Two-Particle Bound States at Interfaces and Corners

We study two interacting quantum particles forming a bound state in $d$-dimensional free space, and constrain the particles in $k$ directions to $(0,\infty)^k \times \mathbb{R}^{d-k}$, with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from $k$ to $k+1$. This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all $k$ the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413--1444, 2020) to dimensions $d>1$.


Introduction and Main Results
We consider two interacting quantum particles in d-dimensional space that form a bound state in free space. We constrain the particles in k directions to (0, ∞) k × R d−k for some k ∈ {1, ..., d} and impose Neumann boundary conditions. The goal of this paper is to show that at low energy the particles will stick to the boundary of the domain. In fact, the particles want to be close to as many boundary planes as possible. In particular, they stick to the corner where all boundary planes intersect. Neumann boundary conditions can be interpreted as representing perfect mirrors. It is remarkable that while such boundary conditions are not sufficiently attractive to capture single particles, mutually bound pairs are always attracted to the boundary.
In order to justify the picture of particles sticking to the boundary, we show that introducing a boundary plane lowers the ground state energy. Then it is energetically favorable for the particles to localize at a finite distance to the new boundary plane. Moving the particles away from that boundary plane would reduce the boundary effects and raise the energy to reach the previous ground state energy, which is strictly higher. Since moving just one of the particles to infinity would increase the potential energy between them, both particles stick to the boundary.
This problem was already studied (for particles with equal masses) in the case d = k = 1. Kerner and Mühlenbruch [9] considered a hard-wall interaction between the particles. (For a higher-dimensional version of this problem, which is different from the one we consider here, however, see [3].) More general interactions were studied by Egger, Kerner and Pankrashkin in [6]. Additionally, they showed that the Hamiltonian has only finitely many eigenvalues below the essential spectrum. We show here that this also holds true for particles with different masses and all dimensions d and numbers of boundary planes k. The finiteness of the number of bound states is a consequence of the fact that the effective attractive interaction with the boundary decays exponentially with distance, a decay that is inherited from the corresponding one of the ground state wave function in free space. Let x a and x b be the coordinates of the particles. The Hamiltonian of the system is where V : R d → R is the interaction potential. We change to relative and center-of-mass coordinates y = x a − x b and z = max a +m b x b M , where M = m a + m b is the total mass. The conditions x a j > 0 and x b j > 0 for 1 ≤ j ≤ k result in the coordinates (z 1 , ..., z k , y 1 , ..., y k ) lying in the domain Q k = (z 1 , ..., z k , y 1 , ..., y k ) ∈ R 2k | ∀j ∈ {1, ..., k} : z j > 0 and − M m b z j < y j < M m a z j , (1.2) while (z k+1 , ..., z d ) and (y k+1 , ..., y d ) lie in R d−k . In these coordinates, the Hamiltonian becomes is the reduced mass. Separating the variables (z k+1 , ..., z d ) from the rest, we write the Hamiltonian as H = H k ⊗ I + I ⊗ q, where q = − 1 2M ∆ on H 2 (R d−k ) and with domain D[h k ] = H 1 (Q k × R d−k ). Due to the free part of the kinetic energy q, the Hamiltonian H has no discrete spectrum if k < d. We remove this free part and work with H k instead of H. We impose the following conditions on the interaction potential V .
Remark 1.2. Condition (i) implies that in the quadratic form h k the interaction term is infinitesimally form bounded with respect to the kinetic energy, see Proposition A.3 in the Appendix. The KLMN theorem (see e.g. Theorem 6.24 in [13]) then guarantees that there is a unique self-adjoint operator H k corresponding to h k , which is bounded from below. Assumption (ii) means that the particles form a bound state in free space. Condition (iii) is a rather strong form of decay of the negative part at infinity. Presumably some weaker assumptions would be sufficient, but in our proofs this version is convenient. Also the assumptions on the positive part of V can probably be relaxed. Assumption (iv) is imposed for convenience as it implies that it is irrelevant which coordinates are restricted, and without loss of generality we pick the first k. However, our methods easily extend to the general case.
Our first result is that the ground state energy strictly decreases upon adding a Neumann boundary that cuts space in half, i.e. when going from k → k + 1. Moreover, the essential spectrum after dividing space starts at the previous ground state energy. Theorem 1.3. Let V satisfy Assumptions 1.1. Then for every k ∈ {1, ..., d}, the bottom of the spectrum of the operator H k is an isolated eigenvalue E k = inf σ(H k ). Moreover, the essential spectrum of H k is σ ess (H k ) = [E k−1 , ∞). In particular, the ground state energies form a decreasing sequence E d < E d−1 < ... < E 0 < 0.
Our second result is that the operators H k have only finitely many bound states.
Then H k has a finite number of eigenvalues below the essential spectrum.
In the one-dimensional case d = k = 1 with equal masses m a = m b , Theorems 1.3 and 1.4 were proved in [6]. While we follow their main ideas, several new ingredients are needed to extend the results to general d and k. In particular, the localization procedure in the proofs is more complicated and requires several additional steps.
Remark 1.5. At various places it will be convenient to switch back to the particle coordinates in the first k components, while keeping the relative coordinate in the last d− k components. We shall from now on use the notation x a = (x a 1 , ..., x a k ), x b = (x b 1 , ..., x b k ) for the first k components of the particle coordinates andỹ = (y k+1 , ..., y d ) for the remaining components of the relative coordinate. In this notation, y = (x a − x b ,ỹ) and Remark 1.6. By Corollary 5.1 in [7], if H k has a ground state, it is non-degenerate and we can choose the corresponding wave function to be positive almost everywhere.
The remainder of this paper is structured as follows. Section 2 contains the proof of Theorem 1.3. In Section 3, we prove Theorem 1.4. The Appendix contains an explicit example for d = 1 in A.1, the proof of Lemma 2.3 in A.2, as well as technical details of the proofs in A.3. The exponential decay of Schrödinger eigenfunctions needed in the proof is discussed in Appendix B by Rupert L. Frank.

Proof of Theorem 1.3
We shall prove the following two statements.
The assumption E k−1 ≤ ... ≤ E 0 in the first Proposition holds as a consequence of the second Proposition. These two propositions combined yield Theorem 1.3.
Proof of Theorem 1.3. We proceed by induction. The claim is that H k has a ground state, and that the ground state energies form a strictly decreasing sequence E d < ... < E 0 . For k = 0 the former is true by Assumption 1.1(ii). For the induction step we apply Propositions 2.1 and 2.2. Assuming that the claim is true for k − 1, Proposition 2.2 implies that H k has spectrum below E k−1 . By Proposition 2.1 this part of the spectrum must consist of eigenvalues. Since H k is bounded from below by Proposition A.3, it must have a ground state. The ground state energy E k is strictly smaller than E k−1 by Proposition 2.2.

Proof of Proposition 2.1
In order to compute the essential spectrum of H k , we follow the proof of Proposition 2.1 in [6]. For the inclusion [E k−1 , ∞) ⊂ σ ess (H k ) we use Weyl's criterion (see Section 6.4 in [13]). For the opposite inclusion, we bound the essential spectrum of H k from below by introducing additional Neumann boundaries. They split the particle domain into several regions. One of them is bounded, so it does not contribute to the essential spectrum. In another, the interaction potential is larger than E k−1 , and hence there is no essential spectrum below E k−1 . In the remaining regions, the Hamiltonian can be bounded from below by approximately H k−1 ⊗ I. For this operator the essential spectrum starts at E k−1 .
For the opposite inclusion σ ess (H k ) ⊂ [E k−1 , ∞), we partition the domain Q k × R d−k into k + 2 subsets. By Assumption 1.1(iii) there is a number L 0 such that for all y ∈ R d with |y| > L 0 the potential satisfies V (y) > E 0 . For L > L 0 and 1 ≤ l ≤ k let These sets are sketched in Figure 1. The set Ω k+1 is bounded. For (z, y) ∈ Ω k+2 , we always have |y| > L. Moreover, in Ω l the range of y l is independent of z l . For 1 ≤ l ≤ k + 2 we define the quadratic forms a l : H 1 (Ω l ) → R as ∈ Ω l , l ≤ k. In particular, the domain of y l is independent of z l . The (red) triangular area 2 corresponds to the domain of z j and y j for (z, y) ∈ Ω l and j < l ≤ k + 1.
Consider now A l with l ≤ k. In order to separate the variable z l from the rest, let q be the quadratic form q[ϕ] = 1 2M ∞ L/δ |ϕ ′ (z l )| 2 dz l with domain H 1 ((L/δ, ∞)). The remaining variables lie in Ω L,l k−1 := (z 1 , ..., z l , ..., z k , y 1 , ..., where the hat means that the z l variable is omitted. Note that for L → ∞ the set Ω L,l k−1 becomes Q k−1 × R d−k+1 with l and k components swapped. Define the quadratic form We can decompose a l = h L,l k−1 ⊗ I + I ⊗ q. (2.17) It is well-known that the self-adjoint operator corresponding to q has purely essential spectrum [0, ∞). Therefore, we obtain inf σ ess (A l ) = inf σ(H L,l k−1 ). Using localization arguments, one can easily prove the following.
The proof of Lemma 2.3 is rather straightforward and follows similar arguments as in the onedimensional case in Proposition A.5 in [6]. For completeness, we carry it out in Appendix A.2.
Collecting all estimates and applying (2.14), we see that inf

Proof of Proposition 2.2
The goal is to find a trial function ψ such that (ψ, H k ψ) < E k−1 ψ 2 2 . Then inf σ(H k ) < E k−1 by the min-max principle.
Proof. Carrying out the integration over z k , we have We rewrite this as Integrating over z k as in the proof of Lemma 2.4, we obtain We pull the function f into the gradients and write Let us write h k [·, ·] for the sesquilinear form associated to the quadratic form h k . The previous equation reads We now simplify the integral in B. By the Sobolev embedding theorem (Theorem 4.12 in [1]), the restriction of an H 1 -function to a hyperplane is an L 2 -function. Therefore, one can restrict the function ψ k−1 to y k = 0 and obtain a finite number J : . Integration by parts with respect to y k gives This holds for all γ > 0. Minimizing with respect to γ yields This yields (2.1).
We are left with showing that J > 0. Suppose that J = 0. Define a new function . Since ψ k−1 and ψ k−1 are linearly independent, this contradicts the uniqueness of the ground state (Remark 1.6). Hence, J > 0 and inf σ(H k ) < E k−1 .

Finiteness of the Discrete Spectrum
In this section we shall give the proof of Theorem 1.4. An important ingredient will be the exponential decay of the ground state wave function ψ k of H k . In fact, the Agmon estimate (Corollary 4.2. in [2]) implies that for any a < inf Strictly speaking, the assumptions on the interaction potential stated in [2] are slightly stronger than ours. However, the Agmon estimate only requires V to be form-bounded with respect to the kinetic energy with form bound less than 1, as shown in Theorem B.1 in Appendix B by Rupert Frank. As we argue in Proposition A.3, this is the case given Assumptions 1.1.
In order to derive (3.1) from Theorem B.1, we remove the boundaries in the particle domain via mirroring and consider the operator H k acting on H 1 (R d+k ) (see Proposition A.1). It suffices to prove the exponential decay for the ground state ψ k of H k . We rescale the variables to remove the masses in front of the Laplacians using the unitary transform U ϕ(z, y) . Switching to relative and center of mass coordinates by Theorem B.1. Hence (3.1) holds.
Definition 3.1. Let n ∈ Z ≥0 and A be a self-adjoint operator with corresponding quadratic form a. We define (3.4) By the min-max principle, if n is larger than the number of eigenvalues below the essential spectrum, we have E n (A) = inf σ ess (A). Otherwise, E n−1 is the n-th eigenvalue of A below the essential spectrum counted with multiplicities.

Definition 3.2. For a self-adjoint operator A and a number
In the case k = d = 1, Theorem 1.4 was already shown in [6]. We generalize the proof using similar ideas. The overall strategy is to construct localized operators A and bound N (H k , E k−1 ) using N (A, E k−1 ). The localized operators fall into three categories. First, they can have compact resolvent or second, the corresponding potential is larger than E k−1 . In these cases, the number of eigenvalues below E k−1 is certainly finite (or even zero). In the third category, the operator is of the form where K is a well behaved error term. One estimates this operator by projecting onto L 2 (R) ⊗ ψ k−1 and its orthogonal complement. This reduces the problem to a one-dimensional operator. Then, (3.1) and the Bargmann estimate [4] imply that the number of eigenvalues is finite.
The boundary of the particle domain consists of k orthogonal d − 1-dimensional hyperplanes. We start by localizing into two separate regions, distinguishing whether there is a particle close to all the hyperplanes, or whether both particles are far from some hyperplane. For R > 0, let We define the functions Note that for all functions ϕ ∈ L 2 (Ω 0 ) we have support supp f R j ϕ ⊂ Ω j . By the IMS localization formula we have for all ϕ ∈ H 1 (Ω 0 ) that where (3.11) Note that there is a constant c 1 > 0 such that W R ∞ ≤ c 1 R 2 . For j = 1, 2, define the quadratic forms For all quadratic forms a j in this proof, let A j denote the corresponding self-adjoint operator. In Lemma A.5, we verify that these operators exist.
is an L 2 -isometry and thus injective. By the min-max principle, we have We are left with showing that N (A 2 , E k−1 ) < ∞. For k = 1, wave functions in the support of A 2 are localized away from the boundary. Effectively, the boundary has thus disappeared and one can directly make a comparison with H k−1 = H 0 . For k > 1, the domain Ω 2 is more complicated and we need to continue localizing in order to effectively eliminate one of the boundary planes. For now, assume k > 1 and let r = R/8. We localize x a in the k sectors In Ω 2 both x a and x b lie outside the square (0, R) 2 . If x a lies below the upper diagonal, the configuration belongs to Ω 3,1 . If x a lies above the lower diagonal, the configuration belongs to Ω 3,2 .
In the sector Ω 3,j , the largest component of x a is x a j up to the constant r. The domains are sketched in Figure 2 for the case k = 2. For the localization, we need functions f r 3,j on Ω 2 which are supported in Ω 3,j , satisfy k j=1 (f r 3,j ) 2 = 1, and their derivatives scale as 1/r. We construct auxiliary functions f 3,j corresponding to the case r = 1 and set The idea behind the construction of the auxiliary functions is as follows. We want that f 3,1 equals 1 on Ω 3,1 apart from the boundary region which overlaps with other Ω 3,j . The expression max{x a 2 , ..., x a k } − x a 1 measures the distance to the boundary of Ω 3,1 and is large outside Ω 3,1 . Hence, to define f 3,1 , we apply χ 1 to this expression (up to some constants). For the sum condition to hold, the remaining f 3,j will contain the corresponding factor χ 2 . This χ 2 factor takes care of the behavior at the boundary towards large x a 1 . For the next function f 3,2 , we proceed analogously to before, but ignoring the x a 1 direction. Inductively, for x a ∈ (0, ∞) k and 1 ≤ j ≤ k − 1 we define where the product in the first line has to be understood as 1 for j = 1. Note that for all 1 ≤ j ≤ k the derivatives are bounded, i.e. (∇f 3,j ) 2 ∞ < ∞. By construction, we have k j=1 (f 3,j ) 2 = 1. That the functions f r 3,j indeed have the correct support is the content of the following Lemma, which is proved at the end of this section. Moreover, For 1 ≤ j ≤ k, define the quadratic forms We localize x b close and far from the domain of x a . Define the sets For k = 2, they are sketched in Figure 3. Let f r . By the IMS formula, we have for all ϕ ∈ D[a 3,k ] (3.31) For j = 4, 5, define the quadratic forms In Ω 3,2 , the first particle's coordinate x a lies in the shaded area, while the second particle at x b lies outside the square (0, R) 2 . If x b lies above the lowest diagonal (blue), the configuration belongs to Ω 4 . If x b lies below the middle diagonal (red), the configuration belongs to Ω 5 . Note that for any configuration in Ω 5 , the particles are separated by at least distance r/ √ 2.
For k = 1, we set F r = G r = 0 and a 4 = a 2 . For any choice of k ≥ 1, we now just need to show N (A 4 , E k−1 ) < ∞. At the boundaries which constrain the kth component of x a and x b , the operator A 4 has Dirichlet boundary conditions. The idea is to extend the domain of x a k and x b k to R, which leads to the new operatorÂ 4 defined below. InÂ 4 , the boundary hyperplane in the kth direction has disappeared. This makes it possible to compare the operator A 4 to the Hamiltonian H k−1 of the problem with k − 1 boundary hyperplanes. Let us write Let us change to relative and center-of-mass coordinates y = (x a − x b ,ỹ) and z = max a +m b x b M . Then . Note that we can separate z k from the other variables and write the corresponding operator asÂ 4  Using the Schwarz inequality, we estimate Since E k−1 is a discrete and non-degenerate eigenvalue of H k−1 , we have E k−1 In total, we havê in ran Π. Then N (Â 4 , E k−1 ) ≤ N (B 1 , E k−1 ) by the min-max principle.
To bound Z R , first use that K R is bounded to obtain where By construction, I(z k ) = 0 for z k < 0. We shall show that I(z k ) decays exponentially for z k ≥ 0. In fact, if z k is large and K R (z, y) = 0, then necessarily one of the remaining coordinates z 1 , ..., z k−1 , y 1 , ..., y d has to be large as well. This is essentially the content of the following Lemma.

The Agmon estimate (3.1) tells us that there is a constant a > 0 such that
We apply Lemma 3.4 with this constant a and conclude that χ supp K R (z, y) ≤ e −c 4 (z k −2R) α(z, y) for z k ≥ 2R and suitable constant c 4 > 0. In particular, for z k ≥ 2R. Recall that Z R vanishes on (−∞, 0) and Z R ∞ < ∞. With (3.51) we thus conclude the desired exponentially decaying bound.
It remains to give the proof of Lemmas 3.3 and 3.4.

Proof of Lemma 3.4.
Recall the definitions of W R , F r and G r in (3.11), (3.25) and (3.31), respectively. Since supp K R ⊂ supp W R ∪ supp F r ∪ supp G r , we estimate α on each of these three sets. In supp W R , at least one particle is close to the corner, i.e. in the hypercube (0, 2R) k . If z k is large, this means that the two particles are far apart and y k is large. To be precise, using which implies the desired bound on α.
For k = 1, both F r and G r are identically zero, hence to estimate α on their support we can restrict our attention to the case k > 1. Observe that in supp F r every coordinate x a j for 1 ≤ j ≤ k is smaller than or similar in magnitude to the largest of the other coordinates x a i , i = j; in particular, this applies to j = k. Intuitively, for large z k either x a k or |y k | needs to be large. If x a k is large, also some other x a j with j < k has to be large. Phrased precisely, by Lemma 3.3 we have The constraint in S F can be written as z k − r ≤ ( √ M z, √ µy) · e for a vector e ∈ R k+d . A simple Schwarz inequality therefore shows that on the set S F we have as long as z k ≥ r, which yields the desired bound on α.
Similarly to the previous case, in supp G r the coordinate x b k is of similar magnitude as the largest of the other coordinates x b j . We have Analogously to before, on the set S G we have This concludes the proof.
for the factor χ 1 to be non-zero. This is equivalent to max{x a j+1 , ..., x a k } ≤ x a j + r k . Thus, for any 1 ≤ j ≤ k we have max{x a j , ..., x a k } ≤ x a j + r k on the support of f r 3,j . Let us argue inductively why max{x a 1 , ..., x a k } ≤ x a j + r. Suppose we know for some 1 < l ≤ j that max{x a l , ..., Inductively, we see that for every j we have max{x a 1 , ..., x a k } ≤ x a j + j r k ≤ x a j + r.

A.1 Explicit example in one dimension
To illustrate the effect of a boundary on two-particle bound states, we present an explicit example in one dimension. We consider particles with equal masses m a = m b = 1 2 and with deltainteraction V (y) = −αδ(y) for α > 0. The full Hamiltonian is either on L 2 (R 2 ) or on L 2 ((0, ∞) 2 ) with Neumann boundary conditions. In the first case, corresponding to k = 0, we look at the operator H 0 = −2 ∂ 2 ∂y 2 − αδ(y) on L 2 (R). It has the ground state ψ 0 (y) = e − α 4 |y| with corresponding energy E 0 = − α 2 8 . The second case corresponds to k = 1. To compute the ground state of H = H 1 on L 2 ((0, ∞) 2 ), we mirror the problem along the x a = 0 and x b = 0 boundaries, and look for the ground state of the modified Hamiltonian . This is exactly the operator considered in Proposition A.1. Switching to relative and center of mass coordinates y = x a − x b and z = x a +x b 2 , we obtain The ground state ofH 1 is ψ 1 (y, z) = ψ 0 (y)e − α 2 |z| , which decays exponentially away from the Neumann boundary. The ground state energy E 1 = − α 2 4 is strictly lower than E 0 .

A.3 Technical details
By mirroring along the x a j = 0 and x b j = 0 hyperplanes, we can relate H k to an operator H k defined in L 2 (R d+k ).
Proposition A.1. Let H k be the operator defined by the quadratic form

Moreover, the function ψ k is a ground state of H k if and only if the function
is a ground state of H k .
Proof. The operator H k commutes with all reflections along the x a j = 0 or x b j = 0 hyperplanes. Reflections along different hyperplanes commute as well. Therefore, the Hilbert space H = L 2 (R d+k ) splits into subspaces H = r H r characterized by the eigenvalues ±1 of these reflections. We can write H k = r H r k , where H r k is the restriction of H k to H r . For the spectrum, we obtain inf σ( H k ) = min r inf σ( H r k ) and inf σ ess ( H k ) = min r inf σ ess ( H r k ). The subspace that is symmetric under all reflections corresponds to Neumann boundary conditions on [0, ∞) 2k × R d−k . The other subspaces H r are antisymmetric under at least one reflection, so they have Dirichlet boundary conditions along the corresponding hyperplane. Thus, the domains of the quadratic forms for H r k satisfy D[h r k ] ⊂ D[h sym k ]. By the min-max principle, E n ( H r k ) ≥ E n ( H sym k ). Therefore, both inf σ( H k ) = inf σ( H sym k ) and inf σ ess ( H k ) = inf σ ess (H sym k ). Note that the map U : The next lemma follows from the Sobolev inequality, see e.g. Sections 8.8 and 11.3 in [10].
Let Ω ⊂ R d be a domain satisfying the cone property (as defined in [10]) with radius R and opening angle θ. Let V satisfy Assumption 1.1(i). Then, for any 0 < a < 1 there is a constant b ∈ R (depending only on d, R, θ, V and a) such that Proof. The quadratic form q k : is closed and bounded from below. In order to apply the KLMN theorem, we need to show that there are constants a < 1, b ∈ R such that for all We have ψ 2 2 = ψ 2 2 and ∇ ψ 2 2 = ∇ψ 2 2 . Moreover, ψ and 2 k ψ agree on [0, ∞) 2k × R d−k . Hence, Since the integrand is nonnegative, extending the domain of integration from [0, where we changed to coordinates w = x a +x b 2 and y. For almost every w ∈ R k , the function f (y) = ψ(w + (y 1 , ..., y k )/2, w − (y 1 , ..., y k )/2,ỹ) lies in H 1 (R d ) by Fubini's theorem. By Lemma A.2, for any 0 <ã there is a constant b independent of f such that R d |V ||f | 2 ≤ã ∇f 2 2 + b f 2 2 . Integrating over w then gives For 1 ≤ j ≤ k, ∂ y j ψ(w + (y 1 , ..., y k )/2, w − (y 1 , ..., y k )/2,ỹ) Lemma A. 4 Proof. In all cases we prove that the potential term in the quadratic form is infinitesimally bounded with respect to the kinetic energy term. The claim then follows from the KLMN theorem. Let us begin with the quadratic form h l,L k−1 in (2.16). The idea is to use the same mirroring argument as in Prop. A.3 for the coordinate components from l + 1 to k. In the first l − 1 components, we extend the triangular domain in Figure 1 via a suitable mirroring, in order to be able to apply Lemma A.2. To be precise, we define the map φ taking (0, Let us use the notation φ = (φ 1 , φ 2 ). Note that for a function f defined on the triangular domain, we have where one contribution of f 2 2 comes from the triangular domain, and the second f 2 2 is the sum of the contributions with x b < 0 and x a < 0. In the region with x b < 0 we have Moreover, if f ∈ H 1 , then f • φ ∈ H 1 by the Lipschitz continuity of φ. Let us work in center of mass and relative coordinates in the first l components, and with the x a and x b coordinates in components l + 1 to k. The kinetic part of h l,L k−1 is then For ψ ∈ H 1 (Ω l,L k−1 ) define ψ on Ω l,L k−1 := (z 1 , . . . z l−1 , x a l+1 , . . . x a k , y 1 , . . . , y l , x b l+1 , . . . x b k ,ỹ)|∀j < l : z j ∈ (0, L/δ), as ψ(z, y) = 1 2 (l−1)/2 for any a > 0 and a suitable constant b. As in (A.22) we have Since a can be arbitrarily small, the interaction term is infinitesimally bounded w.r.t. q l,L k−1 . Let us now consider the quadratic form a l in (2.13). For l = k + 2, the potential term is bounded from below since |y| > L, and is hence infinitesimally bounded w.r.t the kinetic energy.
for any a > 0 and a suitable constant b. Hence, Since a can be arbitrarily close to zero, the interaction term is infinitesimally bounded w.r.t. q k+1 . Proof. The quadratic forms a j with j ∈ {1, 2, 4, 5} in Eqs. (3.12) and (3.32) and the forms a 3,j for 1 ≤ j ≤ k in (3.26) have the form for some bounded potential V ∞ . The quadratic form q j : H 1 (Ω j ) → R given by is closed and bounded from below. Using that ϕ ∈ D[a j ] vanishes outside Ω j and applying Proposition A.3, we obtain for some a < 1 and b ∈ R. By the KLMN theorem, there is a unique self-adjoint operator A j corresponding to a j . Forâ 4 in (3.38), note that K R is bounded. Adapting the argument in Proposition A.3, we show that the interaction term is infinitesimally bounded with respect to the kinetic part q : We have ψ 2 2 = ψ 2 2 and ∇ ψ 2 2 = ∇ψ 2 2 . Following the same steps as in Proposition A.3 from (A.19)-(A.23) with this adapted choice ofψ, we obtain that for any 0 < a there is a b such that By the KLMN theorem,â 4 corresponds to a self-adjoint operator. Since b 1 in (3.45) differs from a 4 by a bounded term, it also corresponds to a self-adjoint operator. For b 2 in (3.50) and a 1,ext in (3.18), the potential is bounded. Thus, these forms also correspond to self-adjoint operators.  Figure 4. Mirroring the domain along the x a j = 0 and x b j = 0 hyperplanes, we obtain the setΩ sketched in Figure 5.

B Exponential decay of Schrödinger eigenfunctions (by Rupert L. Frank 1 )
It is a folklore theorem that eigenfunctions of Schrödinger operators corresponding to eigenvalues below the bottom of their essential spectrum decay exponentially. This was raised to high art by Agmon [2] and others; see, for instance, the review [12]. It may be of interest to note that the most basic one of these bounds holds under rather minimal assumptions of the potential. This is what we record here. Let V ∈ L 1 loc (R d ) be real and set V ± := max{±V, 0}. Given α ∈ [0, 1], we say that V − is −∆-form bounded with form bound α if there is a C α < ∞ such that for all ψ ∈ H 1 (R d ) .
In this case, we define a quadratic form h by D[h] := ψ ∈ H 1 (R d ) : This quadratic form is lower semibounded in L 2 (R d ) and, if α < 1, closed. Thus, it corresponds to a selfadjoint, lower semibounded operator, which we denote by −∆ + V . We abbreviate Theorem B.1. Assume that V + ∈ L 1 loc (R d ) and that V − is −∆-form bounded with bound < 1. (B.1) We emphasize that E ∞ may be equal to +∞, in which case E ′ may be taken arbitrarily large. If E ∞ < ∞, the decay exponent √ E ′ − E can be any number < √ E ∞ − E. Note that under the assumptions of the theorem, ψ is not necessarily bounded, so one cannot expect pointwise exponential decay bounds. The bounds in the theorem control the quantities that are natural from the definition of the operator in the form sense.
In order to prove Theorem B.1, we use a geometric characterization of the bottom of the essential spectrum due to Persson [11]. Let K ⊂ R d be a compact set and define Clearly, E 1 (−∆ + V | R d \K ) is nondecreasing in K and therefore its supremum over all compact K ⊂ R d exists in R ∪ {+∞}.
We first assume Theorem B.2 and show how it implies Theorem B.1. Then we will provide a proof of Theorem B.2 under our assumptions on V .
Proof of Theorem B.1. Fix E ∞ > E ′′ > E ′ . By Theorem B.2, there is an R ′ > 0 such that for all u ∈ D[h] with u ≡ 0 in B R ′ /2 . Next, for an R > 0 to be specified, we choose two smooth, real-valued functions χ < and χ > on R d such that supp χ < ⊂ B 2R and supp χ > ⊂ R d \ B R (B.2) and such that χ 2 < + χ 2 > ≡ 1 on R d . By scaling an R-independent quadratic partition of unity, we may assume that |∇χ < | 2 + |∇χ > | 2 ≤ CR −2 (B.3) with a constant C independent of R. By increasing R ′ if necessary, we can make sure that C(R ′ ) −2 ≤ (E ′′ − E ′ )/2 =: ǫ with C from (B.3). Let f : R d → R be a bounded Lipschitz function and take ϕ = e 2f ψ ∈ D[h] as a trial function in the quadratic form version of the equation (−∆ + V )ψ = Eψ to obtain, after an integration by parts, Thus, in view of the IMS formula (see, e.g., [5,Theorem 3.2]), |∇(e f χ > ψ)| 2 +Ṽ |e f χ > ψ| 2 dx withṼ := V − |∇f | 2 − |∇χ < | 2 − |∇χ > | 2 . For R ≥ R ′ we bound the terms on the right side from below by Thus, we are left with proving Theorem B.2. We use the following abstract characterization of the essential spectrum. This lemma is classical. The proof in [8,Lemma 1.20] shows the first assertion and, in the case of finiteness, the existence of a normalized sequence with a[ξ j ] → inf σ ess (A) and ξ j ⇀ 0. Since this sequence is bounded in D[a], a subsequence converges weakly in D[a] and, since D[a] is continuously embedded into the Hilbert space, the weak limit is necessarily zero, as claimed.
Proof of Theorem B.2. We abbreviate E ′ ∞ := sup K compact E 1 (−∆ + V | R d \K ). We begin by proving E ∞ ≥ E ′ ∞ . We may assume that E ∞ < ∞ and we shall show that for all R > 0, for then the claimed inequality follows as R → ∞. Fix R > 0 and let χ < and χ > be as in the proof of Theorem B.