## Essential self-adjointness of Schrödinger type operators on manifolds (2002)

Venue: | RUSS. MATH. SURVEYS |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Braverman02essentialself-adjointness,

author = {Maxim Braverman and Ognjen Milatovic and Mikhail Shubin},

title = {Essential self-adjointness of Schrödinger type operators on manifolds},

journal = {RUSS. MATH. SURVEYS},

year = {2002},

volume = {57},

pages = {641--692}

}

### OpenURL

### Abstract

We obtain several essential self-adjointness conditions for the Schrödinger-type operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV. These results generalize the

### Citations

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Sobolev spaces
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Citation Context ...ighborhood of supp(dψ) is in W 1,2 loc , then ψu ∈ Dom(HV,max). 6.6. Proof of Proposition 6.1. Fix an open neighborhood U ⊂ W of suppV such that the closure U of U is contained in W. Let φ, ˜ φ : M → =-=[0,1]-=- be smooth functions such that φ2 + ˜ φ2 ≡ 1, the restriction of φ to U is identically equal to 1, and suppφ ⊂ W. It follows that supp ˜ φ ⊂ M\U. By IMS localization formula (cf. [23, §3.1], [75, Lemm... |

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Citation Context ...ver, if n>3, then p>2 and s1 > 1. Now we can improve s1 as follows. Let us observe that (4.1) implies D∗Du = f − Vu= h ∈ L s1 loc (E). If s1 > 1, then by the standard elliptic regularity results (cf. =-=[87]-=-, Sect. 6.5) we obtain u ∈ W 2,s1 loc (E). Therefore by the Sobolev embedding theorem (cf. [1, Th. 5.4] or [36, Th. 4.5.8]) we conclude that u ∈ L t2 loc (E), where 1 = t2 1 − s1 2 1 = + n t1 1 2 − p ... |

321 |
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Citation Context ... functions on R n and on manifolds. C.1. Stummel classes. Uniform Stummel classes on R n were introduced by F. Stummel [84]. More details about Stummel classes and proofs can be found in Sect. 1.2 in =-=[23]-=-, Ch. 5 and 9 in [72], and also [2, 83, 84].sESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 41 The (uniform) Stummel class Sn consists of measurable real-valued functions V on R... |

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Pseudodifferential Operators and Spectral Theory
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Citation Context ...en hV (u) ≥ �u� 2 , for all u ∈ W 1,2 comp(E). (8.9) The proof of the proposition occupies the rest of this section. The following simple “integration by parts” lemma follows e.g. from Theorem 7.7 in =-=[78]-=-. Lemma 8.8. The equality (Du,v) =(u, D ∗ v) holds if one of the sections u, v has compact support and u ∈ L 2 loc 1,2 (E), v ∈ Wloc (F) or, vice versa, u ∈ W 1,2 loc (E), v ∈ L2 loc (F). Here (·, ·) ... |

291 |
Heat kernels and Dirac operators
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- 2004
(Show Context)
Citation Context ...D)= |ξ| 2 Id, where |ξ| denotes the norm on T ∗ M defined by the scalar product (2.4). In this situation we say that the operator D ∗ D has a scalar leading symbol. It follows from Proposition 2.5 of =-=[6]-=-, that there exists a Hermitian connection ∇ on E and a linear self-adjoint bundle map F ∈ C ∞ (EndE) such that D ∗ D = ∇ ∗ ∇ + F. Proposition 7.1. Let D be as above and consider the operator HV = D ∗... |

273 |
Foundations of Differentiable Manifolds and Lie Groups, volume 94 of Graduate Texts in Mathematics
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Citation Context ...), and 〈α ∧ β〉 ∈ Ω i+j (M,C). If one of the forms α,β belongs to Ω 0 (M,E) ≡ C ∞ (E), then we will omit “∧” from the notation and write simply 〈αβ〉. Let ∗ :Λ i → Λ n−i be the Hodge-star operator, cf. =-=[89]-=-. This operator extends naturally to the spaces Λ i ⊗ E and Ω i (M,E). The formula 〈α,β〉 Λ i ⊗E := ∗ −1 〈α ∧∗β〉 ∈Λ 0 ∼ = C, α,β ∈ Λ i ⊗ E, i =0,... ,n defines a non-degenerate Hermitian scalar product... |

154 |
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Citation Context ...α|x|α−1 sgn x + |x| 2α . Let V = −|x| 2α , and let HV = D∗D + V . Then HV = − d2 dx 2 − α|x|α−1 sgn x. This operator is not essentially self-adjoint since α − 1 > 2 (cf. Example 1.1 in section 3.1 of =-=[5]-=-). However, for any δ>1, δD ∗ D + V = −δ d2 dx 2 − δα|x|α−1 sgn x +(δ − 1)|x| 2α . Since 2α>α−1 and δ−1 > 0, there exists Cδ > 0 such that −δα|x| α−1 sgn x+(δ−1)|x| 2α > −Cδ for all x ∈ R. Hence, δD ∗... |

152 | Eigenfunction Expansions Associated with Second-Order Differential Equations", S econd Edition - Titchmarsh - 1962 |

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Sobolev spaces
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Citation Context ...g. an example provided in [10]. In fact, if w ∈ L1 loc is fixed, then the condition that wv is in L1 1,2 loc for every v ∈ Wcomp is equivalent to the inclusion |w| ∈W −1,2 loc (this follows e.g. from =-=[57]-=-, Theorem 2 in Sect. 8.4.4). 8.6. Applying Lemma 8.4 to the bundle map A = −V− and using Lemma 8.2, we see that � 〈V−u, u〉dµ =(V−u, u) (8.8) is finite for all u ∈ W 1,2 comp(E). Consider the expressio... |

128 |
Riemannian center of mass and mollifier smoothing
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Citation Context ...any complete Riemannian manifold) to construct cut-off functions χk satisfying all the conditions except the uniform estimate for ∆Mχk (i.e. the second estimate in (B.9)). This was done by H. Karcher =-=[45]-=- (see also [79]). But the estimate of ∆Mχk presents a difficulty. The following sufficient condition provides an important class of examples. Proposition B.3. Let (M,g) be a manifold of bounded geomet... |

77 |
Brownian motion and Harnack inequality for Schrödinger operators
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Citation Context ...C.1. Stummel classes. Uniform Stummel classes on R n were introduced by F. Stummel [84]. More details about Stummel classes and proofs can be found in Sect. 1.2 in [23], Ch. 5 and 9 in [72], and also =-=[2, 83, 84]-=-.sESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 41 The (uniform) Stummel class Sn consists of measurable real-valued functions V on Rn , such that � � lim sup |x − y| r↓0 x |x−... |

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Direct methods of qualitative spectral analysis of singular differential operators, Translated from the Russian by the IPST staff, Israel Program for Scientific Translations
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Citation Context ...ERMAN, OGNJEN MILATOVIC, MIKHAIL SHUBIN Here the expression HV u is understood in the sense of distributions. To prove the theorem, it is enough to show that u = 0 (cf., e.g., Theorem X.26 of [64] or =-=[32]-=-). Arguing ad absurdum, let us assume that u �=0. Since the restriction V | M\U of V to M\U vanishes, (6.4) implies that ∇ ∗ ∇u| M\U = −bu| M\U ∈ L 2 (E). Hence, u| M\U ∈ W 2,2 loc by elliptic regular... |

70 |
Schrödinger operators with singular potentials
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Citation Context ....4) ∞ c (Rn ) we have (TJ ρ v)(x) =b(x) ∂ ∂xi � � jρ(x − y) v(y)dy = b(x) ∂ ∂xi � jρ(x − y) � v(y) dy � � = b(x) − ∂ ∂yijρ(x � � � − y) v(y)dy = − ∂ ∂yi � b(x)jρ(x − y) �� v(y) dy. 3 It is assumed in =-=[47]-=- that aik and bi are in C 1 but the same argument can be carried through if we only assume that they are Lipschitz.sHence, ESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 35 (J ρ... |

67 |
Linear operators part II spectral theory, self adjoint operators in a hilbert space
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Citation Context ...ely related to it, we concentrated on results about operators on manifolds. A comprehensive review of self-adjointness results for one-dimensional operators can be found in N. Dunford and J. Schwartz =-=[25]-=-. About the multidimensional case (for operators on Rn ) the reader may consult M. Reed and B. Simon [64, Ch. X], D. E. Edmunds and W. D. Evans [26], and also review papers by H. Kalf, U.-W Schminke, ... |

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Essential self-adjointness of powers of generators of hyperbolic equations.J.Funct.Anal.12(1973
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Citation Context ... in domains G ⊂ M of a complete Riemannian manifold M if the potential V (x) is estimated below by γ(dist(x, ∂G)) −2 with an appropriate constant γ>0. Cordes uses a “stationary” approach. P. Chernoff =-=[16]-=- used the hyperbolic equation approach to establish similar results. (In a later paper [17] he extended his results to the case of singular potentials V .) We refer the reader to H. O. Cordes’ book [2... |

61 | Spectral theory of elliptic operators on noncompact manifolds
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Citation Context ... a manifold of bounded geometry with arbitrary measure dµ having a positive smooth density. Then Conjecture P holds. Proof. For the definition and properties of manifolds of bounded geometry see e.g. =-=[67, 74]-=-. In particular, a construction of cut-off functions χk satisfying all the necessary properties on any manifold of bounded geometry can be found in [74], p.61. Remark B.4. Note that the bounded geomet... |

49 |
Nonlinear analysis on manifolds: Sobolev spaces and inequalities
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Citation Context ... that (b +∆M) −1h = 0, hence h =0. Note that the space � H2 (M) is different (at least formally) from the space H2 2 (M) whose norm includes arbitrary second-order covariant derivatives (see E. Hebey =-=[33]-=-, Sect. 2.2). It seems still unknown whether C∞ c (M) is dense in H2 2 (M) (see Section 3.1 in [33]). Now we need to know whether it is true or not that � (ν,ψ)S = ψν (B.8) (the integral in the right ... |

46 |
Simader: Schrödinger operators with singular magnetic vector potentials
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Citation Context ...ρ lim (Au) ,v dµ. = 〈Au, v〉dµ. ρ→0+ For every R>0 define the truncation uR of u by the formula � u(x), uR(x) = R if |u(x)| ≤R; u(x) |u(x)| , if |u(x)| >R. It follows from Theorem A of the Appendix in =-=[52]-=- that uR ∈ W 1,2 comp(E) for all R>0 and that uR → u as R →∞in W 1,2 comp(E). Hence, � (Au, u) = lim R→∞ (Au, uR) = lim R→∞ 〈Au, uR〉dµ, (8.7) where the last equality follows from (8.6). By our assumpt... |

33 |
An index theorem on open manifolds I
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Citation Context ... a manifold of bounded geometry with arbitrary measure dµ having a positive smooth density. Then Conjecture P holds. Proof. For the definition and properties of manifolds of bounded geometry see e.g. =-=[62, 69]-=-. In particular, a construction of cut-off functions χk satisfying all the necessary properties on any manifold of bounded geometry can be found in [69], p.61. Remark B.4. Note that the bounded geomet... |

32 | Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators - Levitan, Sargsjan - 1975 |

29 | Semiclassical asymptotics on covering manifolds and Morse inequalities. Geom. Funct. Anal - Shubin - 1996 |

26 | Schrödinger operators in the twentieth century
- Simon
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Citation Context ...C.1. Stummel classes. Uniform Stummel classes on R n were introduced by F. Stummel [84]. More details about Stummel classes and proofs can be found in Sect. 1.2 in [23], Ch. 5 and 9 in [72], and also =-=[2, 83, 84]-=-.sESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 41 The (uniform) Stummel class Sn consists of measurable real-valued functions V on Rn , such that � � lim sup |x − y| r↓0 x |x−... |

25 |
Lifting in Sobolev spaces
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Citation Context ... all x ∈ M. The lemma follows now from the monotone convergence theorem.sESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 27 Remark 8.5. More general results can be found e.g. in =-=[10]-=-, [11], [15]. Note that the statement of the lemma is not completely trivial. For example, if w,v are scalar functions, w ∈ W −1,2 loc ∩L1 loc and v ∈ W 1,2 comp, then it might happen that wv is not i... |

24 |
A special Stokes’s theorem for complete Riemannian manifolds
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Citation Context ...stood as either the scalar product in L2 or the duality between W −1,2 comp and W 1,2 loc extending the L2 -scalar product from C ∞ c (E) by continuity. The following well-known lemma (cf. M. Gaffney =-=[30]-=-), whose proof we reproduce for completeness, provides us with a sufficient amount of “cut-off” functions to be used later. Lemma 8.9. Suppose that g is a complete Riemannian metric on a manifold M. T... |

23 | Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds
- Shubin
- 2001
(Show Context)
Citation Context ... = D ∗ D + V is semi-bounded below on C ∞ c (E). Then HV is essentially self-adjoint. For the case when HV is a magnetic Schrödinger operator acting on scalar functions, this result was formulated in =-=[79]-=-. Unfortunately, in case of singular potentials not all details of the proof were presented. These details, however, can be found in the present paper. Theorem 2.7 immediately implies the following Co... |

22 |
Generalized Functions - Vol.4, Applications of Harmonic Analysis
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(Show Context)
Citation Context ...ll that a distribution ν on M is called positive (notation ν ≥ 0), if for every non-negative function φ ∈ C ∞ c (M), we have (ν,φ) ≥ 0. It follows that ν is in fact a positive Radon measure (see e.g. =-=[31]-=-, Theorem 1 in Sect. 2, Ch.II). We write ν1 ≥ ν2 if ν1 − ν2 ≥ 0. The main result of this section is the following Theorem 5.7. Assume that u ∈ L1 loc (E) and ∇∗∇u ∈ L1 loc (E). Then ∆M|u| ≤Re〈∇ ∗ ∇u, ... |

20 | Spectral theory of linear differential operators and comparison algebras - Cordes - 1987 |

20 |
Fundamental properties of Hamiltonian operators of Schrödinger type
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(Show Context)
Citation Context ...|xi − xj| −1 are dominated by the Laplacian in R 3N both in operator sense and in the sense of quadratic forms. This can be proved either by “separation of variables” as in the classical Kato’s paper =-=[46]-=-, or by use of Stummel and Kato classes – see Example C.2 in Appendix C for the Stummel classes (the corresponding argument can be repeated verbatim for the Kato classes). Note that neither of these t... |

18 |
Spektraltheorie halbbeschrankter Operatoren
- Friedrichs
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(Show Context)
Citation Context ...oint on both ends corresponds to essential self-adjointness. Among the first authors who wrote about multi-dimensional Schrödinger operators HV = −∆+V in Rn we find T. Carleman [14] and K. Friedrichs =-=[28]-=- who independently proved the essential self-adjointness in case when V is locally bounded and semibounded below. (Carleman’s proof was reproduced in the book by I.M. Glazman [32, Ch. 1, Theorem 34].)... |

18 |
The identity of weak and strong extensions of differential operators
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(Show Context)
Citation Context ...richs mollifiers. We will now deduce Theorem 5.7 from Proposition 5.9 using the Friedrichs mollifiers technique. For reader’s convenience we will review basic definitions and results of K. Friedrichs =-=[29]-=-. Suppose that j ∈ C∞ c (Rn ), j(z) ≥ 0 for all z ∈ Rn , j(z) = 0 for |z| ≥1, and � Rn j(z)dz =1. For ρ>0 and x ∈ Rn , define jρ(x)=ρ−nj(ρ−1x). Then jρ ∈ C∞ c (Rn ), jρ ≥ 0, jρ(x)=0if Rn jρ(x) dx =1. ... |

15 |
Uniqueness of the self-adjoint extension of singular elliptic differential operators
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(Show Context)
Citation Context ...mplified by M. Shubin [76], and then the result was extended to magnetic Schrödinger operators in [77]. A first Sears-type result for Hb,V with locally singular V was obtained by T. Ikebe and T. Kato =-=[37]-=- (still with a radial minorant for V ). The paper [77] extends this result to complete Riemannian manifolds, allows non-radial minorant of V with Oleinik-type completeness condition, though requires V... |

15 |
On the essential self-adjointness of the Schrödinger operator on complete Riemannian manifolds
- Oleinik
- 1994
(Show Context)
Citation Context ...16. Most recent history. Here we provide references for the most recent papers which were our source of inspiration. For more history see Appendix D. Theorem 2.7 generalizes recent work of I. Oleinik =-=[58]-=- and M. Shubin [77]. The latter author considered the scalar magnetic Schrödinger operator HV = −∆A+V on a complete Riemannian manifold, where ∆A = d ∗ A dA, A is a real sufficiently regular 1-form, a... |

13 |
On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Lecture
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(Show Context)
Citation Context ...sional case (for operators on Rn ) the reader may consult M. Reed and B. Simon [64, Ch. X], D. E. Edmunds and W. D. Evans [26], and also review papers by H. Kalf, U.-W Schminke, J. Walter and R. Wüst =-=[43]-=-, H. Kalf [41] and B. Simon [83]. We tried not to repeat these sources unless this was relevant for the main text of our paper. As many good stories in mathematics, this one was initiated by H. Weyl (... |

13 | Essential self-adjointness of Schrödinger operators bounded from below - Simader - 1978 |

11 |
Kato’s inequality and spectral distribution of Laplacian on compact Riemannian
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- 1980
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Citation Context ...sh a “smooth version” of the Kato inequality, cf. Proposition 5.9. In the case when dµ is the Riemannian volume form on M, a similar inequality was proven in H. Hess, R. Schrader and D. A. Uhlenbrock =-=[35]-=-. Then, in Sect. 5.15 we prove “L1 loc ” version of the Kato inequality. 5.1. A pairing on the space of bundle-valued forms. Let E be a Hermitian vector bundle over M and let E denote the complex conj... |

11 | On the expansion of arbitrary functions in terms of eigenfunctions of the operator −∆u + cu - Povzner - 1953 |

11 | Essential self-adjointness of Schrödinger operators with singular potentials - Simon - 1973 |

11 |
Singuläre elliptische Differentialoperatoren in Hilbertschen Räumen
- Stummel
- 1956
(Show Context)
Citation Context ...view definitions and the most important properties of Stummel and Kato classes of functions on R n and on manifolds. C.1. Stummel classes. Uniform Stummel classes on R n were introduced by F. Stummel =-=[84]-=-. More details about Stummel classes and proofs can be found in Sect. 1.2 in [23], Ch. 5 and 9 in [72], and also [2, 83, 84].sESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 41 T... |

10 |
Self-adjointness of powers of elliptic operators on non-compact manifolds
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(Show Context)
Citation Context ...ss of the Laplace-Beltrami operator on a complete Riemannian manifold was established by M. Gaffney [30] (not only in L 2 (M) but also in the standard L 2 -spaces of differential forms). H. O. Cordes =-=[21]-=- (see also [22, Ch.4]) established essential self-adjointness of powers of Schrödinger-type operators with positive potentials (in L 2 spaces defined by a measure which is unrelated with the metric, a... |

10 |
Note on the uniqueness of the Green’s function associated with certain second order differential equations
- Sears
- 1950
(Show Context)
Citation Context ... may be not semibounded below. The first result as in Theorem 2.7 (but on M = R n with standard metric and measure, and with q = q(|x|)) for HV = −∆+V , where V ∈ L ∞ loc (Rn ), is due to D. B. Sears =-=[73]-=- (see also [86, Sect. 22.15 and 22.16], [5, Ch. 3, Theorem 1.1]), who followed an idea of an earlier paper by E. C. Titchmarsh [85]. F. S. Rofe-Beketov [68] was apparently the first to allow the minor... |

9 |
Expansion in Eigenfunctions of Self-Adjoint Operators, Naukova
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(Show Context)
Citation Context ...Sect.1.7]. More details about the hyperbolic equation method can be found in the paper by Yu. B. Orochko [61] which also contains a good review with many relevant references. Yu. M. Berezansky’s book =-=[4]-=- also contains an extensive discussion on self-adjointness of operators generated by boundary value problems for elliptic and more general operators. Up to now we assumed at least that V ∈ L 2 loc (Rn... |

9 |
Self-adjointness of the Beltrami-Laplace operator on a complete paracompact manifold without boundary
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(Show Context)
Citation Context ...’s arguments from [16] to prove the essential self-adjointness of powers of HV , where HV ≥−a −b|x| 2 and V is smooth (in case HV = −∆+V on M = R n with the standard metric and measure). A. A. Chumak =-=[18]-=- used the hyperbolic equation method to prove essential self-adjointness of semi-bounded below operators Hg,0,V on complete Riemannian manifolds. Let us also mention a subtle result of H. Donnelly and... |

9 | Classical and quantum completness for the Schrödinger operators on non-compact manifolds, Geometric Aspects
- Shubin
- 1998
(Show Context)
Citation Context ...ifolds. He also provided a geometric self-adjointness condition relating it to the classical completeness in the situation which is not radially symmetric. Oleinik’s proof was simplified by M. Shubin =-=[76]-=-, and then the result was extended to magnetic Schrödinger operators in [77]. A first Sears-type result for Hb,V with locally singular V was obtained by T. Ikebe and T. Kato [37] (still with a radial ... |

9 | Essential Self-Adjointness for Magnetic Schrödinger Operators on Non-Compact
- Shubin
- 1999
(Show Context)
Citation Context ...tory. Here we provide references for the most recent papers which were our source of inspiration. For more history see Appendix D. Theorem 2.7 generalizes recent work of I. Oleinik [58] and M. Shubin =-=[77]-=-. The latter author considered the scalar magnetic Schrödinger operator HV = −∆A+V on a complete Riemannian manifold, where ∆A = d ∗ A dA, A is a real sufficiently regular 1-form, and dAu = du + iuA i... |

8 | On self–adjointness of a Schrödinger operator on differential forms
- Braverman
- 1998
(Show Context)
Citation Context ...ete Riemannian manifold, where ∆A = d ∗ A dA, A is a real sufficiently regular 1-form, and dAu = du + iuA is a deformed differential, while V ∈ L∞ loc (M). Keeping the L∞ loc assumption, M. Braverman =-=[7]-=- generalized the work of I. Oleinik to Schrödinger operators on differential forms. Later, M. Lesch [53] noticed that one does not need to restrict oneself to Laplacian-type operators with isotropic s... |

8 |
On essential self-adjointness of semi-bounded second order elliptic operators without completeness of the Riemannian manifold
- Brusentsev
- 1995
(Show Context)
Citation Context ...possible that incompleteness of the metric is compensated by a specific behavior of V , so that the operator Hg,b,V is self-adjoint even though the metric g is not complete (see e.g. A. G. Brusentsev =-=[12, 13]-=-, H. O. Cordes [22, Ch.4] and references there). Let us make some comments on Schrödinger-type operators which may be not semibounded below. The first result as in Theorem 2.7 (but on M = R n with sta... |

8 |
On the essential self-adjointness of Schrödinger operators with locally integrable potentials
- Kalf, Rofe-Beketov
- 1998
(Show Context)
Citation Context ...efined and H ∗ V,min = HV,max. More general operators of the form (D.1) can also be considered. We refer the reader to I. Knowles [50], M. Faierman and I. Knowles [27], H. Kalf and F. S. Rofe-Beketov =-=[42]-=- and Yu. B. Orochko [61] for results and references in this direction. In particular, Orochko treated locally integrable potentials by the hyperbolic equation method. Let us discuss some essential sel... |

8 |
On a theorem of Titchmarsh and
- Levitan
- 1961
(Show Context)
Citation Context ...reasing magnetic field at infinity allows faster fall off of the scalar potential V to −∞. (This fall off can be in fact arbitrarily fast depending on the growth of the magnetic field.) B. M. Levitan =-=[54]-=- suggested a new proof of the Sears theorem, based on consideration of the hyperbolic Cauchy problem ∂2u ∂t2 + Hu =0, u|t=0 ∂u� = f, � = g, (D.2) ∂t t=0 where H is the symmetric differential operator ... |