1. Introduction 514

2. Eigenvalues and eigenfunctions 516

2.1 Spectral asymptotics 516

2.2 The isoperimetric problem 518

2.3 The Rellich inequality 519

2.4 Numerical studies of the rectangular plate 519

2.5 The ground state 521

2.6 Stability properties 522

2.7 The ideal column 523

3. Solving elliptic equations 524

3.1 One-dimensional results 524

3.2 Constant-coefficient operators 524

3.3 Green functions of bounded regions 525

3.4 The Poisson problem 526

3.5 The Dirichlet problem 526

3.6 Conical points 527

4. The semigroup e−Ht 528

4.1 Elliptic systems 528

4.2 Extensions to Lp 529

4.3 Generalized Schrödinger operators 530

4.4 Kato class potentials 530

5. Heat kernel bounds 531

5.1 General properties of heat kernels 532

5.2 Gaussian upper bounds 532

5.3 Sharp constants 533

5.4 Complex time bounds 534

5.5 Pointwise lower bounds 535

6. Lp spectral theory 536

6.1 Functional calculus 536

6.2 Helffer–Sjöstrand calculus 537

6.3 Spectral independence 538

7. Non-self-adjoint operators 538

7.1 Second-order operators 538

7.2 Lp multiplier theory 539

7.3 H∞ functional calculus 540

7.4 Applications of the H∞ functional calculus 540

References 541