## COMPACTNESS OF SCHRÖDINGER SEMIGROUPS

### BibTeX

@MISC{Lenz_compactnessof,

author = {Daniel Lenz and Peter Stollmann and Daniel Wingert},

title = {COMPACTNESS OF SCHRÖDINGER SEMIGROUPS},

year = {}

}

### OpenURL

### Abstract

Abstract. This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L2-space). For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic functions of sublevel sets. In the context of Dirichlet forms, we can even characterize compactness of the semigroup for measure perturbations. Here, certain ’averages ’ of the measure outside of compact sets play a role. As an application we obtain compactness of semigroups for Schrödinger operators with potentials whose sublevel sets are thin at infinity.

### Citations

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(Show Context)
Citation Context ...require that it is form small w.r.t. H0 + V+, i.e., there are some q < 1 and Cq ∈ R such that (V−u | u) ≤ q ((H0 + V+)u | u) + Cq‖u‖ 2 for all u. Then H0 + V+ − V− can be defined by the KLMN theorem (=-=[13]-=-, Theorem X.17) and we have Q(H0 + V+ − V−) = Q(H0 + V+). The reader might have noticed that we didn’t require V+, V− to be the actual positive and negative parts of V (thanks to Vitali Liskevich for ... |

210 |
Dirichlet forms and symmetric Markov processes
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(Show Context)
Citation Context ...] + ‖ϕ‖ 2 | ϕ ≥ 1U , ϕ ∈ D} we define the capacity of U, for U open; one can then extend Cap(·) in the usual way to an outer regular setfunction by letting Cap(E) := inf{Cap(U) | U ⊃ E, U open }; see =-=[5]-=- for details. From [5], we also infer that every u ∈ D admits a quasi-continuous version ũ, the latter being unique up to sets of capacity zero. This allows us to consider measure potentials in the fo... |

80 |
Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, 132
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(Show Context)
Citation Context ...or X being Euclidean space or a manifold, the required spatial local compactness of H0 is sometimes easily checked in terms of compactness of Sobolev embeddings, i.e. in variants of Rellich’s theorem =-=[6]-=-, Theorem V.4.4, see also the discussion in Section 4 below. (4) The Laplacian on quantum or metric graphs is spatially locally compact under quite general assumptions, since D(H0) is continuously emb... |

38 |
H p Sobolev spaces
- Strichartz
(Show Context)
Citation Context ...e show that 1E(−∆ + 1) −p ∈ K for some p ∈ N. (Then, 1E(−∆ + V+ + 1) −1 ∈ K as well and the statement follows from Theorem 2.1.) □COMPACTNESS OF SCHRÖDINGER SEMIGROUPS 9 An inequality of Strichartz, =-=[22]-=- and [18], Lemma 4.10, gives that, for any h ∈ ℓ ∞ (L 2 ) and p > d 4 : ‖h(−∆ + 1) −p ‖ ≤ c · ‖h‖2;∞. Therefore, for E as above, we get 1E(−∆ + 1) −p = ‖ · ‖ − lim R→∞ 1E1 BR(0)(−∆ + 1) −p ∈ K, since ... |

37 |
Trace Ideals and their Applications. Second edition, Mathematical Surveys and Monographs, 120
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(Show Context)
Citation Context ...Proof. Forfixedn ∈ N and E = {V+ ≤ n} we show that 1E(−Δ+1) −p ∈K for some p ∈ N. ( Then, 1E(−Δ+V+ +1) −1 ∈Kas well and the statement follows from Theorem 2.1. ) An inequality of Strichartz, [22] and =-=[18]-=-, Lemma 4.10, gives that, for any h ∈ ℓ ∞ (L 2 ) and p> d 4 : ‖h(−Δ+1) −p ‖≤c ·‖h‖2;∞. Therefore, for E as above, we get 1E(−Δ+1) −p = ‖·‖− lim R→∞ 1E1BR(0)(−Δ+1) −p ∈K, since −Δ is spatially locally ... |

35 | Linear Operators in Hilbert Spaces. Graduate Texts - Weidmann - 1980 |

24 |
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(Show Context)
Citation Context ..., to mankind. 1. Relative spectral compactness and all that In this section, H is a Hilbert space and H some selfadjoint operator on H. The following notion is very useful in perturbation theory, see =-=[14, 24, 25]-=-; we need a rather easy special case, where the “perturbation” B is bounded. We write L = L(H) for the bounded operators and K = K(H) for the ideal of compact operators, which is, of course a norm-clo... |

24 |
Lineare Operatoren in Hilberträumen, Teil 2
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(Show Context)
Citation Context ..., to mankind. 1 Relative spectral compactness and all that In this section, H is a Hilbert space and H some self-adjoint operator on H. The following notion is very useful in perturbation theory, see =-=[14, 24, 25]-=-; we need a rather easy special case, where the “perturbation” B is bounded. We write L = L(H) for the bounded operators and K = K(H) for the ideal of compact operators, which is, of course a norm-clo... |

20 | Siudeja: Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes - Kulczycki, B |

18 |
Spektraltheorie halbbeschrankter Operatoren
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(Show Context)
Citation Context ...lication we obtain compactness of semigroups for Schrödinger operators with potentials whose sublevel sets are thin at infinity. Introduction It is a classical fact, going back at least to Friedrichs =-=[4]-=- that a Schrödinger operator −∆+V with a potential V that goes to ∞ at ∞ has only discrete spectrum so that σess(−∆ + V ) = ∅. This fact has attracted some renewed interest in recent years [11, 23, 17... |

18 |
Sobolev spaces. Springer Series in Soviet Mathematics
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- 1985
(Show Context)
Citation Context ...5], we also infer that every u ∈Dadmits a quasi-continuous version ũ, the latter being unique up to sets of capacity zero. This allows us to consider measure potentials in the following way: see [9], =-=[10]-=- for the special case of the Laplacian and locally finite measures, [19] and the references in there. Let M0 = {μ : B→[0, ∞] | μ a measure μ ≪ Cap},where≪ denotes absolute continuity, i.e. the propert... |

15 | Discreteness of spectrum and positivity criteria for Schrödinger operators
- Maz’ya, Shubin
(Show Context)
Citation Context ...edrichs [4] that a Schrödinger operator −∆+V with a potential V that goes to ∞ at ∞ has only discrete spectrum so that σess(−∆ + V ) = ∅. This fact has attracted some renewed interest in recent years =-=[11, 23, 17]-=- where the issue is first to come up with simple proofs and second to explore more general situations. In this paper we add to this discussion with two main goals: first a rather easy method of proof ... |

10 |
On the discreteness of the spectrum conditions for selfadjoint differential equations of the second order, Trudy Mosk
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(Show Context)
Citation Context ...∞” is much harder for the case of a measure µ. This situation is studied in some detail in Section 3 with Theorem 3.1 as the main result. An elegant criterion for the 1-d Laplacian, due to Molchanov, =-=[12]-=-, says that −∆+µ is compact if and only if µ(U +x) → ∞ as x → ±∞ for some nonempty open interval (equivalently all nonempty open) U ⊂ R. It is quite easy to see that an analogous statement is wrong in... |

10 |
Smooth perturbations of regular Dirichlet forms
- Stollmann
- 1992
(Show Context)
Citation Context ...e latter being unique up to sets of capacity zero. This allows us to consider measure potentials in the following way: see [9], [10] for the special case of the Laplacian and locally finite measures, =-=[19]-=- and the references in there. Let M0 = {µ : B → [0, ∞] | µ a measure µ ≪ Cap}, where ≪ denotes absolute continuity, i.e. the property that µ(B) = 0 whenever B ∈ B and Cap(B) = 0. For measures in M0 we... |

8 | Eigenfunction expansions for Schrödinger operators on metric graphs, Integral Equations Operator Theory 62
- Lenz, Schubert, et al.
- 2008
(Show Context)
Citation Context ...e also the discussion in Section 4 below. (4) The Laplacian on quantum or metric graphs is spatially locally compact under quite general assumptions, since D(H0) is continuously embedded in L ∞ , see =-=[8]-=-. (5) For combinatorial graphs, the condition of spatial local compactness is trivially satisfied, as 1E has finite rank in this case. Therefore we get a rather easy and not very subtle criterion in t... |

7 | Scattering by obstacles of finite capacity
- Stollmann
- 1994
(Show Context)
Citation Context ...0, thenH0is spatially locally compact: In fact 1Ee−tH0 factors through L∞ and the little Grothendieck theorem gives that it is a Hilbert-Schmidt operator, in particular compact. See the discussion in =-=[20]-=-, or [3] for the case of positivity preserving semigroups. Therefore, our Corollary 2.6 contains Theorem 2 from [17] and Corolarry 1.2. from [23] as special cases. It seems that our proof is shorter a... |

4 |
Discreteness conditions of the spectrum of Schrödinger operators
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(Show Context)
Citation Context ...f. From Theorem 4.2 and Theorem 1.3 we deduce that 1E : W s,2 (R d ) → L 2 is compact, since W s,2 (R d ) = H s 2 (−∆) in the notation introduced in Section 1. □ The referee kindly pointed references =-=[1, 2]-=- that contain criteria for emptyness of the essential spectrum of Schrödinger operators. The condition in these papers is V = V1 + V2, ∫ (V1 + C) −1 (y)dy → 0 for |y| → ∞, (⋆⋆) B(x,1) where C is a sui... |

4 | Casteren, Trace norm estimates for products of integral operators, and di¤usion semigroups, Integral Equ
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- 1995
(Show Context)
Citation Context ...is spatially locally compact: In fact 1Ee−tH0 factors through L∞ and the little Grothendieck theorem gives that it is a Hilbert-Schmidt operator, in particular compact. See the discussion in [20], or =-=[3]-=- for the case of positivity preserving semigroups. Therefore, our Corollary 2.6 contains Theorem 2 from [17] and Cor. 1.2. from [23] as special cases. It seems that our proof is shorter and easier tha... |

4 |
Störungstheorie von Dirichletformen mit Anwendungen auf Schrödingeroperatoren, Habilitationsschrift
- Stollmann
- 1994
(Show Context)
Citation Context ...ong in dimensions d ≥ 2. In the recent paper [11], Maz’ya and Shubin proved a compactness criterion in arbitrary dimension. Our result here goes back to the second named author’s Habilitationsschrift =-=[21]-=- that gives a criterion for regular Dirichlet forms with ultracontractive semigroups, a setup that is much more general than the Laplacian in euclidean space. Finally, we record the consequences of ou... |

3 |
Eigenwertproblem von ∆u + λu = 0
- Rellich, Das
- 1948
(Show Context)
Citation Context ...le function V+ : R d → [0, ∞] satisfies the condition that all sublevel sets {V+ ≤ n} have finite measure, for n ∈ N. However, this latter condition is too strong, as was observed already by Rellich, =-=[15]-=-, a fact we learned from [17]. In this latter paper, a sufficient condition can be found that covers Rellich’s example. The following condition is obviously weaker: Definition 4.1. Denote by C(k) the ... |

3 |
Compactness of Schrödinger semigroups with unbounded below potentials
- Wang, Wu
(Show Context)
Citation Context ...edrichs [4] that a Schrödinger operator −∆+V with a potential V that goes to ∞ at ∞ has only discrete spectrum so that σess(−∆ + V ) = ∅. This fact has attracted some renewed interest in recent years =-=[11, 23, 17]-=- where the issue is first to come up with simple proofs and second to explore more general situations. In this paper we add to this discussion with two main goals: first a rather easy method of proof ... |

2 | Schrödinger operators with purely discrete spectrum, see preprint 08-191 at http://www.ma.utexas.edu/mp arc/ or http://arxiv.org/abs/0810.3275v1
- Simon
(Show Context)
Citation Context ...edrichs [4] that a Schrödinger operator −∆+V with a potential V that goes to ∞ at ∞ has only discrete spectrum so that σess(−∆ + V ) = ∅. This fact has attracted some renewed interest in recent years =-=[11, 23, 17]-=- where the issue is first to come up with simple proofs and second to explore more general situations. In this paper we add to this discussion with two main goals: first a rather easy method of proof ... |

1 |
Tadeusz Kulczycki and Bartłomiej Siudeja, Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes
- LENZ, STOLLMANN, et al.
(Show Context)
Citation Context ...s that ψn → 0 in norm. Of course this latter is basically well-known, see, e.g., [14], Theorem XIII.64, p. 245. The equivalence of (i) and (vii) in the above corollary immediately gives: Example 1.6 (=-=[7]-=- Theorem 1.1 part two). Let H0 ≥ 0 be a translation invariant, selfadjoint operator on R d , Q(H0) ∩ L 2 (Br(0)) ̸= ∅ for some r > 0 and 0 ≤ V + ≤ M < ∞ on a sequence of disjoint balls with radius r. ... |

1 |
Maz ′ ja, On the theory of the higher-dimensional Schrödinger operator
- Vladimir
- 1964
(Show Context)
Citation Context ...m [5], we also infer that every u ∈ D admits a quasi-continuous version ũ, the latter being unique up to sets of capacity zero. This allows us to consider measure potentials in the following way: see =-=[9]-=-, [10] for the special case of the Laplacian and locally finite measures, [19] and the references in there. Let M0 = {µ : B → [0, ∞] | µ a measure µ ≪ Cap}, where ≪ denotes absolute continuity, i.e. t... |

1 |
spaces, Springer Series in Soviet Mathematics
- Sobolev
- 1985
(Show Context)
Citation Context ..., we also infer that every u ∈ D admits a quasi-continuous version ũ, the latter being unique up to sets of capacity zero. This allows us to consider measure potentials in the following way: see [9], =-=[10]-=- for the special case of the Laplacian and locally finite measures, [19] and the references in there. Let M0 = {µ : B → [0, ∞] | µ a measure µ ≪ Cap}, where ≪ denotes absolute continuity, i.e. the pro... |

1 |
Über die Aulösung linearer Gleichungen mit unendlich vielen Unbekannten
- Schmidt
- 1908
(Show Context)
Citation Context ...hard to establish. In our investigation below we take advantage of a smaller class of operators that is easier to deal with - the Hilbert-Schmidt operators - one of the great gifts of Erhard Schmidt, =-=[16]-=-, to mankind. 1. Relative spectral compactness and all that In this section, H is a Hilbert space and H some selfadjoint operator on H. The following notion is very useful in perturbation theory, see ... |

1 |
von Dirichletformen mit Anwendungen auf Schrödingeroperatoren
- Störungstheorie
- 1994
(Show Context)
Citation Context ...s. 12 D. LENZ, P. STOLLMANN, AND D. WINGERT [11], Maz’ya and Shubin proved a compactness criterion in arbitrary dimension. Our result here goes back to the second named author’s Habilitationsschrift =-=[21]-=- that gives a criterion for regular Dirichlet forms with ultracontractive semigroups, a setup that is much more general than the Laplacian in euclidean space. Finally, we record the consequences of ou... |

1 |
Operatoren in Hilberträumen. Teil 1, Mathematische Leitfäden. [Mathematical Textbooks
- Lineare
(Show Context)
Citation Context ..., to mankind. 1. Relative spectral compactness and all that In this section, H is a Hilbert space and H some selfadjoint operator on H. The following notion is very useful in perturbation theory, see =-=[14, 24, 25]-=-; we need a rather easy special case, where the “perturbation” B is bounded. We write L = L(H) for the bounded operators and K = K(H) for the ideal of compact operators, which is, of course a norm-clo... |

1 |
On the theory of the higher-dimensional Schrödinger operator
- Maz’ja
- 1964
(Show Context)
Citation Context ...rom [5], we also infer that every u ∈Dadmits a quasi-continuous version ũ, the latter being unique up to sets of capacity zero. This allows us to consider measure potentials in the following way: see =-=[9]-=-, [10] for the special case of the Laplacian and locally finite measures, [19] and the references in there. Let M0 = {μ : B→[0, ∞] | μ a measure μ ≪ Cap},where≪ denotes absolute continuity, i.e. the p... |

1 |
Eigenwertproblem von Δu + λu =0in Halbröhren, in: Studies and Essays Presented to R
- Rellich, Das
- 1948
(Show Context)
Citation Context ...ble function V+ : Rd → [0, ∞] satisfies the condition that all sublevel sets {V+ ≤ n} have finite measure, for n ∈ N. However, this latter condition is too strong, as was observed already by Rellich, =-=[15]-=-, a fact we learned from [17]. In this latter paper, a sufficient condition can be found that covers Rellich’s example. The following condition is obviously weaker: Definition 4.1 Denote by C(k) the u... |