It will turn out that the deficiency indices and isometric operators in between the defect spaces are particularly important in the theory of selfadjoint extensions. One can show that the dimension of ker ( * − + ) does not depend on + ∈ ℂ + and the dimension of does not depend on − ∈ ℂ − . However, for fixed + ∈ ℂ + and − ∈ ℂ − and hence, in particular, for + = and − = − , the dimensions of ker ( * − + ) and ker ( * − − ) may be different. According to the second von Neumann formula both dimensions coincide if and only if admits self-adjoint extensions in .

Theorem 4.2. Let be a densely defined closed symmetric operator in . Then there exist self-adjoint extensions of in  if and only if
) .
If, in this case ∶ ker( * − ) → ker( * + ) is a unitary operator and dom( * ) is decomposed as in (4.1), then the operator defined by is a self-adjoint extension of and, vice versa, for any self-adjoint extension of there exists a unitary operator ∶ ker( * − ) → ker( * + ) such that (4.2) holds.
It is intuitively clear that for a densely defined symmetric operator with infinite deficiency indices (and hence equal deficiency indices as  is assumed to be separable) there is a lot of flexibility for the unitary operators in between the infinite dimensional defect subspaces. This flexibility also allows to construct self-adjoint extensions with various different spectral properties. Now we wish to consider the following particular situation. Let again be a densely defined closed symmetric operator in  with equal infinite deficiency indices and assume that there exists a self-adjoint extension of such that the resolvent is a compact operator. In this situation we shall construct another self-adjoint extension of with prescribed essential spectrum in Theorem 4.3 below. In this context we also mention the following variant of a remarkable result due to J. Brasche, which can be viewed as the final outcome of a long and fruitful investigation originally initiated by H. Neidhardt and his coauthors: For a densely defined closed symmetric operator in  with equal infinite deficiency indices and a (finite or infinite) gap there exists a self-adjoint extension with arbitrary spectrum within this gap; cf. [19,Theorem 27] for more details. In particular, for any closed set Σ ⊂ ℝ with Σ ≠ ℝ there exists a self-adjoint extension of in  such that ess ( ) = Σ.
Proof. Let 0 be a self-adjoint extension of in  such that the resolvent ( 0 − ) −1 is a compact operator for some, and hence for all, ∈ ( 0 ). Let us fix some point ∈ ℝ ∩ ( 0 ) ∩ (Ξ). Note that this is possible since we have assumed ℝ ∩ (Ξ) ≠ ∅ and the spectrum of 0 is a discrete subset of the real line due to the compactness assumption. In the present situation the spaces ker ( * − + ) and ker( * − − ) for ± ∈ ℂ ± are both infinite dimensional and one can show that here also the space ker( * − ) is infinite dimensional; this follows, e.g., from the direct sum decomposition dom( * ) = dom ( 0 )+ ker( * − ) (4.4) and the fact that * is an infinite dimensional extension of 0 . Moreover, it is no restriction to assume that the Hilbert space  in the assumptions of the theorem coincides with ker( * − ) since any two separable infinite dimensional Hilbert spaces can be identified via a unitary operator. Now observe the orthogonal sum decomposition  = ker( * − ) ⊕ ran( − ) and with respect to this decomposition we consider the bounded everywhere defined operator We claim that −1 is a well-defined operator. In fact, if ℎ = 0 for some ℎ ∈  then (4.5) implies and since the left-hand side belongs to dom( 0 ) and the right-hand side is nonzero only in ker( * − ) it follows from the direct sum decomposition (4.4) that ℎ = 0. This confirms that −1 , and hence also is a well-defined operator. It is clear from = ( − ) −1 that ∈ ( ) and is self-adjoint in  since the same is obviously true for in (4.5). In order to determine the essential spectrum of recall the Weyl theorem (see, e.g., [83,Theorem XIII.14]) which states that compact perturbations in resolvent sense do not change the essential spectrum. In the present situation we have that is a compact operator and hence the essential spectrum ess ( ( − ) −1 ) coincides with the essential spectrum of the diagonal block matrix operator, that is, This implies (4.3). in (4.6). On the other hand, if = 0 + ∈ dom( 0 )+ ker( * − ) is such that (4.7) holds then one can verify in a similar way that there exists ℎ ∈  such that = ℎ, and hence ∈ dom( ). Summing up we have shown the representation (4.6).
Finally we note that the explicit form (4.6) of comes via a restriction of the adjoint operator * and the decomposition (4.4); the domain of is described by an abstract boundary condition depending on the choice of the operator Ξ. This abstract result can of course be formulated in various explicit situations, e.g., for infinitely many ′ -interactions as in Section 3 or for the Laplacian on a bounded domain as in Section 2, where the boundary condition in (4.6) can be specified further.
Furthermore, the self-adjoint extensions and 0 can be described in the formalism of von Neumann's second formula in Theorem 4.2. If one fixes a unitary operator 0 ∶ ker( * − ) → ker( * + ) for the representation of 0 in (4.2) then the unitary operator ∶ ker( * − ) → ker( * + ) corresponding to the self-adjoint extension can be expressed in terms of 0 and the parameter Ξ. The technical details are left to the reader.

A C K N O W L E D G E M E N T S
The authors are most grateful to the anonymous referee for many useful comments which improved the manuscript considerably. In particular, the referee kindly provided us with an elegant proof of Proposition A. 3

APPENDIX A: CONTINUOUS DEPENDENCE OF THE EIGENVALUES ON VARYING DOMAINS
In this appendix we establish an auxiliary result on the continuous dependence of the eigenvalues of the Neumann Laplacian on varying domains, which is useful and convenient for the proofs of Theorem 2.4 and Theorem 2.6.
For our purposes it is sufficient to consider the following geometric setting: Let Ω ⊂ ℝ 2 be a bounded Lipschitz domain and assume that also the subdomains are (bounded, nontrivial) Lipschitz domains. Furthermore, we assume that the set (see Figure A1, left). Note that Ω , = Ω and Ω , = Ω − ∪ Ω + = Ω ⧵ Γ. We denote by ⊂ [ , ] × [ , ] the set of all admissible pairs { , }, that is, Since the domain Ω , in (A.1) has the cone property (see, e.  Figure A1 (right, here = 3). In this case = { 1 , … , For the proof of Theorem A.1 we will make use of a variant of an abstract result from [56] (originally formulated and proved for operators in varying Hilbert spaces), which is here obtained as Corollary A.4 below. The next proposition and its elegant proof were communicated to us by an anonymous referee. Proposition A.3. Let , ∈ ℕ, and be compact normal operators in a Hilbert space . Assume that the following conditions hold: Proof. Let ∶= − , ∈ ℕ, and note that, due to the compactness of the operator , Let , ∈ ℕ, be an eigenfunction of corresponding to and normalized by (A.12) Due to (i) and (A.11) the right-hand side of (A.12) tends to 0 as → ∞; a contradiction. □ From Proposition A.3 we immediately get the following useful corollary.
The strategy is to apply Corollary A.4 to the resolvents of the Neumann Laplacians Ω , and Ω , . More precisely, we consider the operators which are compact nonnegative operators acting in  = 2 (Ω) = 2 ( Ω , ) . We show below that these operators satisfy the assumptions (i) and (ii) in Proposition A.3. Then it follows that (A.13) holds for each ∈ ℕ and from we conclude (A.14). In order to check condition (i) let ∈ 2 (Ω) and set = , ∈ ℕ. For ∈ 1 ( Ω , ) it follows from the definition of the Neumann Laplacian Ω , that satisfies In particular, using (A.15) for = we get and therefore We set ± = ↾ Ω ± . Below we shall use the same ±-superscript notation for restrictions of other functions onto Ω ± . It follows from (A. 16) that ( ± ) ∈ℕ is a bounded sequence in 1 ( Ω ± ) and hence there exist ± ∈ 1 ( Ω ± ) and a subsequence → ∞ such that as → ∞, where ± Γ ± stands for the restriction of the trace of the function ± ∈ 1 ( Ω ± ) onto Γ. Next we introduce the set of functionŝ ∶ ∃ = ( ) > 0 such that = 0 in -neighborhoods of (0, ) and (0, ) } .
It is known thatˆ1 ( Ω , ) is dense in 1 ( Ω , ) (this is due to the fact that the capacity of the set {(0, ), (0, )} is zero; we refer to [80] for more details). Now let ∈ˆ1 ( Ω , ) . It is clear that for sufficiently large we also have ∈ 1 As ± ∈ 1 ( Ω ± ) this implies ∈ 1 ( . This also shows that the limit function is independent of the subsequence and hence we conclude that (A.21) holds for any subsequence . Thus, as → ∞. We have verified assumption (i).
To check assumption (ii) in Proposition A.3 let ( ) ∈ℕ be a bounded sequence in 2 (Ω). The same arguments as in the proof of (i) (cf. (A.16)) show that the sequence ( ) ∈ℕ is bounded in 1 (Ω ⧵ Γ), and hence contains a weakly convergent subsequence in 1 (Ω ⧵ Γ). Since the embedding 1 (Ω ⧵ Γ) ↪ 2 (Ω ⧵ Γ) = 2 (Ω) is compact (again we use Rellich's embedding theorem) we conclude that there is a strongly convergent subsequence in 2 (Ω), that is, condition (ii) is satisfied. □ Remark A.5. Besides the continuity of the function { , } ↦ ( Ω , ) one can also conclude that it decreases (resp., increases) monotonically with respect to (resp., with respect to ). This follows easily from the min-max principle (see, e.g., [31,Section 4.5]). Note, that, in general, when one perturbs a fixed domain Ω by removing a subset ( ∈ ℝ is a parameter) the monotonicity of the eigenvalues of the Neumann Laplacian in Ω ∶= Ω ⧵ does not follow from the monotonicity of the underlying domains with respect to , i.e., even if Ω ⊂ Ω˜, it does not mean that (Ω ) ≥ (Ω˜) (see [72, Section 2.3] for more details). This is in contrast to Dirichlet Laplacian, where the monotonicity is always present -see, e.g., [51,72,77,80] for the properties of Dirichlet eigenvalues in so perturbed domains. However, in our configuration monotonicity nevertheless holds for Neumann eigenvalues. This is due to a special structure of the removed set having the form of two walls with zero thickness.

APPENDIX B: CONVERGENCE RESULTS FOR MONOTONE SEQUENCES OF QUADRATIC FORMS
We recall a well-known convergence result for a sequence of monotonically increasing quadratic forms from [85] which is used in the proof of Theorem 3.1.
Consider a family { } >0 of densely defined closed nonnegative sesquilinear forms in a Hilbert space . For simplicity we assume that the domain of is the same for all , and we use the notation dom ( ) =  1 . Let  be the nonnegative self-adjoint operator associated with the form via the first representation theorem. Now assume, in addition, that the family { } >0 of forms increases monotonically as decreases, i.e. for any 0 < <˜< ∞ one has One verifies that 0 is a well-defined nonnegative symmetric sesquilinear form (which is not necessarily densely defined) and, in fact, by [85] the limit form 0 is closed. Let us now assume that dom( 0 ) is dense in , so that one can associate a nonnegative self-adjoint operator  0 with 0 via the first representation theorem. We wish to mention here that in the situation where the limit form 0 is not densely defined one associates a self-adjoint relation (multivalued operator) via the corresponding generalized first representation theorem for nondensely defined closed nonnegative forms; see [32,85,88] or the more recent paper [15] for details and related convergence results. According to [85] one then has convergence of the corresponding nonnegative self-adjoint operators in the strong resolvent sense (see also [32,Proposition 4.1] or [15,Theorem 4.2]): Theorem B.1 (Simon, 1978). For each ∈  one has Now let us assume, in addition, that the spectra of the self-adjoint operators  and  0 are purely discrete. We write ( (  )) ∈ℕ and ( (  0 )) ∈ℕ for the eigenvalues of these operators counted with multiplicities and ordered as nondecreasing sequences. In this case one can conclude the following spectral convergence: Proof. The discreteness of the spectra of  and  0 is equivalent to the compactness of the resolvents