Ja n 20 20 SELF-ADJOINT EXTENSIONS OF BIPARTITE HAMILTONIANS

We compute the deficiency spaces of operators of the form HA⊗̂I + I⊗̂HB, for symmetric HA and self-adjoint HB. This enables us to construct selfadjoint extensions (if they exist) by means of von Neumann’s theory. The structure of the deficiency spaces for this case was asserted already in [IMPP14], but only proven under the restriction of HB having discrete, non-degenerate spectrum. Introduction In quantum mechanics, the dynamics of a system is governed by the Schrödinger equation ∂tψt = −iHψt, where H is a self-adjoint operator on a Hilbert space H, called the Hamiltonian, and ψt ∈ H is the wave function at time t. Its time evolution is given by ψt = e ψ0. In many situations however, physical reasoning yields merely a symmetric, rather than self-adjoint, operator, defined on a subspace of sufficiently regular functions. It is then natural to ask whether this operator has self-adjoint extensions, and if so, how many. This question was completely answered by von Neumann, whose extension theory states the following: A symmetric operator H has self-adjoint extensions if and only if the dimensions of the deficiency spaces N (H− i) and N (H+ i) coincide, and in this case, the self-adjoint extensions of H are parametrized by the unitary operators from N (H − i) to N (H + i). Now, given two systems A and B, we can consider the composite system AB. It is modeled on the tensor product of the Hilbert spaces of the individual systems A and B. In the simplest case when there is no interaction between A and B, the time evolution of the composite system is separable, that is, ψt = (e ⊗̂e)ψ0. The corresponding Hamiltonian is HAB = HA⊗̂I + I⊗̂HB. In the light of the discussion above, one is lead to study self-adjoint extensions of operators of this form when HA and HB are merely symmetric (in general, the time evolution generated by such a self-adjoint extension will not be separable). In the case when one of the operators is self-adjoint, this problem was considered by Ibort, Marmo and Pérez-Pardo. They state the following result ([IPP15, Theorem 2.3], [IMPP14, Theorem 1]). Theorem. Let HA be a symmetric operator on HA, HB a self-adjoint operator on HB and define HAB on HAB = HA⊗̂HB by HAB = HA⊗̂I + I⊗̂HB. Date: January 16, 2020. 1 2 LENZ, WEINMANN, AND WIRTH Let NA± = N (H A∓ i) be the deficiency spaces of system HA. The deficiency spaces NAB± = N (H AB ∓ i) of HAB then satisfy NAB± ≃ NA±⊗̂HB. However, in their proof they restrict themselves to the case when the spectrum of HB consists solely of simple eigenvalues, and only state that the general case can be treated by a judicious use of the spectral theorem. In this article, we give a complete proof of this theorem for general self-adjoint HB. Let us outline the strategy. First, the spectral theorem allows us to view HB as a multiplication operator Mφ on L (Ω, μ) for some measure μ. The tensor product HA⊗̂L(Ω, μ) can be identified with the Bochner Lebesgue space L(Ω, μ;HA). Under this identification, the operator HAB acts as (HABψ)(ω) = HAψ(ω) + φ(ω)ψ(ω). These identifications make it possible to reduce the asserted isomorphism for the deficiency spaces to a similar computation as in the proof of Ibort, Marmo and Pérez-Pardo. There are two main difficulties to overcome. First, while the identification of tensor products of operators on HA⊗̂L(Ω, μ) with operators on L(Ω, μ;HA) is fairly obvious in the bounded case, we deal with unbounded operators and, as usual, more care is required to determine the correct domains. This is done in Section 2. Second, the asserted isomorphism for the deficiency spaces comes from fiberwise isomorphisms. Then one has to prove that these isomorphisms can be chosen so that they depend measurably on the base point. In the discrete case this is of course obvious, but it becomes non-trivial in the general case at hand. This problem is resolved in Section 3. Finally, the the proof is completed in Section 4. The self-adjoint extensions of operators of the formHAB as above were also described in [BBM18], using a completely different approach based on boundary triplets. More precisely, given a boundary triplet for H A, they construct a boundary triplet for H AB that respects the tensor structure. This article arose from the second author’s Bachelor’s thesis under supervision of the remaining two. Acknowledgments. M. W. gratefully acknowledges financial support by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes). T.W. thanks PAO Gazprom Neft (ПАО Газпром нефть) and ORISA GmbH for their financial support in form of scholarships during his Master’s and Bachelor’s studies respectively. The authors want to thank Mark Malamud for pointing out the reference [BBM18] to them. 1. Preliminaries Throughout this article, all Hilbert spaces are assumed to be separable. An operator A is called symmetric if it is densely defined and A ⊆ A. Definition 1.1 (Deficiency Spaces). Let A be a symmetric operator. We call N±(A) = R(A± i) ⊥ the deficiency spaces of A and their dimensions d±(A) = dimN±, the deficiency indices of A. SELF-ADJOINT EXTENSIONS OF BIPARTITE HAMILTONIANS 3 In case we can rule out confusion, we shall write N± instead of N±(A). Next we state the central theorem of von Neumann’s extension theory (see, for instance, [Wei00][Chapter 10] for an extensive treatment). In the following, U+̇V denotes the algebraic direct sum of subspaces U and V , not necessarily orthogonal, while U ⊕ V is reserved for the orthogonal direct sum of closed subspaces. Theorem 1.2 (von Neumann’s Extension Theorem). Let A be a closed, symmetric operator on H. Then A has self-adjoint extensions if and only if d+(A) = d−(A). In this case, let U : N+ → N− be unitary. Define the operator B by D(B) = D(A)+̇(U + I)N+ B(f + Ug + g) = Af + ig − iUg. Then A ⊆ B = B, and all self-adjoint extensions of A arise this way. This theorem yields a one-to-one correspondence between the set of self-adjoint extensions of the operator A and the set of unitary operators N+ → N−, hence reduces the problem of finding self-adjoint extensions of A to constructing unitary operators between the deficiency spaces. We will therefore be interested in computing the deficiency spaces of symmetric closed operators. We also recall the spectral theorem in multiplication operator form (see for example [Wei00][Chapter 8]). Theorem 1.3 (Spectral Theorem). Let H be a self-adjoint operator on the Hilbert space H. There exists a σ-finite measure space (X, μ), a measurable function φ : X → R and a unitary operator U : H → L(X, μ) such that H = UMφU. Let H and K be Hilbert spaces. We denote by H⊗K their algebraic tensor product and by H⊗̂K their Hilbert space tensor product. Definition 1.4. For operators A on H and B on K, we define the operator A⊗ B on H⊗̂K by D(A⊗ B) = D(A)⊗D(B), A⊗ B( ∑